Journal of Mathemat i ca l Sciences, Vol. 98, No. 2, 2000

H I L B E R T C*- A N D W*-MODULES A N D T H E I R M O R P H I S M S

V. M. Manui lov and E. V. Troitsky UDC 517.986

A Hilbert C*-module is a natural generalization of the Hilbert space arising by replacing the field of scalars C by a C*-algebra. For commutative C*-algebras, such a generalization was described for the first time in the work of I. Kaplansky [30]; however, the noncommutative case at that time apperared to be too complicated to study. The general theory of Hilbert C*-modules appeared 25 years ago in the basic papers of W. Paschke [52] and M. Rieffel [56]. This theory has proved to be a very convenient tool in the theory of operator algebras, allowing one to study C*-algebras by studying Hilbert modules over them. In particular, a series of results on such classes of C*-algebras as AW*-algebras and monotoneously complete C*-algebras was obtained [21]. In terms of Hilbert modules, the important notion of Morita equivalence was formulated for C*-algebras [14, 57]. This notion also has applications in the theory of group representations. It turned out to be possible to study group actions with the help of Hilbert modules arising from such actions [54, 58]. Some results on conditional expectations of finite index [7, 69] and on completely positive maps of C*-algebras [4] were also obtained.

The theory of Hilbert C*-modules may also be considered as a noncommutative generalization of the theory of vector bundles [19, 36]. This was the reason that Hilbert modules became a tool in topological applications, namely, in the index theory of elliptic operators, in the K-theory and in the KK-theory of G. G. Kasparov [32 35, 46, 48, 66], and in noncommutative geometry on the whole [15, 16].

Among other applications, it is necessary to emphasize the theory of quantum groups and unbounded operators [5, 6, 70] and some physical applications [2, 39].

Along with these applications, the theory of Hilbert C*-modules itself has been developed, too. A number of results concerning the structure of Hilbert modules and operators on them were obtained [22, 41 44, 65]. In addition, an axiomatic approach in the theory of Hilbert modules based on the theory of operator spaces and tensor products was developed [10, 11].

A detailed bibliography of the theory of Hilbert C*-modules can be found in [24]. A significant part of the results presented here was only announced in the literature, or the proofs

were discussed only briefly. We have tried to fill in such gaps. We cannot discuss here all aspects of the theory of Hilbert modules, but we have tried explicitly to present the basic notions and theorems of this theory, a nmnber of important examples, and also some results related to the authors' interests.

Most of the material presented formed the content of a lecture course presented by the authors at the Department of Mechanics and Mathematics of Moscow State University in 1996.

We are grateful to A. S. Mishchenko for introducing us to the theory of Hilbert C*-modules. Together with Yu. P. Solovyov, he has acquainted us with the circle of problems related to applications in topolog~y.

While working in this field and in the process of writing the present text, we were significantly influenced by our friend and co-author, M. Frank.

We have discussed a number of problems of the theory of Hilbert C*-modules with L. Brown, A. A. Irmatov, G. G. Kasparov, R. Nest, G. K. Pedersen, and W. Paschke; some applications were also considered with J. Kaminker, J. Cuntz, V. Nistor, J. Rosenberg, A. Ya. Helemskii, and B. L. Tsygan.

The research was partially supported by RBRF Grant No. 96-01-00182-a.

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 53, Functional Analysis-6, 1998.

1072-3374/00/9802-0137 $25.00 (~) 2000 Kluwer Academic/Plenum Publishers 137

1. Basic D e f i n i t i o n s

1.1. C * - a l g e b r a s

The basic information on C*-algebras can be found in [17, 29, 55, 62]. We will present some results

on C*-algebras which will be necessary for us further on.

Recall tha t an involutive Banach algebra is called a C*-algebra if for each of its elements a the

following relation is fulfilled:

Ila*all = I1<12 Any such algebra can be realized as a norm-closed subalgebra of the algebra of bounded operators B(H) on a Hilbert space H. We do not assume the presence of the unit in C*-algebras. By A + we denote

the C*-algebra obtained from the C*-algebra A by unit izat ion (taking the direct sum with complex

numbers).

We also need the notion of a positive element of a C*-algebra. First of all, we recall that the spectrum of an element a of a unital C*-algebra is the set Sp(a) of complex numbers z such that a - z - 1 is not

invertible. If a C*-algebra A has no unity, then the spec t rum of the element a E A is its spec t rum in

the C*-algebra A + D A. The spec t rum is a compact subset of C. An element a E A is called positive (we write a _> 0) if it is Hermitian, i.e., satisfies the condit ion a* = a, and if one of the following

equivalent [17, 1.6.1] conditions is realized:

(i) Sp(a) C [0, ec);

(ii) a = b*b for some b E A;

(iii) a = h 2 for some Hermit ian h E A.

The set of all posit ive elements P+(A) forms a closed convex cone in a and P+(A) N ( - P + ( A ) ) = O.

Among the elements h existing by item (iii), there exists only one positive element, called the positive square root of a (we write h = al/U).

We also recall that a linear functional p " A -+ C is called positive if ~(a) _> 0 for any positive

element a E P+(A). A positive linear functional is called a state if II~ll = 1. We have Ilall = snp~(a ) ,

where a _> 0 and sup is taken over all states.

A C*-homomorphism of an algebra A into the C*-algebra B(H) of all bounded operators on a Hilbert

space H is called a representation. A vector ~ E H is called cyclic for the representation re �9 A --+ B(H) if the set of all vectors of the form re(a)~, a r A, is dense in H . The vector { E H is called separating for the representat ion re : A ~ 13(H) if the equality re(a){ = 0 implies a = 0.

We can associate with each positive linear functional w on a C*-algebra A a unique (up to uni tary

equivalence) representat ion re~ of the algebra A on some Hilber t space H~ and a vector ~ E H~ such that

co(a) = ( re~(a)~ , ~ ) for all a E A and the vector {~ is cyclic. The construct ion of such a representat ion

is called the GNS-construction.

An approximate unit of a C*-algebra A is an increasing net e~ < A, a E A, such that Ite ll < 1 and

lira [I a - a e ~ ] l = 0 for any a E A. Each C*-algebra has an approximate unit ea such that ea ___ 0 and

ea _> e;3 for a >_/3 [17].

Def in i t ion 1.1.1 . A C*-algebra possessing a countable approximate unit is called ~r-unital.

Def in i t ion 1.1.2 . An element h E A is called strictly positive if for any positive nonzero p (or, equally,

for any state) one has ~(h) > 0.

R e m a r k 1.1.3. These two conditions are equivalent. It is possible to consider ei >_ 0. Then h := ~ e~/2 i

is strictly positive. Conversely, e, := h */~ is a countable approximate unit. A separable algebra always

satisfies these conditions. The details can be found in [55].

138

We will often use the following s ta tements .

L e m m a 1.1.4 ([55, L e m m a 1.4.4]). Let x, y, and a be elements of a C*-algebra A such that a >_ 0 and

x* x < a s, yy* < a ~, c~ + /3 > 1 .

Then the sequence un = x[(1/n) + a]- l /2y is norm convergent in A to u such that Ilutl < IIa(a+,2-1)/211.

P r o o f . Pu t d~,~ := [ ( l /n ) + a] -~/2 - [ ( l / m ) + a] -1/2. Then

]]Un - - u l] 2 = ]]xdn~,~y][ 2 = Ily*dn,~x* xd,~,yl] <_ []y*dnmaadnmY][ = ]]aa /2dnmY][ 2

= [la /2dn,.,yy*dn. &/21l < lla /2d . a d, ma /2ll = 2.

Studying the convergence of a monotone sequence on a spec t rum a, we ob ta in by the Dini theorem tile uni form convergence of

[ ( l / n ) > t t E Sp(a) .

Therefore, tld~,~a(~+~)/211--+ O, so that , by the Cauchy criterion, {u~} is n o r m convergent to an element u E A. Then, reasoning as above, we obtain

Ilunll = IIx[(1/n) + a]-i/2yll < 11&/2[(Un) + a]-l/2aB/211 < Ila(~+~-~)/211,

so tha t Ilull-< Ila(~+~-~)/211- []

P r o p o s i t i o n 1.1.5 ([55, Propos i t ion 1.4.5]). Let x and a be elements of a C*-algebra A such that a > 0

and x*x < a. For any 0 < a < �89 there exists an element u �9 A such that [lull < I[a�89 and x = ua ~.

P r o o f . Let U S d e f i n e ~t n : = X[(1/It)@ a]-�89189 -c~. By Lemma 1.1.4, {u~} is norm convergent to an element u ~ A such tha t

I (~

Ilull _< Ila�89 = II II- Furthermore , the Dini t heo rem applied to appropr ia te functions on the s p e c t r u m implies

II x - Una~]l 2 = IIx(1 - [ ( l / n ) +a] - l /2a l /2) l12 < Ilal/2(1 - [ ( l /n) + a]-Z/2aX/2)]l ~- > O, n ~ oc.

Thus, x = ua ~. []

1.2. P r e - H i l b e r t m o d u l e s

Let 34 be a module over a C*-algebra A. We denote the action of an element a E A on M by x. a, where x E 3//.

D e f i n i t i o n 1.2.1. A pre-Hilbert A-module is a (right) A-module 3A equ ipped with a sesquilinear form (., .} : M x 34 -+ A satisfying the following propert ies:

(i) (x,x} _> 0 for any x E A4; (ii) (x,x} = 0 only in the case where x = 0;

(iii) (y, x} = (x, y}* for all x, y E .hi; (iv) ( x , y . a } = ( x , y } a for all x, y C 3 4 , a E A .

The map (., .} is called an A-valued scalar (or inner) product.

Let us consider a few examples.

E x a m p l e 1.2.2. If J C A is a right ideal, t hen J can be equipped wi th the s t ructure of a pre-Hilbert A-module if we define the inner product of e lements x, y E J by the equal i ty (x, y) := x*y.

E x a m p l e 1.2.3. If {Ji} is some countable set of right ideals in a C*-algebra A, then the linear space M of sequences (xi), xi c ,],~, satisfying the condit ion ~ [[xi[[ 2 < oc, becomes a right A-module if

i

139

( x i ) - a := (xia) for (xi) E At, a E A, and becomes a pre-Hilbert A-module if we define the inner product X* of elements (xi), (Yi) �9 M by the equality ((xi), (Yi)} := E iY~-

i

Let K: be a right A-module equipped with a sesquilinear map [., .] �9 /(; • /(; -+ A, satisfying all properties of Definition 1.2.1 except (ii). Pu t

X := {x e ~ " [x,x] = 0}.

For each positive linear functional ~z on the C*-algebra A, the map (x, y) ~ p([x, y]) is a (degenerate) inner product on ~ , and hence the set N~ = {x �9 ~ : F([x, x]) = 0} is a linear subspace in /~. By taking the intersection of all such subspaces, we obtain tha t N = ('1N~ is also a linear subspace in /~. From

properties (iii) and (iv) of Definition 1.2.1, it follows that N . A C N; therefore, N is a submodule in/C. The quotient module 34 = I~/N is equipped wi th the obvious s t ructure of a pre-Hilbert A-module with the inner product (x + N, y + N) := [x, y].

Let 3/1 be a pre-ni lber t A-module, x �9 Ad. P u t Ilxll~ := ll(x,x)ll ~/2. We will omit the index 34 if

it will not lead to any confusion of norms.

P r o p o s i t i o n 1.2.4 ([52]). The function I111~ is a norm on .A4 and satisfies the following properties:

(i) Ilx "allM -< Hxll~" Ilall for all x �9 A4, a �9 A; (ii) (x,y) (y,x> < Ilyll~ (x,x) for all x , y �9 .M;

(iii) II(x,y}H < ]lxl]~ [[YHM for all x ,y �9 34.

P r o o f . For any positive linear functional ~ on A, the function x ~ p((x ,x)) 1/2 defines a seminorm

on 34. For each x �9 Ad,

Ilxll~ = I I (x , x ) t l ~/2 = sup{gz((x,x))l/2},

where the supremum is taken over all states ~ on A. Therefore, II-II~ is a seminorm and by property (ii) of Definition 1.2.1, II'llz4 is a norm on 34. S ta tement (i) follows from the equality

Ilx . a l l ~ = II(x.a,z.a)ll--Ila* (x,x)all <_ Ilall 2 I I (x ,x) l l - - I lx l l~ Ilall 2-

To prove statement (ii), we will take x ,y ~ 34 and a positive linear functional ~ on A. Applying the Cauchy Bunyakovskii inequality for the (degenerate) inner product ~((., .)) on 34, we obtain

~((~, y} (y, x}) = ~((x,y. (y, x)}) _< ~((:~, ~})~/~. ~((y. (y,x) ,y. (y, ~}})~/~

= y)((x,x)) 1/2. ~((x,y} ( y , y ) ( y , x ) ) 1/2 < ~((x ,x)) 1/2. II(y,y)ll ~/2. ~((x, y ) (y ,x ) ) 1/2.

Thus, for any positive linear functional ~ we have ~ ( ( y , x ) ( x , y ) ) < IlYlIM" ~( (x , x } ) Hence statement (ii) is proved. It obviously implies (iii). []

We will call the inequality (ii) (and also its consequence, the inequality (iii)) of Proposit ion 1.2.4 the Cauchy-Bunyakovskii inequality for Hilbert modules.

R e m a r k 1.2.5. For any C*-pre-Hilbert module, or more precisely, for any sesquilinear form (-, .), the following polarization equality is obviously satisfied:

3

4 (y,x} = ~ i ~ (x + iky, x + iky) for all x, y �9 3/t. k=0

1.3. Hi lbert C * - m o d u l e s

D e f i n i t i o n 1.3.1. An A-module 34 which is at the same time a Banach space with a norm I]'11 satisfying the inequality IIx-all _< Ilxll Ilall, x E Ad, a E A, is called a Banach A-module.

140

D e f i n i t i o n 1.3.2. A pre-Hilbert A-module AA which is complete wi th respect to the norm II-]]~ is

called a Hilbert C*-module.

If AA is a pre-Hilbert A-module , t hen the act ion of the C*-algebra A and the inner product on J~A

extend to the complet ion AA, which thus becomes a Hilber t module. Let us consider some examples.

E x a m p l e 1.3 .3 . If J C A is a right ideal, then the pre-Hilbert module J is complete with respect to the no rm []']].I = ]l']]. In part icular: the C*-algebra A itself is a free Hilbert A-module with one generator .

E x a m p l e 1.3.4. If {Adi} is a finite set of Hilbert A-modules ; then it is possible to define their direct sum ( ~ M i . The inner p roduc t on | is defined by the formula (x, y} := E (x~, y~), where x = (xi),

i

y = (y~) E G M ~ . We will denote the direct sum of n copies of a Hilbert Module M by M ~ or Ln(AA).

E x a m p l e 1.3.5. If {Adi}, i E N, is a countable set of Hilbert A-modules, t hen it is possible to define their direct sum (~ A4~. We define the inner p roduc t on the A-module O M~ of all sequences x = (xi), x~ E A/t~, such tha t the series ~ (x~, x~} is norm convergent in the C*-algebra A, by the formula

i

Ix, y/:-- for Y i

Let us show tha t the ment ioned series converges. F rom the convergence of series ~ (xi, xi} and y~ (Yi, Yi} i i

it follows tha t for any c > 0 there exists a number N such that for all n > 0 the following es t ima te

holds:

NLn(Xi,9;i} < s , N~a(yi,Yi) < a. i : N i=-N

Then

N•_•n N + n N ~ ~

<- Z ly ,y l i=N i=N i=N

This proves tha t the inner p roduc t is well defined.

Let us verify the completeness of the module G . h 4 i . Let x (n) , (n)~ ~ A / t i be Cauchy = ~x i ) E a sequence. T h e n for any c > 0 there exists a number N such tha t for all n, m _> N

- x i ,xi - x ~ <C. (1) i

Since all s u m m a n d s in (1) are positive, the inequali ty

- - ;z i , ; z i - - X ~ E

holds for each number i separately. But then the sequences _(n) E M i are the Cauchy sequences and

they have limits xi = limx~ n) E AAi. Let us verify t ha t a series ~ (xi, x~} is n o r m convergent in A. Fix

5 > 0. There exists a number n > N such tha t the es t imate (1) is valid. Let us choose a n u m b e r K satisfying the condit ion

k-~'~t; / (n) (n) \ \ x i ,x i / < c.

141

Then for any k > 0 we have

Therefore,

K + k ( / x ( m ) ,~(m) <x~n) (m) (m)\ ,/'~!m) ~(n) x~m)> _~_

i=K

- - ;Li : : c i - - ; L i - - _ _ - - ;Li : & i - - ; ~ i < s i=K

K+k

V ~ 'xl i=K

< 2 ~ + i=K i=K

!m ) x~m) > 1/2 z __< 2s -~- 2 x ~ n ) _ xi-(m) , xi-(n) - - X ~ m )

i=K

I (~) (m)\ 1/~ _< 2C+2C 1/2 \X i ,X i / .

Now, by solving the square inequality, we obtain tha t

i=K

Passing in the inequality (2) to the limit m -+ oc, we obtain

k(xi, xd < 8e. i=K

This proves that the series ~-~i (xi,x~) is norm convergent.

We shall denote the direct stun of a countable number of copies of a Hilbert module A,4 by 12 (A4) or

H34. We call the Hilbert C*-module 12(A) (another notation is HA) the standard Hilbert module over

A. If tile C*-algebra is unital, then the Hilbert module HA possesses the s tandard basis {ei}, i C N,

where e~ = (0 , . . . , 0 , 1 , 0 , . . . , 0 , . . . ) with the unit at the ith place.

E x a m p l e 1.3.6. Let B C A be a C*-subalgebra of a C*-algebra A having a common unit. Let us

assmne that there exists a linear map E : A --+ B not increasing norms and being a projection, tha t is,

E 2 = E. Such a map is called a conditional expectation from A to B. A conditional expectat ion is a

positive map, i.e., E(a*a) > 0 for all a E A, and it satisfies the equality

E(blab2) = biE(a)b2 for a E A, bl,b2 E B

(see [62]). A conditional expecta t ion is called exact if for any positive element a E P+(A) the equality

E(a) = 0 implies a = 0. In the case where the conditional expectation is exact, it is possible to introduce

the s tructure of a pre-Hilbert B-module on the C*-algebra A by putt ing

{x,y} = E(x*y), x ,y E A.

We will give a condition for this nlodule to be a Hilbert module (i.e., to be complete) in Sec. 4.4.

Let H C 3// be a closed submodule of a Hilbert nlodule A~. We define the orthogonal complement J~• by the equality

.M "• = { y E Y U I : (x,y} = 0 for all x e A / ' } .

Then Af • is a closed submodule of the Hilbert module M too. However, the equality Ad = A / | Af • is not always fulfilled, as the following shows:

142

E x a m p l e 1.3.7. Let A = C[0, 1] be the C*-algebra of all continuous functions on the segment. Let

us consider in the Hilbert A-module 54 = A the submodule N" = C0(0, 1) of functions vanishing at the

end points of the segment [0, 1]. Then, obviously, N "• = 0.

If 34 is a Hilbert A-module, then we denote by 54 �9 A the closure in 54 of the linear span of the

elements of the form x - a, x 6 A4, a 6 A.

L e m m a 1.3.8. 54 - A = 54.

P r o o f . Let e~ 6 A be an approximate unit. Then for any x 6 54

llx- x-<ll ~ = ll(x- x . e ~ , x - x. e.>ll < (1 + lle;ll)ll(x,x>- (x,x> e~{l ~ O,

which proves that the elements of the form x �9 e~ are dense in 54. []

We will often use the following useful s tatement .

L e m m a 1.3.9. For any x E 54,

Let (x, x) = a; then P r o o f .

x = l i m x ( x , x } ( ( x , x } + e ) -~. S - - + O

) Ita(t + c) = - tl =

I1-<x,x> (<x ,x> + c) - 1 - xll 2 = It <x(<x ,x> ( < x , x } + c) - 1 - 1), x((x, x} ((x, x} + e) -1 - 1)} 11

= Ila(a2(a + c) -2 - 2a(a + e) -1 + 1) l l - - Ilaa(a + c) -2 - 2a2( a + 8) -1 + all > 0,

since the following inequalities hold under the condition t > 0:

- 1 = (t + s ) 2 ] = s

and

ct + 2t 2 3

IF'(t + c) < - tJ t~+ ~ = <C. []

The following s ta tement is an analog of the polar decomposition for Hilbert modules. We will see

below that, as in the case of algebras, the exact polar decomposition exists only in the case of Hilbert

modules over W*-algebras.

P r o p o s i t i o n 1.3.10 ([38]). Let 54 be a Hilbert A-module, x 6 54, and 0 < c~ < 1/2. Then there exists

an element z 6 54 such that x = z . (x, x} ~.

P r o o f . For n E N, let us define functions

{ n -~/2, i f A _ < l / n ,

g~(A) = Aa/2 ' if A > 1/n.

Then, by the spectral theorem,

II ~- (g,~((x, x } ) - g.~((x, x}))l I = If(x, x} ( g n ( ( X , X } ) - g.~((X, X}))~][ v~

= s u p { ] A ( g n ( A ) - g,.,(A))[ : A 6 Sp ( (x , x})}.

Therefore, the sequence x . gn((x, x}) is a Cauchy sequence; hence it has a limit z 6 54. Then

lim Hx.gn( (x ,x} ) (x ,x} a - x N= lim ]]x(g~((x,x}) (x ,x} ~ - l)H n --~. o o n - & o o

lim sup{]A1/2(g~(A)A ~ - 1)]" A 6 Sp (x,x}} = 0 .

llz <*,*>* - xll =

z

This completes the proof. []

143

A Hilbert C*-module 3/l is called finitely generated if there exists a finite set {xi} C AA such tha t Ad coincides with the linear span (over C and A) of this set. A Hilbert C*-module 34 is called countably generated if there exists a countable set {x~} C jr4 such that 34 coincides with the norm-closure of the linear span (over C and A) of this set.

1 .4. T h e s t a n d a r d H i l b e r t m o d u l e HA

T h e o r e m 1.4.1 (Kasparov stabil ization theorem [34]).

generated Hilbert A-module. Then Ad | HA -~ HA. Let A be a C*-algebra and A4 be a countably

P r o o f . We s tar t by proving the theorem for the case of a uni ta l C*-algebra A. For this purpose , it is convenient to use the procedure of a lmost-or thogonal iza t ion [20]. An element x of the Hilber t module N" is called nonsingular if the element {x, x} ~ A is invertible. The set {x~} EAf is called orthonormal if {x~, x i ) = (~.ij. It is called a basis of the module N" if finite sums of the form ~ x ~ . a i , ai ~ A, are dense

in A/.

L e m m a 1.4.2 ([20]). Let N" be a Hilbert A-module containing the orthonormal elements e~ , . . . , e~,

x ~ A[, c > O. If an element y E A[ satisfies {y, y} = 1 and y• e~, . . . , en}, then there exists an element en+l E N" such that

(i) the elements e l , . . . , en ,e~+l are orthonormal;

(ii) en+l E S p a n A ( e l , . . . ,en, x ,y ) ; (iii) dist(x, S p a n A ( e l , . . . ,e~+l)) _< c.

P r o o f . Let

T h e n

n

x' = x - E e i (ei, x}, x" = x' + cy. i = 1

(x", x"} = (z ' , x'} + z 2 >_ z 2 > 0;

therefore the element x" is nonsingular. Pu t e~+l = x " . (x", x"} -1/2. Then

en+l E SpanA(x ' , y ) • , en}.

Therefore, the elements e l , . . . , en, en+~ are o r thonormal . Since

x' E SpanA(x, e l , . . . ,e~) and en+l E SpanA(x ' ,y ),

we obta in s ta tement (ii). Finally, let us put

n

w = e~+l {x",x"}l/2 + E e i {ei, x) E S p a n A ( e i , . . . , en+t); i = 1

t hen the equality Ilw - xll = II x' ' - :r'll = I I cy l l = c proves (iii). []

We return now to the proof of Theorem 1.4.1. Let {y,~} be the sequence of generators of the module 3//. By {en} we denote the s tandard basis of the module HA. Let {x~} C {en} 0 {y~} be a sequence in which one encounters each element en and each e lement Yn infinitely many times. Then the set {x~} is generat ing for the module Ad | HA. We will prove the theorem by induction. Let us assmne tha t the o r thonormal elelnents e l , - . . ,gn E AJ | HA and the number re(n) _> n are already cons t ruc ted in such a way tha t

(i) c SpanA ,... (ii) dist(x~, SpanA(gl , , ~ ) ) < 1 1 < /c < n.

144

Since each element xi is equal to ej or Yk, it is possible to find a number m' > re(n) such t ha t em,•

�9 . . , xn+~}. Since em, • ,era(n)}, it follows f rom condi t ion (i) tha t

By Le lnma 1.4.2, there exists an element

en+s E SpanA(eZ, . . . , en, Xn+l, era,) (3)

such t ha t the elements g l , . - - , e~, en+l are o r thonorma l and

dist (x~+,, Span A (gt, �9 �9 �9 , g~+l)) _< - - 1

n + l

It follows from (3) and from condit ion (i) tha t

{ ~ 1 , �9 - �9 , en+l} C S p a n a ( x z , . . . , x,~+l, e l . . . , e,~,).

By pu t t i ng m(n + 1) = m ' we complete the s tep of the induction�9 Thus, an o r thonormal sequence g~ satisfying propert ies (i) and (ii) has been cons t ruc ted . But proper ty (ii) means tha t this sequence generates the whole modu le A// | HA; therefore Ad | HA ~- HA.

Thus, Theorem 1.4.1 is proved for unital C*-algebras. Let A be a C*-algebra wi thout uni t and A + be its unit ization. By defining the action of A + on the Hilbert A-module M by the formula x . (a, A) := x - a + xA, x E 34, (a,A) ~ A +, k E C, we tu rn 3// into a Hilbert A+-module. Let us consider the

A+-modu le HA+ and denote by HA+A the closure in HA+ of the linear span of elements of the form z �9 a, x E HA+, a E A. It is easy to see tha t HA+A = HA. The i somorphism A.4 | HA+ ~ HA+ implies the isomorphism

.A4 | HA = M A | HA+ A = (34 | HA+ )A ~ HA+ A = HA.

This completes the proof of the theorem. []

D e f i n i t i o n 1.4.3. Let M be a Hilbert A-module such tha t there exists a Hilbert A-module iV" for

which the relation A4 | Af ~ Ln (A) holds wi th finite n. Then 3// is called a finitely generated projective A-module .

The following two theorems of Dupr~ and Fi lhnore show tha t finite-dimensional project ive submod- ules in Hilbert modules have the simplest construct ion.

T h e o r e m 1.4.4 (Dupr5 Fil lmore [20])�9 Let A be a unital C*-algebrn and Ad be a finite-dimensional

projective A-submodule in the standard Hilbert A-module HA. Then

(i) the nonsingular elements of the module M • are dense in AJJ-;

(ii) HA = M | AdZ;

(iii) M • ~- HA.

P r o o f . We begin the proof of the theorem wi th the case where M ~ L~(A). Let g l , . . . ,g~ be an o r thonormal basis in A4. We fix c > 0. For each m, let us put

n

' E % = - g i {gi , e , , j ,

i = 1

, A d i where e.~ E . Then n

{e~n, e~ } = 1 - ~ {em,gi} {9i,em). i = l

Since the equality (x, e,~} -+ 0 holds for each x E HA, w e conclude tha t {e~m, e ' } -+ 1; therefore, there I exists a number mo such tha t for m _> m0 the e lements e m are nonsingular. Hence it is possible to define

I I I e I = ( m , - 1 / 2

145

such that ' " " ' ~em, era? = i. Let x E Ad • Then

( e " , x } ( < , e t } -1/2 ( & , x ) , , -1/2 = = <e.~, e~) <e~,x) -* 0

Let us select a number m _> mo such that ]] ~{e'm, x} ]I < c and put x t = x + ~ em." It is easy to see that

II ' - x l l = ( 4 )

Let us show that the element x' is nonsingular. Pu t

I, (e" x~ , , , tt x } + c u = X - - e m ~ m, / , v=ern~iem, 1).

Then u l v (since u l e ~ ) and x' = u + v. Therefore,

<x', = + v> : + (<&', x> + 1)*(<&', x> + i), (5)

e H X and the right-hand side of (5) is invertible since ]]( m, }H < & Therefore, (x' ,x') is invertible too.

Together with the est imate (4), this proves s ta tement (i).

Let {x~} be a sequence in which each element em is repeated infinitely many times. Let us put 7~

x = x~ - ~ 9~ (g~,xj . T h e n (taking c = 1) it is possible to find an element gn+~ E 3/l • such that i = 1

(gn+l, gn+~} = 1, d i s t (x ,g~+lA) _< 1, and therefore dist(xl , SpartA(g1,. . . ,g~+~)) <_ 1. At the next step,

we replace the module M by SpanA(g l , . . . ,g~+l) , z i by xh, and z = 1 by e = 1/2. Going on with the

indicated procedure, we will obtain an or thonormal basis {gk}, k E N, extending the basis g l , . . . , gn of

the submodule At; then {gk : k > n} is a basis of the module A/t • This proves statements (ii) and (iii).

We pass now to the case of an arbi t rary finitely generated projective module A4. Let M| ~ L,.(A). By Theorem 1.4.1, A/" | HA TM HA holds, and therefore

L~(A) ~ Af | Yt4 C A/'| HA TM HA.

Hence, if K is the orthogonal complement to the submodule N' | ill the module Af| then ~ ~ HA and A / | M | = N" | HA. But, obviously, /C = A4 • is the orthogonal complement of the submodule

A4 in the module HA. []

T h e o r e m 1.4.5 ([20]). Let A be a unital C*-algebra and let 2t4 be a finitely generated projective Hilbert submodule in an arbitrury Hilbert A-module Af. Then A[ = M | 3A •

P r o o f . Similarly to the previous theorem, the proof can be reduced to the case where Ad = Ln(A) ?%

is a flee module. If { g l , . . . , g,~} is the s tandard basis of M and x E N', then put x' = x - ~ g~ (g~, x}. i = l

Then x' E M and x - x' E M • therefore Af = 3,t | AA • []

2. O p e r a t o r s o n H i l b e r t M o d u l e s

2 .1 . B o u n d e d o p e r a t o r s a n d o p e r a t o r s a d m i t t i n g a n a d j o i n t

Let fl4,Af be Hilbert C*-modules over a C*-algebra A. The bounded C-linear A-homomorphisms

from the module AJ to the module A / a r e called operators from ~4 to Af. We denote by HomA(Ad,Af) the

set of all operators from the module 3,t to tile module A/. If A / = A/l, then EndA(fl4) = H o m A ( M , M ) is obviously a Banach algebra. However, we shall soon see that there is no na tura l involution on this algebra.

Let T E HomA(A4,Af). We say tha t T admits an adjoint if there exists an operator T* E HomA(Af, AA)

such that for all x E A8, y E J~,

(Tx, v> = (x, T'y).

146

L e m m a 2.1.1.

for any x, y E Ad

Let A,'[ be a Hilbert A-module, and T" A,'[ -+ A4 and T* �9 A4 -+ Ad be maps such that

(x, Ty) =/r*~, ,~) . Then T (and T* as well) is a bounded homomorphism from EndA(Ad). Therefore, T C End~t(A/l ).

P roo f . For any x, y, and z from 3.4, w E C, and a E A, one has

( z ,T ( x + y)) = (T*z,x + y) = (T* z ,x) + (T* z,y} = (z, r x ) + (z, Ty} = (z, T x + Ty) ,

(z,T'wx) = (r* ~,~) w = (z, Tx) w = (z, w r z ) ,

(z,T(xa)) = <T*z, xa) = <T*z,x)a= (z, T x } a = (z , (Tx)a) .

Since z is an arbitrary element, it follows that

T(x + y) = Tx + Ty, T(wx) = wTx , T(xa) = (Tx)a,

and the linearity properties hold.

To prove the continuity of T, we should verify that its graph is closed. Let x~ ~ x, T(x~) --~ y in A/l, and z E JM be an arbitrary element. Then

0 = ( T * ( y - Tx),xc~} - ( T * ( y - Tx) ,xa}

= ( y - Tx , T(x~)} - ( f * ( y - Tx ) , x~} > ( y - Tx, y) - ( T * ( y - Tx) ,x ) = ( y - Tx , y - Tx ) .

We now show that there exist operators without adjoint.

E x a m p l e 2.1.2. Let A be a unital C*-algebra. As above, the standard basis of the Hilbert module HA consists of the elements ei = ( 0 , . . . , 0, 1, 0 , . . . ) , where 1 is at the ith place. It is possible to associate with each operator E EndA(HA) its matr ix with respect to this basis:

IIt~jll, t~j = (e~, Tej).

Then the adjoint operator has the matr ix IIt~ill.

Let A = C([0, 1]), and let the flmctions ~ E A, i = 1, 2 , . . . , be defined by the equali ty

0 on[0 ] l 1 1

~ = 1 at the point x~ = ~ + ,

is linear o~ [~@l,X~l ~ d [x~,} ]

Let an operator T E EndA(HA) have the matrix

0 0 00

(actually, it is an operator from the module HA to A, i.e., is bounded. But the operator T* is not well defined, since

an A-functional). It is easy to verify that T it should have the matrix

( ~; 0 0 . . . ) ~ 0 0 ~; 0 0

but the image of the basis element ei should have the first column as its coordinates, i.e., an element of HA; however, the series ~ ~i~* does not converge with respect to the norm in tile C*-algebra A.

147

We denote the set of all operators from the module 34 to the module A / a d m i t t i n g an adjoint by H o m ~ ( A 4 , ~ ) . T h e algebra End},(A4) = Hom~(A4 ,34 ) is a Banach involutive algebra. Moreover, it is a C*-algebra; this follows from the es t imate

IIT*TII _> sup {(T*Tx, x)} = sup {(Tx, Tx)} = IITII 2, xEB~ (M) x~B~ (~4)

where the uni t ball in the module Ad is denoted by B I ( M ) . The following s ta tement will be used by us frequently w i thou t special reference.

P r o p o s i t i o n 2 .1 .3 . For an operator T : A4 ~ 34, the following conditions are equivalent:

(i) T is a positive element of the C*-algebra End*(34) ; (ii) for any x E 34, the inequality (Tx, x) >_ 0 holds, i.e., this element is positive in the algebra A.

P r o o f . The first condit ion is equivalent to the equali ty T = S*S for some S E End*(A4). Therefore,

(Tx, x} = (Sx, Sx} > 0 for any x ~ Ad.

Now let (Tx, x} >_ 0 for all x E M . Then

(Tx, x} = (Tx, x}* = (x, Tx} for all x E 2M.

The map (x, y) ~-~ (Tx, y) defines a sesquilinear form on 3,1; therefore, by the polar izat ion equality 1.2.5, (rx, y} = (x, Ty) for all z and y f rom 3,t. This means, by Lemlna 2.1.1, t ha t T E End*(Ad) and T = T*. So, T is a self-adjoint e lement of the algebra End* (A4) and, therefore (see [17, 1.6.5]), it can be represented in the form of the difference T = T+ - T _ of two elements of End*(34) , T+ _> 0, T_ > 0, and T+T_ = T_T+ = 0. Then (T_y, y) <_ (T+y, y} for any y E 34 . In part icular ,

= < = 0 .

On the other hand , T_ > 0 and T_ a > 0; therefore (T3_x,x} > 0 (as the s t a tement in this direction has already been proved). Thus, we have the following unique possibili ty: (T~x, x} = 0 for any x ff 34. By the polarizat ion equality, this implies t ha t (Tax, y} = 0 for all x and y from A4, and T 3 = 0, T_ = 0.

Hence, T = T+ _> 0. []

T h e o r e m 2 .1 .4 ([52]). Let 34 and N" be Hilbert A-modules and T 34 ~ A[ be a linear map. Then the following conditions are equivalent:

(i) the operator T is bounded and A-linear, i.e., T(x . a) = Tx . a for all x E 34, a E A; (ii) there exists a constant K >_ 0 such that for all z E A4 the operator inequality (Tx, Tx} <_ K (x, x)

holds.

P r o o f . To obta in the second s t a t emen t from the first one, let us assume tha t T(x �9 a) = Tx �9 a and

IITII _< 1. If the C*-algebra A does not contain a unit , then we can consider the modules A4 and Af as modules over the C*-algebra A + obta ined from A by unit izat ion. For x E Ad and n E N, let us pu t

~ X n ~ X �9 a n . an x, x) + ~/

Then (Xn,X~} =a ~ {x , x )a~= {x,x} ((x,x} + ~ ) -1 < 1; therefore, Iixnll G 1, hence IiTxnll G 1. Then for all n E N the opera to r inequality (Txn, Txn} _< 1 is valid. But

1 (Tx, Tx} = a~ ~ (Tx~,Tx~)a~ 1 <_ a~ 2 = (x,x} + - . (1)

'Tt

Passing in the inequal i ty (1) to the limit as n -+ oc, we obta in (Tx, Tx) <_ (x, x). To derive the first s ta tement f rom the second one, we assume tha t for all x C M the inequality

(Tx, Tx} < (x, x} is fulfilled. This obviously implies tha t the opera tor T is bounded , IITll __ 1. Let

148

x E M , y E A/'. Let us define a map r : A + -+ A + by the equali ty

r (a) = ( y , T ( x - a)}.

Then

r(a)*r(a) = (T(x .a) ,y} ( y ,T (x .a ) } <_ [[yll 2 ( T ( x . a ) , T ( x . a ) ) < Ilyll 2 < z . a , x . a )

= ]ly]] 2a* (x ,x}a _< 11yl]2 ][xll2a.a.

To complete the proof, we use the following s ta tement .

Lemma 2.1.5 ([28, 52]). Let A be a unital C*-algebra and let r : A -+ A be a linear map such that for some constant K >_ 0 the inequality r(a)*r(a) <_ Ka*a holds for all a E A. Then r(a) = r (1)a .for all a E A. []

Therefore, r ( a ) = r(1)a, i.e.,

( y ,T (x .a ) } = (y, Tx>a= (y, T x . a }

for all y EAf, x E J~4. This implies the s t a tement of the theorem. []

Corollary 2.1 .6 . Let ,/~4,Af be Hilbert A-modules and T C EndA(AA,Af). Then

I l r l l - - inf{ K 1 / 2 : (Tx, Tx} <_ K (x,x> Vx e M}.[~

Let A.t = A f | 1 6 3 be a decompos i t ion into an o r thogona l direct sum of Hilbert modules. Example 2.1.7. We define an opera to r P : M -+ A/t as the opera tor of project ion onto a submodule Af along the module s Then P is bounded , l]Pll = 1, and P* = P; therefore P ff End~(Ad) .

2.2. Compact operators in Hilbert modules Let AJ ,Af be Hilbert A-modules, x c Af, y E A/t. Let us define the act ion of an opera tor 0x,y : M --~

Af on an element z E Ad by the formula

0x,y(z) := x ( y , z ) (2)

Operators of the form (2) are called elementary operators. They obviously satisfy the following equali-

ties:

(i) (0x,y)* = 0y,x;

(5) Ox,yO~,~ = Ox<y,~,>,v = O~,,<~,y) for u C M , v E Af; (iii) TOx,y = OTx,y for T E HomA(Af, s (iv) Ox,yS = Ox,S*y for ~q E HOln~(Z~,M).

We denote the closed linear span of the set of alI e lementary opera tors by KJ(]vI,Af). We shall call the elements of ]C(M,A/') compact operators. In the case Af = A4, equali t ies (i) (iv) mean tha t the algebra I~(M) = I C ( M , M ) is a closed two-sided ideal in the C*-algebra E n d ~ ( M ) . Compac t operators acting on Hilbert modules are not compact operators in the usual sense if t lmy are considered as operators from one Banach space to another. However, they are a natural general izat ion of compac t operators on a Hilbert space.

Proposition 2.2 .1 . Let H A be the standard Hilbert module over a unital C*-algebra A, and It Ln(A) C HA be the free submodule generated by the first n basis elements. An operator K C End~(HA) is compact if and only if the norms of restrictions of the operator K onto orthogonal complements to submodules Ln ( A ) tend to zero.

149

P r o o f . Let us denote by P~ the projection in HA onto the submodule L,(A) • Then for any zLL~(A) we have

II0~,~(~)ll ~ = II<o~,~(z),O~,~(z)lll = I]<y,~}* (x,x> (y,z>]l _< [[x[[ e [[<y, z>[I 2 = HxH 2 [[<pny, z)[[ 2

_< Ilxll ~ . Ilpnyll 2. Ilzll 2 .

Since IIr~yl[ tends to zero, the same is t rue for the norm of the restriction of the opera tor 0~,y to the

submodule L~(A) ~- and, therefore, for the n o r m of any compact operator. Let us assume now tha t for

some operator K the relation II- 0 holds. Then, since 2 Ke,~ <e~,z} = 0 for any z-J-Ln(A),

we have for Ilzll _< 1 and z iL~(A),

sup Kz - ~ Ke,~ (am, z) z m=l

= s u p IIK~ll -+ 0 (a) z

?%

as n ~ ec. If z E L~(A), then Kz = ~ K e ~ (era, z). This means tha t (3) also holds if the supremum m = t

is taken over the unit ball of the whole module HA; therefore, the operator K is a norm limit of the n

operators K~ = ~ 0~:~.~,~ . [] m = l

Let us note tha t in the case of modules over C*-algebras without unit, the s ta tement of 2.2.1 is not

valid.

We denote the C*-algebra of compact operators of a separable Hilbert space H by 1(2. Since the

algebra ~ is nuclear [37], there is a unique C*-seminorm on the algebraic tensor product o f / ~ by any

C*-algebra A, and we denote by /~ | A the completion with respect to this seminorm. We denote by

AJ~(A) = 3A~ | A the C*-algebra of all n • over A.

P r o p o s i t i o n 2.2.2. There exist the following natural isometric isomorphisms:

(i) ]C(A) ~ A; (ii) ]C(L,~(A)) ~- AJ~(A);

(iii) IC(HA) ~ ]C | A.

P r o o f . If a C*-algebra is unitai, then (i) is obvious. In the general case, we consider a map p "

Spanc(0~,b �9 a, b E A) --+ A defined by the formula

AiOa~,b~ = A~aib~. \i=l i=l

Let us verify tha t tiffs map is well defined. If ~ AiO~,,b~ = ~ #jOcj,dj, then ~ Aia~b~x = ~ #jcjd3x for i j i j

any x E A; therefore, ~ A~a,~b~ = ~ pjcjd~. The map ~ is multiplicative and involutive: i j

~(0a,b 0c,d) = ~(6ab*,dc*) = ~(0a,b)~(0c,d), ~(0~,b) = ~(Ob,a) = ~(0a,b)*.

The surjectivity of ~ follows from the possibility of the representat ion a = u(a*a) 1/a for any a E A (see 1.1.5). If (u~), a c .4, is an approximate unit of the algebra A, then

lim ~iOa~,bi(Ua) = Aiaib~ ; o~

i = 1 i = 1

150

n therefore, II~(k)ll ~ Ilkil for k = E "~iOai,b~ �9 This means tha t the map p can be extended by continuity

i = 1 up to a map from the whole a lgebra/C(A). The inequality

i=~ "~iOa~'bi -~IISx,~P<_ 1 i=~ ~iaib*x ~ i=~1 ~iaib* ~-- ~(i:~l'~iOai, hi)

shows tha t the map ~ is an isometry, so (i) is proved�9 Sta tement (ii) can be proved similarly with the

use of the map

~n : OalQ...Gan,blO"'| ~'} ( alb~ . . . album)

�9 O o ~ o

a~b~ . . . a~b~

Finally, since the isometric map from the linear space U1QLn(A)) to the linear space UA,4n(A) is Tt n

defined, and since these spaces are dense in C*-algebras IC(HA) and K: | A respectively, we obtain (iii). []

L e m m a 2.2.3. Let x E A,4 be an arbitrary element. Then there exist z E A4 and k = O~,v ~ IC(Ad) such that x = kz.

P r o o f . Let us put

u := v := z := l i m x ( e + (z,x>l/a) -1. a -+0

Since s2(e + s) -1 is uniformly convergent to s on bounded sets, in order to prove tha t u is well-defined,

one should note that for t = (z, x} the following relations hold:

< X ( s 71- (:F.,X}I/3)--I __ X (# -~- <X,X}I/3)--I,x ( s ~- <X, x } l / 3 ) - - I __ X (# "~- (X,X}I/3)--I>

= [(~ + t l /3) -1 _ ( , + r + t~/3) -~ _ ( , + t~/3) -~]

= [(c + t~/~) -~ - (~ + t l /~)-~]2(W~)~

The same argmnent shows tha t x = kz. []

Note tha t we have also proved tha t A4 {M, A4} = Ad.

T h e o r e m 2.2.4. A Hilbert A-module Ad is countably generated if and only if the C*-algebra ~(Ad) is (r-unital�9

P r o o f . Let the algebra ]C(M) be c~-unital and xn be a countable approximate unit of it. Then

x = lira x , x for any x E A 4 . (4)

Indeed, by Lemlna 2.2.3, x = kz holds for some k E E(AJ) , z E M . Since xnk > k wi th respect to the

norm, XnX = Xn]~Z ~ ]r = X. By definition, any compact operator is close to a linear combination of e lementary ones. Hence, for

each x~ there exist elements x~ and y~ from 34 such tha t

1 Oz?,v? - aen < n ' n = l ,2, . . . .

Let us show that the countable set x. ~ i = 1, s(n) n = 1,2, generates the module 34. Let Z ' � 9 1 4 9 ' "" " ' us consider an arbitrary element x E 3,t and an arbitrari ly small e > 0. By (4), we can find n so large that

I1~- ~ x l l < ~ and -~ < -'2

151

Then

,n s s - �9 ( y ? , x ) <_ I I x - n( )ll + - < + = c.

Assume tha t the module Ad is countably generated. It can be considered as a module over the algebra

A + obtained by unitization of A (if it was not unital) with respect to the action x . (a, p) := x . a + px ,

x E 3,t, a E A and p E C. If it was countably generated over A, then it should be countably generated

over A + too. Since, in the definition of elementary compact operators, only the A-inner product is

involved, we have ~A(34) = ](:A+ (34). Thus, we can restrict ourselves to the case of a unital algebra A.

So, 34 is a countably generated module over the unital algebra A. By the Kasparov stabilization

theorem, 2M | HA ~- HA. Let ~ : 34 -~ HA be the corresponding inclusion and ~r : HA -+ All be the

corresponding self-adjoint projection. Let {ei} be the s tandard basis of HA. Note that for a C*-algebra

the proper ty of being ~-unital is equivalent to tha t of having a strictly positive element. Let us consider

X :~ ~ 0e~'e~ , r

n n : l

or in mat r ix form (11) x : = d i a g 1 , 2 , 3 . . . .

Then x is a str ict ly positive element in I~(HA). Indeed, on the one hand, by Proposition 2.2.1 we have

x E I~(HA). On the other hand, if p : K ( H A ) ~ C is a s tate such tha t p(x) = 0, then p(O~,~,c,~) = 0 for

any n since all 0 ~ , ~ _> 0. Then, for any x = (x l , x 2 , . . . ) E HA,

p(O~,,xOx,~) = p O~n,ej.~j O~n,~jX3 = P O~n,~j~j ~,ej~j j = l \j-----1 /

j=l j=l

where the last inequality follows from

Thus,

and

(Oen.(CjXj,ejXj),en(Z), z) = (~n " (~jXj,~jXj) (~n,Z),Z) = (Z,~?~) X~Xj (~Tt,Z) ~ I1: 112 (%,~ z , z ) .

for any x, y, and z from 34,

O~,y(z) = x . (y , z ) = O ~ , ~ % , ~ ( z )

Ip (0x, . ) l = tP(O=,enOc,. ,) l _< Pl/2(Ox,enOen,X) PJ"/9'(Oe,n,,yOy,en) = O, since the second mu l t i p l i cand vanishes. So p vanishes on a dense subset, consequent ly everywhere oi1

]C(HA). We have shown that x is a s t r i c t l y pos i t ive element of ]~(HA). Then '.4 n := x 1/n is a countable

approximate unit of ~(HA) and 7c~,~ is a countable approximate unit of /~(Ad) . Indeed, if k E/~(A4),

7r~k = k, then ~krc E K ( H A ) and

2.3. C o m p l e m e n t e d s u b m o d u l e s and p r o j e c t i o n s in H i l b e r t C * - m o d u l e s Let us recall that a closed submodule A / o f a Hilbert C*-module Ad is called orthogonally comple-

mented if 3A = A / | A/• As we have already seen, a closed submodule of a Hilbert C*-module could

be orthogonally uncomplemented.

152

D e f i n i t i o n 2 .3 .1 . A closed submodule Af of a Hilbert C*-modu le 34 is called (topologically) comple-

mented if there exists a closed submodule s in 34 such tha t N" + s = 34, Af M/2 = {0}.

The following example shows tha t there exist topologically complemen ted bu t orthogonally uncom-

plemented submodules .

Let J C A be an ideal such tha t the equal i ty J a = O, a E A , implies a = 0. Let us Example 2.3.2. put 3/t = A | J ,

Then

N ' = {(b,b): b E J}.

H ~ -- { ( c , - ~ ) : ~ e J} .

Therefore, N" | Af • ---- J | J 7 ~ flA. However, the submodule L; = {(a, 0) : a ff A} C 3/t is a topological complement to Af in 3/t.

We denote a nonor thogonal direct sum of Hilbert modules by A/'@s Decomposi t ion into a direct sum 3/I -- Af(~s allows one to define a projec t ion P onto Af along s T h e operator P is A-linear and, by the closed graph theorem, is bounded; therefore, P E EndA(3A). However, as is clear f rom Example 2.3.2,

the project ion P cannot admit an adjoint. But if A4 = Af @ s t hen the corresponding projection is self-adjoint, P E E n d ~ ( M ) . Since it is more convenient to work wi th or thogonal decomposit ions, we would like to know when such a decompos i t ion exists.

T h e o r e m 2 .3 .3 ([46]). Let M and N" be Hilbert A -modu les and T E Hom~(A4, Af) be an operator wi th

closed image. Then

(i) K e r T is an orthogonally complemen ted submodule in ,~4,

(ii) Im T is an orthogonally complemen ted submodule in Af .

P r o o f . (i) Let h n T = A/'0 and let To :d~4 --+ 3/o be an opera to r such tha t its act ion coincides wi th the action of T. By the open mapp ing theorem, the image of the uni t ball of To(B~(d~d)) contains some ball of radius 5 > 0 in Af0. Therefore, for each y E Af0 it is possible to find x c A// such that Tox = y

and Ilxll_< 5 -~ Ilyll. Then

IlTgyll 2 --[[(y, ToT~y)]] < [[yl[- ]]ToT~)yl[,

and hence,

]]yl] 2 = II(To.~,y)II = ll(x, To*y}II <_ Iizit ' ]iT~y$t <_ ~-~ IiyiI " IlyI] ~/2 IIToT~)YII z /2 .

We obtain tha t for any y E A/'0

Ilyll -< ~-2 IIToT~ylI.

Let us show tha t the spec t rum of the opera tor ToT~ does not conta in the origin. Assume the opposite, i.e., tha t 0 E Sp(ToTg). Let f be a cont inuous function on R such t ha t

f(O) = 1 = l l f l l , f ( t ) = 0 if Itl > 1(~ -2.

Using funct ional calculus in the C*-algebra E n d ~ ( M ) , we define the operator S E End~(Ad) by the 1 --2 equal i ty S = I(ToT~). T h e n I]SI[ = 1 and IIToT~Sll _< ~ �9 We can choose an e lement x C M such t h a t

1 Then Ilxll = 1, I lSx l l > ~.

IlZoT~Sx[[ < 15-2 5 -2 - 2 < I lSx l l ;

this contradicts the assumpt ion (with y = Sx) . So 0 ~ Sp(ToT~); therefore, the operator ToT~) is invertible, and, in part icular , surjective. For any z EAd , it is possible to find an element w E Af0 such tha t T o z = ToTg w. Then z - T~) w E K e r T and

z = (z - T~w) + T~w E Ker T + Im To*.

153

Since the module Im T~ is obviously orthogonal to Ker T, it is a complement of Ker T; this completes the proof of (i).

(ii) Since .~4 = Ker T | Im T~, the submodule Im T~ is closed. Let us remark tha t Im T~ = hn T*; therefore, it is possible to apply the proof of (i) to the case of the operator T* instead of T, and this gives the orthogonal decomposition i v = Ker T* | Im T. []

C o r o l l a r y 2.3.4. If P E End~(Ad) is an idempotent, then its image I m P is an orthogonally cornple-

mented submodule in Ad. []

Co ro l l a r y 2.3.5. Let A4 and i v be Hilbert A-modules and F : 34 ~ i v be a topologically inject•

A-homomorphism (i.e., ]]Fx]] > 5 ]]xI] for some 5 > 0 and for all x E M) admitting an adjoint operator. The., F ( M ) F ( M ) • = iv .

Co ro l l a r y 2.3.6. Let A4 be a Hilbert A-module and J : AA --+ A/I be a self-adjoint topologically injective A-homomorphism. Then J is an isomorphism,. []

L e r n m a 2.3.7 ([46]). Let A4 be a finitely generated Hilbert submodule in a HiIbert module i v over a

unital C*-algebra. Then A4 is an orthogonal direct summand in iv .

Proo f . Let x l , . . . ,x~ E A4 be a finite set of generators. Let us define an operator F : L~(A) --+ i v

by the formula F(e~) = x~, where e~ E L~(A) is the s tandard basis, i = 1 , . . . ,n. It is easy to see that the operator F admits the adjoint F* : Af--~ Ln(A) acting by the formula F*(x) = ((xl, x } , . . . , (x~, x}), where x E .M. By Theorem 2.3.3, the module I m F = 3/i is an orthogonal direct summand. []

L e m m a 2.3.8. Let A be a unital C*-algebra, and let HA = AJ~IV, p: HA --+ AJ be a projection, and N" be a projective module. Then HA = Ad | Ad • if and only if p admits an adjoint.

Proo f . If there exists p*, then ( i - p ) * = l - p * exists, too. Therefore, by Theorem 2.3.3, Ker (1 -p ) = Ad is the image of a self-adjoint projection.

To prove the converse, let us verify, at first, that HA = iv • + A/I • By the Kasparov stabilization theorem, it is possible to assume, wi thout loss of generality, that Af = spanA(el, . . . , e~}, iv • = spanA(e~+l, e~+2, . . . }. Let gi be the images of ei under the projection iV onto A/I•

e l - - f ~ + g ~ , . . . , e ~ - - f n + g ~ , f i E A 4 , g i E 3 / / •

Since the project ion realizes an isomorphism of A-modules i v -~ 34 • the elements g l , . - . , 9 , , are free generators and (gk,g~} > 0a. So, if

f ~ = E f ~ e i , then e ~ - f ~ e k = ~ f ~ e i + g t ~ . i----i i#k

On the other hand,

(gk,g ) >0, i.e., the spect rum is separated from the origin. Then the element 1 - .f2: is invertible in A,

e k - 1 - f ~ . f~e i+gk EJV " • • k - - 1 , . . . , r ~ ,

and, therefore N "• + 34 • = HA. Let x E iv • N 3,t • Since any element y E HA = AAOIV has the form y = r n + n , (x,y} = (x, m} + (x, n} = 0; in particular, (x,x} = 0 and, therefore, x = 0. Thus, HA = IVzoJM• Let us consider the following map:

1 on iv •

q = 0 on f14 •

154

which is a bounded projection since HA = N'2@34 • Let x + y E 34OA/', xl + yl E A/'• • Then

(p(x + y),x~ + y~> = ( x , ~ + y~> = (x, x~>,

Therefore, there exists p* = q. []

2.4. Full H i l b e r t C*-module s

Let A/[ be a Hilbert A-module. We denote by (34, A4} E A the closure of the linear span of the elements of the form (x, x}, x E 34. It is obvious tha t the set (A4, 34} is a closed two-sided involutive ideal in the C*-algebra A.

D e f i n i t i o n 2.4.1. A Hilbert A-module is called ful l if {2M, 34} = A.

It is possible to consider any Hilbert module as a full Hilbert module over the C*-algebra (Ad, f14}. The following statement gives an example showing tha t this is a useful notion.

T h e o r e m 2.4 .2 ([38,20]). Let A be a cr-unital C*-algebra, and let M be a.fuIl Hilbert A-module. Then

(i) there exists a Hilbert A-module A/" such that 19_(M) ~- HA OA/'. I f a C*-algebra A is unital, then there exists a number n and a Hilbert A-module AP such that M | | M = M ~ ~- A | A[' ;

(ii) /f the module 34 is eountably generated, then 12(M) @ HA.

P r o o f . Let us consider the following set:

k

A Ilcll _< 1 , c : Z <~, x~> ,k ~ N,x,~ ~ 34} . i----1

For the proof of the theorem, the following two lemmas will be necessary.

L e m m a 2 .4 .3 ([13]). For any a C A, a > O, and any c > O, there exists c ~ S such that II(1 - e)al[ < 6.

Proof .

Let us put

Since the module Jr4 is full, it is possible to find a finite set of elements yi E Ad such tha t

a k Yi} - ~ <y~, < 6/2. (5)

i=i

k k k

x i = y i ( c ~ + E ( y j , y j } ) - l / 2 , i = l , . . . , k ; c = E ( x ~ : , x ~ } , b = E ( y ~ , y i } . j=l i=i i:I

Then IIcll = II(e ~ + b)-l /2b(c 2 + b)-t/21t < 1; therefore, c E S. Let f ( t ) := c4t2(62 + t ) - < Applying this function to the element b, we obtain the estimate

II/(b)ll = 1164b2( 62 -4- b)-211 = 1162( C2 q- b)-lb262( c2 q- b)-lll = II( 1 - c ) b 2 ( 1 - c)ll ;_< 62/4.

Therefore, l l ( 1 - e)bll < 6/2, and, together with the es t imate (5), it proves the lemma. []

L e m m a 2 .4 .4 ([13]). In the module A4, there exists a sequence (x~), x~ c M , such that the sequence

of partial sums of the series H (xi, x,i} is a countable approximate unit of the algebra A. I f it is unital, i = 1

k

then there exist a finite number k and elements x l , . . . , xa C f14 such that H {xi, xi) = 1. i = 1

155

P r o o f . We shall consider at first the case of a unital C*-algebra. Then, by Lemma 2.4.3, it is possible to k

find an element c ~ S such tha t II1 - c l l < 1/2. Therefore, the element c is invertible and c = ~L] <yj,yj} j = l

k for some k and yj E 34. By pu t t ing xj = yj �9 c -1/2, we obtain ~ <xj, xj} = 1.

j= l

In the case of a C*-algebra without unit, let h C A be a strictly positive element. By induct ion we

shall construct a sequence (cj) in S such that

E < 1; 1 - < ~-~. j = l j = l

1 Under the assumption that By Lemma 2.4.3, we can find an element Cl C S such that II(1 - cl)hll < -~. the elements c l , . . . ,ck have a l ready been found, we can find by Lemma 2.4.3 an element d E S such

tha t

and put

(1 ,~ 1/2 1 \ (6)

Ck4_ 1 = -- Cj E Cj - - j = l

k ) t / 2 k Since 1 - ~ cj _< 1 and d E S, we have Ck+l E S. Since [Idll _< 1 and Ok+ 1 < 1 -- ~ Cj, we have

j=z j=l k+l

cj < 1. Finally, it follows f rom the inequality (6) tha t j = l

1 - E c j h = 1 - E c j ) ( l - d ) 1 - E c j j= i j-----1 j= l

< 1 - E c j j : I -d) 1-Ecj)j=I h i

< 2k+---- T;

this completes a step of the induction. Thus, we obtain that

j = l

as k -+ oc. Since the strictly positive element h generates the whole C*-algebra A [55], the lemma is

proved. []

We continue the proof of the theorem. By Lemma 2.4.4, we can choose a sequence (x~) in the module k

A4 such that the partial sums of ~ (xi, x,z} form an approximate unit in A. Define a map T " A --+ 12(Ad) j = l

by the equality

T(a) : ( x la , . . . , x~a , . . . ) , a E A. (7)

156

o ~

Since the series (Ta, T a ) = y~ a* (xi,xi} a = a*a converges uniformly in A, r(a) E 12(M). Moreover, the i = 1

o o

adjoint operator is well defined, T* 12(34) --+ A, T*(y~) = E {x~, y~} E A for (Yi) E 12(34). Since T*T i = 1

acts identically on A, the opera to r T is an isometry and

12(34) = h a T Q KerT* ~ A |

where A / = Ker T*. This finishes the proof of s ta tement (i) of the theorem for the case of a C*-algebra wi thout unit . In the case of a uni ta l C*-algebra, the previous reasoning can be appl ied word by word if we replace the module 12(A//) by 34k and replace the infinite sequence in (7) by a finite one.

We pass to the proof of s t a t emen t (ii). For this purpose, let us renumber the sequences in the module 12(34) with the help of a bijection N ~ N • N. Then elements of the module 12(34) are realized as sequences (rn~j), rn~j C 34 , and for each i E N the set of sequences (mq) , j E N, is isomorphic to the module 12(34). Such a renumber ing defines an i somorphism 12(34) ~ 12(12(M)). By the i somorphism

12(34) ~ A | A/" we conclude tha t

l (34) = 12(A) 12(H)

Note tha t the Hilbert module 12(A/) is countably generated; therefore,

12(M) ~ 12(A) �9 12(~V) = ~ l~(A)

by the Kasparov stabilization theorem. []

2.5. Dual modules. Self-duality For a Hilbert A-module 3//, let us denote by Ad' the set of all bounded A-linear maps from Ad to A.

The s t ructure of a vector space over the field C is in t roduced by the formula ( )~- f ) (x ) := -~f(x), where E C, f E A4 ~, x E 34. This definit ion seems artificial; however, it is convenient, because it allows us

to define a linear inclusion of the module 34 into 34 ' ( there is also the a l te rnate approach: to define M ' as the set of all anti-linear maps from 34 into A). The formula

( f .a)(x) =a* f (x ) ,

where a E A, introduces the s t ruc ture of a right A-module on 34' . This module is comple te with respect to tile no rm ]]f]] = sup{ llf(x) ]] : ]]x[[ _< 1}. We shall call such modules dual (Banach) modules. Tile elements of the module 34t are called functionals on the Hilbert module 34. Let us note tha t there is

an obvious isometric inclusion 34 C 34 ' which is defined by the formula x ~ (x, .} = ~. Sometimes, if it

will not cause any confusion, we will write (.f, x} instead of f (x) .

D e f i n i t i o n 2.5.1. A Hilbert module 34 is called self-dual if 34 = M ' .

The condit ion of au todua l i ty is very strong. Below, we will see that there exist only a few self-dual modules: each module over a C*-algebra A is self-dual iff A is finite dimensional . Auto-dual Hilbert C*-modules behave exactly as Hilbert spaces. In the same way as in the case of Hilbert spaces, the

following s ta tements can be obviously checked.

P r o p o s i t i o n 2 .5 .2 ([52]). Let M be a self-dual Hilbert A-module, A[ be an arbitraT~y Hilbert A-module, and T : 34 --+ N" be a bounded operator, T E HomA(34,Af) . Then there exists an operator T* : Af --+ 34 such that for all x E 34, y E A/', the equality (x, T'y} = (Tx, y} is valid. []

Corollary 2.5.3. Let ./M be a self-dual Hilbert A-module. Then EndA(34) = End~(Ad) .

P r o p o s i t i o n 2.5.4. Let 34 C N" and 34 be a self-dual Hilbert A-module. Then A/" = 34 | Ad •

P r o o f . Since 34 is self-dual, i : 34 -+ N" is an isometric inclusion admi t t ing an adjoint. Therefore, 34 = i34 has an or thogonal complement by L e m m a 2.3.5. []

157

If A is a unital C*-algebra, then the Hilbert module L~ (A) is obviously self-dual. For an a rb i t ra ry

module this is not true; moreover, the Banach module A4 ~ cannot admit a s t ructure of a Hilbert module

at all. A description of the dual module for the s tandard Hilbert module HA is given by the following:

P r o p o s i t i o n 2.5.5. Consider the set of sequences f = (fi) , fi E A, i E N, such that the norms of N

partial sums ~ f~,f,,; are uniformly bounded. I f A is a unital C*-algebra, then this set coincides with

HA, the action of f on elements of the module HA is defined by the formula

f ( x ) = ~ f ; x ~ , (8) i=1

where x = (xi) E HA, and the norm of f is defined by the equality

Itfll: = sup . (9)

Let f E H~, e~ be the s tandard basis in HA. Let us put fi = (f(e~))*. We shall show tha t the Proof . sequence (f i) determines a functional f in a unique way. Assume tha t there exists a functional g such tha t

g 7/= f , g(ei) = f (e i) . Let us choose x E HA such that IIf(x) - g(x)l I = C ~ O. Denote by x (N) the image of x under the projection onto the submodule LN(A) C HA, X (N) N = }-~i=l eixi = ( x l , . . . ,XN, O,. . . ) . Let

i = N + I C = x*x Ill/2 < 2(lIfll § Ilgll)

us find a number N such tha t

x - x (x)

Since f ( x (x)) = g(x(N)), we have Ilf(x - x (N)) - g(x - x(N))l] = C. But, on the other hand, one has

C f ( x - x (N)) - g(x - x (N)) << (ll.fll + Ilgll) x - x (N) < (llfll + Ilgll) 2(ilfll + Iigll) -< c / 2 .

The contradict ion obtained shows tha t f = 9. The Cauchy-Bunyakovski i inequality

f [ x i < * x~xi i = i i : l i : i

(10)

IIfll 2 _ supN ~-~{=1 f?f{ " (n)

~ i f ? f i Ill/~-, then the equali ty is reached in (10). Let us put f(N) = ( f i ,

shows that

Note that if we take x.i = .fi/

�9 . . , fN, O, . . . ) , f(N) E LN(A) ' ~ LN(A) . It is obvious tha t

Ilfll > f(N) . (12)

But tt.f(N)tt 2 = i=lf,*fi ; there~bre, (9) follows from (11) and (t2). The convergence of the series (8)

follows from the fact that for any e > 0 it is possible to find a number N such that for all n > 0 the

following es t imate holds:

f~ ~ <_ *.fi x~xi <[Ifl] 2 x*x~ <ll f l l2c. [] i = N i = N i = N i = N

158

N

Note that for the functional f = (~e) from Example 2.1.2, the partial sums ~ ~ '~e are uniformly i = 1

bounded; however,-the appropriate series is not convergent.

Let us describe an interesting example of a dual module.

E x a m p l e 2.5.6 ([23]). Let A = B(H) be the algebra of all bounded operators on a separable Hilbert space H. Let us consider pairwise orthogonal projections pi ~ A, i ~ N, such that the series }--~,pi

i converges w*-weakly to 1 E A, and each projection pi is equivalent to 1. We can consider H = @ H,i as

an orthogonal sum of Hilbert spaces isomorphic to H, ui : H --+ Hi being isometrics, so that

P i = u i u i

As was shown above (see Proposition 2.5.5),

12(A)' = {{ai} ai E A , i E N,

U i ~ti.

N E N

is an A-Hilbert module with respect to the inner product

({ai}, {be}} := w*-lim S aib*. i

The maps

S : A--+ 12(A)', S : a ~ a. {ui},

S--1: /2(A)' --+ A, S - 1 : {ae} ~ w* - lira E aiu~ i

define an isometric isomorphism between A and 12(A)'.

Let ~ be a positive linear functional on A. If 54 is a Hilbert A-module and if N~ = {x E A d : p((x, x}) = 0} is its linear subspace, then A/t/N~ is a pre-Hilbert space with the inner product (-,-)~ given by the formula

. , y e M . We denote the norm defined by this scalar product by II'II~, and we denote the Hilbert space obtained by completion of .A4/N,~ with respect to this norm by H~. Let f E A4'. In accordance with Proposition 2.1.4, we have for all x E Ad

f ( x ) * f ( x ) < Ilfll 2 (x,x} ;

therefore, if x E N~, then

Hence, the map

g~(f(x)* f ( x ) ) = 0 = 9~(f(x)).

x § N~ ~ p ( f ( x ) ) (13)

defines a linear functional on Ad/N~. Since

I~(f(x))l _< I1~111/2 qz(f(x)*f(x)) 1/2 <_ [l~ll 1/2 II.fH g)((x,x}) ~/2= II~N 1/2 Ilfl[ Itx § N~II~,

the functional (13) is bounded. Then there exists a unique vector .f~ E H~ such that [I.f~l]~, < Ilfll II~l] 1/2 and (f~, x § N~)~ = ~ ( f ( x ) ) for all x E 34. For x E M , we shall denote by ~ the functional (x, .} E 3//'. Let us note that ff~ = y + Nv for all y E 3//.

Let ga be another positive linear functional on A such that ~b _< p. Then N~ C iV, and the natural map x § N~ ~+ x + N~ can be extended to the map

IIV #ll -< 1.

159

It is easy to see tha t V ~ , ~ ( ~ ) = ~ . The same holds for all functionals on Ad t.

P r o p o s i t i o n 2.5.7. Let 34 be a Hilbert A-module, ~ and ~ be positive linear funetionals on A, and < ~. Then, V~,r = fr for any functional f E 34'.

P r o o f . Let f E A4 t. Since the quotient space M / N ~ is dense in Hv, it is possible to choose a sequence {yn + N , } of the e lements f rom 34/N~ such t ha t ]]Yn + N~ - f~[], --+ O. T h e n

Vv,r = limV~,~(yn + N ~ ) = l~ l (yn + Ne) .

To prove the s ta tement , it is sufficient to show t h a t ~((Yn, x}) -+ ~(.f(x)) for all x E 3,4. But

]~((y~, x } - f (x ) ) l 2 < 11r r (x, yn} - (yn, x} f (x)* - f ( x ) (x, yn) 4- f ( x ) f ( x )* )

-< ]l~[I" ~((y~,x} (X, yn} --(yn,x} f(x)* -- f ( x ) (x,y~} 4- f ( x ) f ( x )* )

for each n ff N. Since

~((yn, x} f (x)*) = (p((yn, x . (f(x))*}) --+ ~ ( f ( x . (f(x))*)) = ~ ( f ( x ) f ( x )* ) ,

it will be sufficient to show tha t ~((yn ,x ) (x , yn} - f ( x ) ( x , yn}) -+ 0. Note t ha t

~((yn, x) (x,y~) - f ( x ) (x, yn)) = ~( (y~ ,x . (x,y~)) - f ( x . (X, yn)))

= (y~ + N~ - f ~ , x . (x,y~) + N~)~,

and the sequence {x. (X, yn} 4- N~} is norm b o u n d e d with respect to the n o r m ]l-II~- Indeed,

Ilx. (x, yn) 4- N~ll~ = ~((x" (x, yn)), (x" (x, yn))) = ~((yn,x) (x,x) (x,y~))

I]XI[ 2 *~((Yn, 2C) (X, Yn)) ~ ]]X]] 2 "~(I[XI] 2 " (Yn,Yn)): I[XI] 4" ]]Yn 4- Xqoll 2,

and the sequence {Yn + N~} is bounded. Since IlYn 4- N~ - ~ o, the s t a t emen t is proved. []

2.6. C*-Eredholm operators. Index The material of this section is taken mainly f rom [48]. Let us first recall the definition of K-groups.

D e f i n i t i o n 2.6.1 ([31, Sec. II.1]). Let M be an Abelian monoid. Let us consider the direct product M • M and its quot ient mono id wi th respect to the following equivalence relation:

(re, n) ~ (m' ,n ') r 3p, q " (re, n) 4- (p,p) = (m',n') 4- (q,q).

This quotient monoid is a group, denoted by S(M) , and is called the symmetrization of M. Let us consider now the addi t ive category ~P(A) of project ive modules over a uni ta l C*-algebra A, and let us denote by [E] the i somorphism class of an object 34 from 7)(A); then the set 6)(7)(A)) of these classes has the structure of an Abel ian monoid with respect to the operat ion [34] 4- [N~ = [34 | In this case, the group S(~(~P(A))) is deno ted by K(A) or Ko(A) and is called the K - g r o w of A or the Grothendieek grow of the category 7~(A). If A has no unit, t hen the natural map A + -+ C induces a map of K-groups, and we put

Ko(A) := Ker(K0(A +) -+ K0(C)) .

The groups Kn(A) := Ko(A | C0(Rn)) for na tura l n appear to be 2-periodic in n, and the definition can be extended i b r n ~ Z by periodicity.

For unital algebras, we could use classes of stable (after adding the direct summand) homotopy project ions in A n ins tead of classes of isomorphic projective modules. More precisely, the project ions p �9 A n -+ A ~ and q : A m -+ A "~ are equivalent if there can be found m t and n ~ such t ha t n 4- n ~ = m + m / = s and the project ions

A ~ = A ~ o A n' ~ A n | n ' = A s , A ~ = A ' ' | "/ A ~ | " ~ ' = A ~

160

can be connected by a norm-continuous homotopy in the set of projectors from End(AQ = End*(A ~) (A unital!). It is possible to consider also equivalence classes of projections in the algebraic sense. The details can be found in [9, 31, 50, 72].

Let us recall the following well known assertion.

L e m m a 2.6.2 (cf. [59, Theorem 4.15]). The set of epimorphisms is open in the space of bounded linear

maps of a Banach space E to a Banach space G.

L e m m a 2.6.3 ([48, 1.4]). Let A/" be a finitely generated A-module over a unital A and a l , . . . a ~ be its !

generators. Then there exists a number z > 0 such that if for some elements a~ , . . , a~ E A f the inequalities

below hold,

liar-a ll < c, 1 , . . . s ,

then the elements a l~ , . . . a~' also generate N'.

Proof . The map

f : L~(A) --+ A[,

is an epimorphism. Hence, by Lemma 2.6.2, epimorphism. Let

g: L~(A) --+ AF,

Then for anyc~ = (c~l , . . . , as ) E Ls(A) with

8

a i -ai)o~i < sa.

Hence g is an epimorphism and the elements Q , . . . a~s generate N'.

( 0 , . . . , 0 , 1 ,0 , . . . , 0 ) ~ ai /

there exists e > 0 such that if IIg - f l l < sc, then g is an

( 0 , . . . , 0 , 1 , 0 , . . . , 0 ) ~+ a~. i

the norm II ll-< 1

[ ]

In this section, we assume that the algebra A is unital. Let us recall the definition of a Fredholm operator [48].

De f in i t i on 2.6.4. A bounded A-operator F : HA --~ HA is called Fredholm if

(i) the operator F admits an adjoint; (ii) there exist decompositions of the domain HA = 341OAfl and range HA = M20N'2 (here Adl,

M2, A/l, A/2 are closed A-modules, A/1,Af2 have a finite nmnber of generators), such that the

(F001 0 ) with respect to these decompositions, operator F has the following matrix form: F = F2

where F1 : A41 -+ .A42 is an isomorphism.

T h e o r e m 2.6.5 ([48]). Let HA ~ M~)A;, where M and A[ are closed A-modules and N" has a finite

number of generators a l , . . . , as. Then 32 is a pwjeetive A-module of finite type.

Proof . By Lemma 2.6.3, there exists c > 0 such that if !

I l a ; - a k l l < c , k = l , . . . , s ,

then {a~} generates Y . Let P �9 HA --+ Y be the projection along the summand A4 in the module HA. Then P is a bounded A-operator. Therefore, there exists 5 > 0 such that if Ilxll < 5, then I l r z l l < e. Let us choose a number no such that

Ila - kll < 5, k = 1 , . . . , s,

where g~ is the projection of ak onto Lno along L • Let us represent ak in correspondence with the n 0 �9

decomposition HA = 2V/OA/as ! I f ! I I

~k = ak + ak, a~ E./~, a k E .M.

161

Then a k - a' k = P ( a k - ak). Therefore, I la~- a~tt < c and {a~} generates A/. Let ~ be tile module

generated by {g~}. Then HA is equal to the sum Ad + ~ . Indeed, if x E HA, then

x Z -~ -- A ak) + Z "~kgk"

Let us consider the bounded A-operator Q, the project ion onto L~ o along L • Then y t 0 "

Q(a~) = ~k, Q ( H ) = IV,

' P ( ~ ) = H . P(~k) = ak,

' the composition of A-operators Since ak are close to ak,

h ; 2 ~ P N

is an A-isomorphism. Therefore, Q �9 N" -+ ~ and P " H --+ Af are A-isomorphisms. In part iular , if 8 8

,kka~ = 0 hotds, then ~ Akg~ = 0. Therefore, M N Af = 0, i.e., HA = .M@~. It is clear tha t A / i s k = l k = l

a closed A-submodule in HA and

L~ o = ( M n L ~ o ) O ~ .

Indeed, (AA NL~o) hA/" = 0; on the other hand, if z E L~ o then x = x' + x " , x' E N', x" E Ad. Since

A / c L~o, we have x" ~ Lno, i.e., x" E M N L ~ o. Thus, N" is isomorphic to a direct summand in the free

finitely generated A-modute Lno. []

T h e o r e m 2.6.6 ([67]). In the decomposition introduced in the definition of a Fredholm operator (see 2.6.4), it is always possible to assume that the modules Ado and 2t41 are orthogonally complemented. More

precisely, there exist decompositions for F,

0 F4 : HA = VoO~Vo -+ VI@W1 = HA,

such that Vo ~ | Vo = HA, V1 • @ gl ~- HA, or (which is the same by Lemma 2.3.8) that projections VoGWo --+ Vo and V10W1 -+ 171 admit an adjoint.

P r o o f . Let W0 = A/0, Vo = Wd L. The orthogonal complement exists by Theorem 2.3.7. The restr ict ion

F[w d_ is an isomorphism. Indeed, if xn C I/V~, then let Xn = x'~ + x~, Xr~ E Ado, x~. E I4/o, IIx~ll = 1.

Let us assume that IIFxnll --+ 0; then IIFx~ + Fx~_[[ --+ 0; since Fx~ E Adl , Fx~ E N't, AdlGN'I = HA, this means tha t IIFx~ll -~ 0 and IIFz~II -+ 0. The opera tor F1 is an isolnorphism; therefore, IIx~ll -~ 0.

If a l , . . . , a~ are generators for Wo = Ho, then 0 = (xn, aj) = (x~, aj} + (x~, aN},

II(x~,aj)H = II(x~,aj}ll <_ Iix~{ll Ilajl I --+ o, n--+ oc, j = 1 , . . . , s .

Since x~ ~ No, we have x~ ~ 0, n ~ oo, xn = x~ + x~ --+ 0; this contradicts IIx~ll = 1. Hence Flwo~ is

an isomorphism.

Let V~ = F(Vo). Since Wo = A/o, it is possible to assume that W~ = N'~. Indeed, any y ~ HA has

the form y = mz +n~ = F(mo) +n~, where 'rn~ ~ JM~, n~ ~ N'~, mo ~ 3//0. In turn, m o = vo + n o , where

Vo ~ Vo, no ~ Wo = A/o, and

y = F(~o + no) + n~ = F(Vo) + (v ( , ,0 ) + ,,,,~) ~ v~ + H~.

Thus, HA = V~ + W~. Let y ~ 171NWt = Vz NN'~, i.e., n t = y = F(vo), n~ ~ N't, vo ~ Vo. Let us decompose vo = m o + n o ,

where mo ~ Ado, no ~ No. Then

r,,~ = F(mo) + F(no), F(mo) = n~ - F(no) , F(mo) e A4~, rh - F(no) e N'~,

162

whence we obtain F(mo) = O, nl - F ( n o ) = O. Since F : Ado ~ A41, we have mo = O. Further,

vo E Vo = N'o l and therefore

0 = <~o, ~o> = <-~o + ~o, , ,o> = <no, ~o>, ~o = 0.

Thus, Vo = mo + no = 0, y = F(vo) = 0. Hence V1 n W1 = 0 and HA = VieW1. By Corollary 2.3.5, the module V1 has the orthogonal complement V~, V1 | V~ = HA, and this

completes the proof. []

R e m a r k 2.6.7. If we do not require tha t the operator F have an adjoint, then it is possible to state

tha t there exists a decomposit ion F : A/'~ @)No --+ 3 4 ~ L n , where Ln = spanA(e~, . . . , en), but 3d~ does

not necessarily have an orthogonal complement. This result was obtained in [27].

D e f i n i t i o n 2.6.8. Let the conditions of Definition 2.6.4 hold. By Theorem 2.6.5, A/1 and A/'2 are

projective A-modules, and we can define the index:

index F = [Al l ] - [Af2] E K ( A ) .

T h e o r e m 2.6.9. The index is well defined.

P r o o f . It is necessary to check tha t the index does not depend on the decompositions of range

and domain involved in Definition 2.6.4. Let Pm be the project ion onto L,~ along L~. Let F be an A-

Fredholm operator, HA = AdlO.N'I be a decomposition of the domain, HA = . /~2(~J~2 be a decomposition

of the range, 0) 0 F4 '

where F1 : A4~ --+ A/t2 is an isomorphism. According to the proofs of Theorems 2.6.5 and 2.6.6, it is

possible to assume tha t

A;1 c Ln, L~ = 2 r Ad~ = P~ ~ L~,

where P~ is a projective finitely generated A-module. Let another decomposit ion of a domain and range

be given: H A I ~ t l ~ t = A42| .Ad l O . ~ ~ , H A =

Then there exists m > n such that

where P[ is a projective finitely generated A-module. This exact ly follows from the proof of Theo-

rein 2.6.5. Let us show that there exists m > n such that if

LL = F ( L ~ ) + H~, and

F ( L • ~ then is the projection on L~n along L~, = ~ m J, I

L~ = "]P2OQm (.Af2) ,

Q~. : H A -+ H A

Qq(H2) - H2,

where P2 is a projective finitely generated A-module, and t~ ! ! ! ! L~ = P~| Q,~(Af~) =Af~,~ '

where 79~ is a projective finitely generated A-module. Indeed, HA = L ~ | m. If a l , . . . , ak are generators

of the module N'2, then I I ! l I H I !

a j = a j + a j , a j E L,~,, aj E L~, j = 1 , . . . , k.

1This project ion is well defined, since L ~ C -/~1 for rn ~ n and hence FIL~ m is an isomorphism; whence L ~ ~- HA is a t - - I t closed A-module, L~n N L ~ = 0, L ~ + L ~ = HA; therefore, HA = Lm•L m is a direct sum of closed A-modules and O~m is

a bounded A-operator.

163

• is a pro jec t ion of x onto " = F(xam), where x is arbitrary, x m For m --+ co, we have Ilayll-+ 0, since aj L~, and IIx.~ll-+ 0 for .~ -+ ~ . Then, for large enough m, we have

Q ~ ( H 2 ) = ;V~ L~ = (L~ AA.42)| ' "

( the proof of this fact repeats the proof of T h e o r e m 2.6.5). Similarly,

L~ (L~ ' - ' ' ~ ' = n M J e Q ~ ( H ~ ) , Q ~ ( H I ) = H~.

A/10~l , where ~ l is a finitely genera ted projective A-module. F rom the Since m >_ n, we have L.~

equalit ies

F(Pl) = F ( L . ~ M I ) = P2, 7 ) 1 C M 1 ,

we obtain tha t F : P~ ~ 722, and it follows f rom relations F(79~) = P~, P~ C A41 tha t F : P~ - P~.

Therefore, we have the following equalities in K(A) :

[ H I ] - ] - [ ~ 1 [ = [.IV';]--~ ['P~] = [ L m ] , [ ~ l ] : [~)2],

[&] + [P~] [H~] + [v;] ' ' = = [L~], [ / ) j = [7)~].

Thus , [ A / J - [ H 2 ] = [Af t ] - [A/~], and we have proved that the index is well defined. []

L e m m a 2.6.10. Let an operator F : HA --+ HA be A-Fredholm. Then there exists a number c > 0 such that any bounded A-operator D satisfying the condition. [IF - DI[ < z and admitting an adjoint is an A-Fredholm operator and index D = index F .

P r o o L By the definit ion of the Fredholm property ,

H A = M i C A / i , HA=M2Q.A[2, F = ( F 1 0 ) 0 F 4 '

F1 : M i -~ M2. T h e n ]]F~II < ][F[]; moreover, if D : HA -+ Ha is an arbi t rary bounded A-operator , then

D = ( D 1 D2) D3 D4 '

and there exists a constant C such tha t IID~II _< CIIDI! (cf. [48, p. 842]). Therefore, if D is an arbi t rary A-opera tor satisfying the est imate IIF-Dll < c, t hen [IF1- Dlll< C .c. Since FI is an i somorphism, we can find 5 > 0 such tha t if I]F1 - D1 [[ < 6 and D1 is an A-operator , then D1 is also an A-isomorphism. By pu t t ing c = 6/C, we obtain that for the opera to r

D = Da D4

the element D1 is an isomorphism. Then

U2DUI = ( Di 0 ) 0 D 4 - D 3 D i i D 2 '

where

U2 = _ D 3 D i 1 : (M2~A;2) ~ (A//25A;2),

i : ( : ~ l ~ J v l ) -+ ( 2 d l ~ H 1 )

are A-isomorphisms. W i t h the help of U1 and U2, we obtain a new decomposi t ion of the domain and range into direct sums:

Ha = M~H~,'- ' MI' = U~(M1), N : = U~(H1), H A I - I t = M2eH~, M2 = W~(M2) , x~ = W~(X2).

164

W i t h respect to the new decomposit ion, the mat r ix of the operator D is equal to U2DUi. Thus, the

opera to r D is Fredholm wi th index

[Ui (A/ i ) ] - [U2-1(N'2)] = !A/ i ] - [Y2] = i ndexF . []

L e m m a 2 .6 .11 . Let F and D be A-Fredholm operators,

[z : HA -+ HA, D : HA -+ HA.

Then D F : HA -+ HA i8 an A-Fredholm operator and index D F = index D + index F .

P r o o f . Let us consider for F and D decomposi t ions from the definit ion

HA = M 1 O N ' I F> 3,42~N'2 ~- HA,

HA t ~ r t ~ f ~ = M2@Af ~ = HA,

where 0) 0) 0 F4 ' 0 D4 '

F i and Di are isomorphisms, and A/i, A/2, A/~, A/~ are projective finitely generated A-modules. As earlier, w i thou t loss of generali ty it is possible to assume tha t

H_9 C L~, L~ = A/207 ), M 2 = L~ �9 T'.

Moreover, since F1 " Adl -+ 342 is an isomorphism, it is possible to change decomposi t ions into direct sums by pu t t i ng

~ 1 = F-I[L• \ n ] , ~ 1 = FFI(~[))@-]~I, -~2 = L~, ~ 2 = Ln.

Thus, a number n can be chosen as large as necessary. Let us choose n in such a way that

P'~" ' H " P' ' L~ = ~Pn[ i), = Ad z A Ln, pn(N'~) = N'I,

where, as earlier, Pn "HA --+ HA is the projec t ion on Ln along L~. T h e n

Let us pu t M 2 = L~, T 2 = P@A/~. W i t h respect to the new decomposi t ion, HA = ~ 2 O T 2 , the ma t r ix of the opera tor F has the form

and F1 is an isomorphism. Then

1 --.Fl-- 1/V2

D e n o t i n g b y U t h e m a t r i x ( ~ - F l l F 2 ) - - - - 1 , pu t M i = U(3d i )

decompos i t ion of the space, HA = A 4 i O ~ i , and the mat r ix F

has the former diagonal form. Let us consider the project ion

T : HA = M2oT) 'GA/~ --~ M 2 | 9'.

0 ) 0 F4 "

, A/1 = U(A/i) . We have obta ined a new

for the decomposi t ions

= HA

Since HA ~ AJ~)A[~, the restriction T ] ~ i : AJ[ --~ 3 , t2 | is an isomorphism. Let us consider the mat r ix D with respect to the decomposi t ion

HA = (- 25P')SH; D> Ad2 N = HA.

165

( D 1 0 ) , where is Put This matrix has the form D = Da D4 D1 a n isomorphism.

0) V D : = -D3D~ 1 D3 D4 = 0 D4 "

Therefore, it is possible to change the decomposition in the range

2| .M 2 H A 3 4 ' ~ - - ' ' V ( 3 4 ; ) , - - ' = = H ~ = v c ( ; ) ,

in such a way that the matrix of the operator D with respect to the new decomposition

HA = 3 4 ' ---'2~H2, 34 ~' = V ( M ; ) , X:; = V(H~),

has the diagonal form. Let us change the decomposition in the range once again:

3 4 ; = n ( j ~ ) , Jv~ = n ( ~ ' ) 5 ~ ; .

The matrix D with respect to the new decomposition

= M 2 0 . / ~ 2 = HA

has the diagonal form. Then the composition DF with respect to the decomposition

HA = ~ l g ~ ~ M ' ~ ' 2 @ ~ 2 = HA

has the form D F = ( (DF)I 0 ) 0 (DF)4 ' and (DF)~ is an isomorphism. Taking into account the fact

that End* HA is a C*-algebra, we conclude that DF is an A-Fredholm operator and

index F = [~1 ] -- [~2], index D = [~2] - [A/'2],='

index DF [~1] = ' = - W ~ ] .

Hence, we obtain that index DF = index D + index F. []

L e m m a 2.6.12. Let K : HA --+ HA be a compact operator. Then 1 + K is an A-Fredholm operator and index(1 + K) = 0.

P roof . It is obvious that 1 + K adnfits an adjoint. Let us choose a number n such that the inequality

IIKtL~n ]l < 1 is fulfilled. With respect to the decomposition HA = L~n | L~, we have the following matrix presentation:

(K1 K2) ( I _ t _ K ) = ( I + K 1 K2 ) K = / (3 /s ' / (3 1 + / - ( 4 "

By the estimate IIKIL~II < 1, the operator 1 + K~ is invertible; hence, as earlier, there exist invertible operators U1 and U2 such that

U~(I + K)UI = ( 1 + K1 0 ) - 0 (1 + K ~ ) - K a ( 1 + K~)-~K2

With respect to the new decomposition HA = All@A/'1 + M2~N'2 = HA, where A41 = UI(L~:), A/', = Ua(L,,), A42 = U~I(L~), & = U~S(Ln), the operator (1 + K) has the diagonal form and, therefore, is an A-Fredholm operator and

index(1 + K) = [UI(Ln)] - [U2-1(L~)] = 0. []

L e m m a 2.6.13. Let us consider an A-Fredholm operator F : H A --+ H A and let K E ]~A. Then the operator F + K is A-Fredhohn and index(F + K) = index F.

166

Proof .

the diagonal form:

HA = Ad I OA[1

Without loss of generality, we can assume that

Let us consider decompositions of the space HA in direct sunls such that the matrix F has

F> .Ad20.A/'2 = HA.

rn M15Pl, M1 l = = Ln |

where 791 is a finitely generated closed A-module. Let us choose a large enough number n in such a way that I[K[L~ I[ < I]FI-~I[ -1. Let us consider the new decomposition of the space HA:

A41 = L~, f i x = L,,, ~ 2 = FL~, N2 = F(79~)QN'2.

Let F = ( F10 F40 ) a n d K = (K1/(3 I(4K2) be matrices ~ F and K with respect t ~ 1 7 6 1 7 6 1 7 6

HA = AJ1Offl --+ Ad2Qff2 = HA. Then

F + K = ( FI + K1 K2 ) K3 F4 -t- K4 '

and the operator F1 + K1 is invertible. By repeating the construction of Lemma 2.6.10 (on operators close to a Fredhohn operator), we obtain

index(F + K) = I T 1 ] - [Y2] = [Ln] - [ F ( P l ) + N'2] = [N'I] + [791]- [791]- [Af2] = index F. []

T h e o r e m 2.6.14. Let

F : HA --+ HA, D : HA ~ HA, D' : HA -+ HA

be bounded A-operators admitting an adjoint and

F D = IdHA +K1, D'F = IdHA +K2, K1, K2 E ]C(HA).

Then F is an A-Fredholm operator.

Proof . Let us consider a decomposition HA for which the operator FD = 1Ha + K1 has the diagonal form (Lemma 2.6.12),

HA = J~15H1 I+KI Ad2@H2 = HA,

and the decomposition of the space HA satisfies the conditions of Theorem 2.6.6. Let us consider the projection

P" HA = Ad25M2 -+ N'2.

It is a compact operator, P E End* HA. The image of the operator (1 - P)(1 + K~) = (1 - P ) F D is exactly equal to AJ2. It is easy to see that up to an isomorphism

( 1 - P ) F D = ( 1 - P ) ( I + K 1 ) = I + ( - P ( I + K 1 ) ) + K I = I + K 1 ,

D ' ( 1 - P ) F = D ' F - D ' P F = 1 + K2,

where K1 E ]CA, /(2 ff ]CA- By Lemma 2.6.13, it is possible to assume without loss of generality that F : HA -+ .Ad2 is an epimorphism. Otherwise, we will pass to the operator (1 - P)F. Let us consider now the decomposition for 1 + K2:

HA = M 1 @ ~ 1 F> ./~2~.~2 D'> ./~2@ff2 = HA.

The composition D ' F [ ~ " Adz -+ Ad2 is an isomorphism. Therefore, since F : HA --+ .A,'t2 is an epimorphism, F " Ad~OTI --+ Ad~ maps Ad~ isomorphically in Ad~ and K e r F C ~ , Ad2 = F(Ad~) + F(ff~). Let us show that F(Ad~) A F(H~) = 0. For this purpose, decompose F into a composition:

Ad~SY~ -~ ( ~ l @ Y ~ ) / K e r F = A d l | ~> Ad2,

167

where F is an isomorphism. Therefore,

A/[2 = F(AJ ~ ) | / Ker F) = F ( ~ ) O F ( f f ~ ) .

Since the A-module H1 is finitely generated, F ( f f l ) is finitely generated, too. We have obtained the decomposition

HA = ~ l ~ f f z --+ F(~I)5[F(~I)SA/2] = HA,

w h e r e F [ ~ 1 : J%41 --+ F(J%41) is an isomorphism. []

L e m m a 2.6.15. If bounded A-operators D, D', and F admitting an adjoint are such that F D and D ' F are A-Fredholm operators, then F is an A-Fredholm operator.

Proof . By the definition of the Fredholm property of F D and D'F, we can find operators T and T' admitting an adjoint such that

(FD)T = 1 + K, T ' (D'F) = 1 + K'.

By Theorem 2.6.14, the operator F is Fredholm. For T, for example, it is possible to take an operator

with the matrix ( ( F j - ~ 0 0 ) w h e r e F D h a s t h e f o r m ( F I O ) 0 ' 0 F2 in the sense of Definition 2.6.4. []

R e m a r k 2.6.16. For applications to elliptic operators, it is important to develop the theory for spaces of the form 12(P). This can be done similarly (see [61]).

2.7. R e p r e s e n t a t i o n s of g r o u p s o n H i l b e r t m o d u l e s

In this section, we assume that G denotes a compact group. First of all, we prove an equivariant variant of the Kasparov stabilization theorem. Let us follow here the original proof [34]. For closely related problems, see also [45].

D e f i n i t i o n 2.7.1. For a C*-algebra B, put OO

i=1

where {V~} is a countable set of finite-dimensional spaces, in which all irreducible unitary representations G are realized (up to isomorphism) and each representation is repeated an infinite number of times; the B-Hilbert completion of the algebraic sum is carried out with respect to the norm given by the following B-inner product on summands:

(xl | bl, x2 | b2) := (xl, X2}y~" bCb2, xl, x2 E V~.

T h e o r e m 2.7.2 ([34]). Let B be a C*-algebra with a continuous action of a group G and g be a countably generated Hilbert G-B-module. The action is assumed to be unitary and agrees with the module structure in the sense that

g(xb) = g(x)g(b), {g(z),g(y)) = g((x,y)), x , y e g, b e B, g e G.

Then there exists an equivariant B-isomorphism preserving the inner product,

g | ~iB ~- 7-tB.

Proof. Let us denote bv E + the module E considered as a B+-module. Assmne that the action of G

on B + is extended from B by the formula g(1) = 1. Assume that we know how to prove the theorem

for unital algebras, so that

s | 7-/B+ = 7/B+,

w h e n c e

g ~ ~ B ~ ((g | ~tB)+)B ~ (g+ | ~/G+)B ~ (7-tB+)B = 7-tB.

168

Thus , we can restr ict ourselves to the case of uni ta l B.

Let {xk} be a countable system of generators of g and {e~} be an or thonormal basis of 7-/B, and

each ek = vk | l u , where vk E V~(k). In o ther words, if {wk} is the union of some o r thonorma l bases of

all V~, then ek = w~ | lB. Let {y~} be a sys tem of elements in g | 7 /u in which each e lement of the form x ~ | and 0 | is repeated an infinite mlmber of times. We can suppose tha t yl = 0 @ e l and put W1 = 0 | V1 | B. Let us assume tha t by induc t ion we have already constructed subspaces W 1 , . . . , Wn satisfying the following conditions:

(i) W,i is a C-finite dimensional G-invariant subspace in g | 7-/8; z 1 (ii) each Wi has a basis ( i , . . . , z ~ (i)) = ( f l , . . . , f p ) such tha t

<z j , z j> : 1B, <z~,z~ s>

(iii) there exists m = m(n) such tha t

= 0 for i # r or j # s ;

WI + " " + W~ c $,~ := $ |

and consequent ly (W1 + ' " + Wn)B C gin, (iv) the distance between Yn and (W1 + ' " + Wn)B does not exceed 1/n.

Note tha t from (i) and (ii) it follows tha t the modules WeB are pairwise or thogonal and G-invariant , as well as (W1 +" �9 �9 + Wn)B. The last module is free, so by L e m m a 2.3.7, it has an or thogonal complement which is G-invariant due to the uni tar i ty of the action.

Let us pass to the construct ion of Wn+l . P u t

P '

Yn+l := fy (fy,Yn+ll " , Y n + l = Y n + l - - Yn+l" j = l

T h e n for any w E (W1 + " " + Wn)B,

<w,Y~'~+l)= fobj,y~+t-- ~-~fj<fj ,yn+l} = ~b;[<.fy,yn+a}-<,f?,y~+l)]=O. j = l j = l / j = l

Since by the definit ion of the sequence yj the element Y~+I lies either in g or in some V/ | B, we have II Y~+I E s for some m' > rn. Let us consider the or thogonal complement S~,m, for (W1 + . . . + W~)B in

g,~,. It is an invariant module and y~+~ E S~,,~,. By the Mostow theorem on periodic vectors [49], the e lements with C-finite dimensional orbits are dense in Sn,~,. Hence one can find a vector z E S~,,~, such

1 t h a t I[z- Yn+l]l" < ----2'2n + and R := Gz is an invariant f inite-dimensional subspace of S~,,~,. Since z is a

total izing vector, R is an irreducible G-module . Therefore, there exists m" > m' such tha t an equivariant i somorphism F : R ~ Vm,, exists. Let { h , , . . . , h~} be an o r thonormal basis of V,~,, and r~ := i = 1 , . . . , k. Then for the corresponding irreducible matr ix representat ion T : G ~ U(k) we have

k k

g(h ) = Z rj(g)hj, = Z a. j = l j----1

Since R C s and m " > m ' , R is o r thogonal to Vm,,. More precisely, each element of R is or thogonal to V~,, | B in g | 7-lB. Hence <ri, hi) = 0 for all i and j . Let

: -- , ' i ~ , ~ ~ c , : - - + 1, r i := ri + (hi | 1B)" (2n + 2)a" i i i=i

169

Then <r~,rj} : {r{,rj) + {(2n + 2)a}-26id and the matrix L : : II <r; ~j> ' , ' Ih , j :~ is positive and invertible k

.II I in Mk(C) C Mk(B). Let D = Ildj*ll := L-l~2 E Mk(B) and ri := E rjdji. Let us take Wn+ 1 equal to j = l

" ' which is the same, since D has the complex linear span of vectors r i , i = 1 , . . . , k, or to the span of r{, complex coeMeients. Then W,~+I C g,~,, and

k

<r~',ry} = <rpdpi,rqdpj} = (D*LD)ij = 5~j. p,q=l

Since all hi and r{ are orthogonal to W1 + ' - " + W n , Wn+l is orthogonal to it, too. Further, let F : = L 1/2, k

/ . . ' ! . . so that r i := ~ rj Fj~. Then j----1

g(r~') = 9 r}dji = (gr})dj{ = grj + (ghj | 1B). (2n + 2)ct dji = j = l j = l

) 1 ) j = l s : l

( 1) = E E T f ( g ) rs O ( h s | lB)" (2n+ 2)(~ e j i = E E T ] ( g ) r : d j { j = l s = l j = l s = l

E E " = = T](g) ~r~ dj~ v ' <

j = l s = l t = i j = l =

k Thus, Wn+l is G-invariant. Let us estimate the distance by put t ing z' = ~ r~(h, so that

i : 1

('ri - r~)ch = (hi (9 1B)" (2n + (2n q2 2)a ,iai i=1

(i,o,,O '-

Therefore,

p(yn+~ ( W l + + v G ) B ) < " , - " - p ( y ~ + l , z ) + . ( z , W n + I B ) < - - 1 1 1

2 n + 2 + p(z'z ') < - - < - . - n + l n

Thus, by induction, the C-subspaces Wi with properties (i)-(iv) are defined for any i. From the explicit expression for r~: we obtain that Wn is isomorphic to some V{, i.e., is irreducible. Further, the B-Hilbert completion of Ad (i.e., the closure in g �9 l /B) of the algebraic orthogonal sum of modules WnB gives the whole COt /B . Indeed, by property (iv) the algebraic sum is dense in s 1 7 4 So, M ~ s Let us prove that A,I is isomorphic to 7/B, i.e., tha t each irreducible representation is repeated indefinitely many times among WnB. Let us suppose the opposite; then A4 @ l /B ~ l /B, or

s | I/ B = S | I/ B | I/ B ~-- M | I/ B ~ I/ S. []

170

Let us now prove the theorem on decomposition of representations [47]. Let J~4 be a Hilbert B- module with a strongly continuous unitary representation G,

T : G --+ U(A/t) C EndS(Jr4 ), 9 ~ Tg,

and assume that the group acts trivially on B. Let {Vs} be a complete collection of pairwise nonequivalent unitary representations of G, ds be their dimensions, and D~q be their matrix elements, which are continuous functions on G. For an invariant normalized Haar measure dg on G, we define an operator

8 8 ~ f 8 P;q : .h4 -+ A/t, Ppq(X) := ds Dpq(g)Tg(X) dg. (14) J G

Since, for a fixed x E A4, a product of a continuous complex-valued function with a continuous module- valued function is integrated and since the group is compact, the integral converges to some element A/t. We obtain a bounded operator. Indeed,

IIP;~xll _< d~ f ~ IITg(x)ll dg < d~ sup Ilxll J g C G G

Therefore,

IIP;qll sup g E G

It is well known [8, I, Sec. 7.1, Theorem 5], that

0, s # s ' , / D g y ( g ) D ~ ( g ) = s'. (15) 1

-~s(~irn(~jn, 8 : G

We need the following Peter Weyl theorem.

T h e o r e m 2 . r . a ([8, 1, Sec. 7.2, Theorem 1]). The functions v~TD;k(g ) form a complete orthonormalized system, in L2(G).

L e m m a 2.7.4. The operators Ppq have the following properties:

(i) Pp~ admits an adjoint and

(ii) the following equality holds:

(iii) the following equalities hold:

= GS; (16)

8 8 t 881 8 . PpqPp, ~, = 5 5qp, Ppq,,

d~ s 8 Tg_Pj~ = Dij(g)P#m ,

i=1

ds

E s P];~Tg = Dmi(g ) Pj~. i=1

(17)

(18)

(19)

Proof . First of all, we note that for unitary operators in A/I the mapping F ~+ F* is continuous in the strong operator topology. In other words, for unitary operators the strong continuity implies the ,-strong one. Indeed,

II(F'* - F*)xll = [[(F '-1 - F-1)xll = IIF'(F '-1 - F<)Fzl l = I l F z - F'zll -+ O.

171

Therefore, it is possible to take T~ instead of Tg in (14), and then take it out of the integral sign. More precisely, the first equality in the chain

(Ppq)* = ds ] Dpq(g)Tg (X) dg = d, / Dpq(g-1)Tg(x) d(g-1) = ds / D~p(g)Tg(x) dg = P ~ qp~

G G G

has to be verified at first at the level of integral sums, and passage to the limit is possible due to the indicated ,-strong continuity. The remaining equalities in the chain are obtained by the invariance of the Haar measure and by the relations T~ = Tg -1 = Tg-~. (i) is proved.

It follows from (14) that

s ~' / ' l s, (g,)TgTvdgdg," PpqPp,q, = d~ds, D~q(g) Dp,q, , ] , J

G G

Since TgTg, = Tgg, , by putting ~ := gg', we obtain from

8 ~ 8 $ ~ 8 I D L ( g ) = DL( f-' ) = = ( g )

and relations (15) that

PpqPp,q, dsds, Dqr(g')Dp:q,(g')d9' D~r@)Tyd~=.7 ~ss' 1 x ~ D~ -= �9 = ( 5 (~qp, P ; q , . ~ s l u -~MUqplUrql l pr

G G

To prove (iii), let us note that

TvP~.~(x) = d, f D~m(h)Tgh(x)dh=d, G

ds

= < ZDJ (g i=1

ds ds

/ D~,~(g-lh)Th(x)dh

G

ds d .

dh= Dji(g )ds D,~m(h)Th(x)dh i=J_ =

8 --i 8 : ~Dj~(g )Pi.~(x)= ~D~y(g)R[.~(x). i=1 i=1

Tile second equality of this item can be proved similarly.

L e m m a 2.7.5. The operators P~ := p~s are self-adjoint pairwise orth, ogonal projections.

Proof . If we rewrite the statement of the lemma as

(Pp)* = P; , P; P;, = @p, p,

then the proof can be immediately obtained from (16) and (17).

Let us put L e m m a 2.7 .6 .

ds ds

EP; p .s : ~ z P"

p = t p=l

The operators ps have the following properties:

(p*)* = p~,

P~P*' = 5ss, PS,

Tgp ~ = PSTg.

In other words, ps are self-adjoint invariant pairwise orthogonal projections in A4.

[]

(20) []

(21)

(22)

(2a)

172

Proof . relation, consider the character of the representation V=,

d=

)c=(g) := Z Dpp(g), p--1

which, as the trace, satisfies the relation x=(g) = xS(hgh-t) . One also has

P= = d= f x=(g)T= @, G

T=P = = d=T= I x=(g')Tg' dg'

G

= ds f xS(gg'g-1)Tgg,g-~ dg'Tg = pSTg.

G

L e m m a 2.7.7.

By the definition of P=, formulas (21) and (22) follow from (20) at once. To verify the third

G

Let us define

34= := P=(34), 34" := ( ~ 3 4 = , (24)

where the sum is assumed to be completed either as a Hilbert sum or (which is the same) as a closure in 34 of the algebraic sum. Then

34" = 34. (25)

Proof . Assume that a C-linear functional f on A4 vanishes on 3//" and that x E 3,t is an arbitrary vector. Then, for any set of indices, we have Pgj(x) c AA', so that

= f(Pi~(x)) = d= f DiS(g)f(Tg(x))dg" 0

G

Therefore, by the Peter-Weyl theorem 2.7.3, f(Tg(x)) = 0 holds ahnost everywhere, and by continuity it vanishes everywhere. In particular, f(Te(x)) = f (x) = 0. Hence, by the Hahn-Banach theorem, 34~ =A4. []

T h e o r e m 2.7.8 ([47]). Let M be a Hilbert B-module with a strongly continuous unitary representation G and let the group act trivially on B. Let {V~} be a complete collection ofpairwise nonequivalent unitary representations of G and

Ads := Homa,c(V=, M) C Home(V=, M ) ~ V~ | M

be a Hilbert B-module, so that the B-product is defined by the formula

dim Vs

(~,~) :-- Z {~(h~),~(h;)}~, 14,...,h~=,,~ i~ an orthobasis in v=. i , j = l

Then for the Hilbert sum we have the G-B-isomorphism cx? (x)

= ~ i 8=1

and

r(v~ | M=) = M =, where Ad s is introduced in (24).

173

Proof. Note that F, are algebraically injective. Indeed, let

0 : Fs (khSj~ = ([9 (~"~h~ogj)

d~ Since, by the Schur lemma, p is either an isomorphism or 0, this equality can be true only if ~ h~cg = 0

j = l d~

o r w = 0 . But then ~ h ~ c g | j = l

By Lemma 2.7.7, it is sufficient to prove only that P~ maps bijectively Vs | 548 to 54~.

Note that by put t ing 54~ := P2(54) = Pi}(54), we obtain by relations (t7) that the operators P~

realize isomorphisms

P;: Mj + 54 . ds

Thus, 54~ = @ 54~ is a sum of isomorphic modules. j = l

Let {h~ , . . . , h}~ } be the orthobasis of V~ with respect to which the matrix elements D~5 were defined. Let us define a homomorphism

r ~ | [541] -~ 54< CS(h~ | x) = U~ (x), (26)

where we have taken 54~ in square brackets to emphasize that there is no action of G on it. By the properties of the operators P]l, the map Cs is an isomorphism. Since, by (18),

d~

Tgr | x) = TgP/I (x ) = E OilY (g) P;I (z) i=l

and

=

\ i = 1 i=i

the map (I)s is equivariant. Further, there is a map

9s: 54[ -+ 548, ~8(x)(v) := ~S(v | x).

Then

Fs o (Idv~ |162 | x) = ~S(v | x).

Since we have an isomorphism on the right and since Fs is algebraically injective, F~ is an isomorphism (see Lemma 2.7.10), whence Cs is an isomorphism. In particular, the images of Fs coincide with Ads and are orthogonal to each other. Hence F is topologically injective and its image coincides with 54. []

R e m a r k 2.7.9. Let tile G-A-module 54 belong to the class P(A) of projective finitely generated modules. Then obviously 548 = Homa(Vs, R4) E P(A). Let us show that in the sum (~ only a finite

8

number of summands do not vanish. Let us denote by a l , . �9 as generators of 54. Let us choose by the Mostow lemma [49] C-periodic vectors b l , . . . , bs so close to a l , . . . , a 8 that they generate 54 as an A- module (see Lemma 2.6.3). Decomposing the linear span of the orbits Gbj, which is a finite-dimensional G-C-module, into irreducible modules, let us choose a new system of generators c 1 , . . . , CN from these G-C-modules. From here it is evident that the nmnber of nonzero summands does not exceed N.

L e m m a 2.7.10. Let F : L --+ M and T : N --+ L be continuous maps of Banach spaces, S = F T be an isomorphism, and Ker F = 0. Then F is an isomorphism.

174

P r o o f . Since S is an isomorphism and F is bounded, T is topologically injective and its image T(N) is closed in L. Assume that it does not coincide with L. Let us choose a vector 0 r x E L \ T(N). Then

0 r F(x) ~ FT(N) . Indeed, let F(x) = FT(y) for some y E N. Since z = Ty E T(N), we have z - x r O, while F ( z - x) = F T ( y ) - F(z) = 0. We have come to a contradiction wi th Ker F = 0. Hence T is a

topologically injective epimorphism, i.e., an isomorphism, and F = ST -1. []

Let us recall some facts abou t integrating operator-valued fnnctions [32]. Let X be a compact space,

A be a C*-algebra, ~ : C(X) --+ A be an involutive homomorphism of unital algebras, F : X --+ A be a

continuous map, and let, for each x E X , the element F(x) commute with the image of ~. In this case,

the integral

F(x) E A d~

X

n 7"1

can be defined as follows. Let X = U U~ be an open covering and ~ c~(x) = 1 be a subordinate i=1 i : 1

par t i t ion of unit. Let us choose a rb i t ra ry points ~i E Ui and form an integral sum

m

=

i=1

If the limit of such integral sums exists, then it is called the integral.

If X is a Lie group G, it is na tu ra l to take as ~ the Haar measure p �9 C(X) --+ C, ~(~) = / o~(g) dg

G

and to define as a norm the cont inuous map Q : G --+ 13(H),

/ Q(g) dg := lim ~ Q(~d / o~(g) dg, G i G

where the algebra A is realized as a subalgebra in the algebra /3(H) of bounded operators on H . If

Q: G ~ P+(A) C 13(H), then, since / c ~ ( g ) d 9 >_ O, we obtain that

G

Q({i) " / c~i(g)dg E P+(A) and / Q(g)dg E P+(A) i G G

(the positive cone P+(A) is convex and closed). Hence we have proved tile following lemma.

L e m m a 2 .7 .11 . Let Q : G --+ P+(A) be a continuous .function. [32] the following inequality holds:

fG Q(g) dg >_ O.

Then .for the integral in the sense of

[]

T h e o r e m 2 .7 .12 ([66]). Let GL = GL(A) be the complete general linear group, i.e., the group of invertible operators f irm Endl2(A), and assume that for the group G a representation g ~-~ Tg (g E G, Tg ~ GL) is given, where the map

G • 12(A) --+ 12(A), (g,u) ~+ Tgu

is continuous. Then there exists an A-inner product on 12(A) equivalent to the initial one (i.e., generating an equivalent non, n) and such that the representation g ~-~ Te is unitary with respect to this new product.

175

P r o o f . Let {., .}' be the initial inner product . For a rb i t ra ry u and v from 12(A) there exists a cont inuous map G --+ A, x ~-+ (Txu, T~v}'. Let us define a new p roduc t by the formula

(u,v} = f (T~u, Txv}' dx,

G

where the integral can be considered in the sense of ei ther of the two definitions in [32, p. 810], since the m a p is cont inuous with respect to the C*-algebra norm. It is easy to see tha t this new produc t sets

an A-Hermi t i an map 12(A) • 12(A) -+ A and that , by L e m m a 2.7.11, (u, u} > 0. Let us show tha t this map is cont inuous. Fix an arbi t rary u E Iz(A). T h e n x ~ Tx(u), G --+ tz(A) is a continuous m a p defined on a compac t space; thus the set {Tx(u) ix E G} is bounded . Therefore, by the principle of uniform boundedness [8, Vol. 2],

uniformly in x E G. If u is fixed, then

and by (27) one has

lim T~(v) = 0 (27) v-+0

IIT~(~)II ~ M~ =const,

/ ,

11(~,~)t1= [(T~(~,),T~(v)/d, <_ M~-volG-suptlT~(~)ll-+0, ~-+0. a o x c G

We have ob ta ined continuity at the point 0, hence on the whole space 12(A) • For Txu = (as(x) ,

a2(x), . . . ) E 12(A), the equali ty (u, u} = 0 takes the form (X) j Z a i ( x ) a * ( x ) d x = O .

G i = 1

Let A be realized as a subalgebra of the algebra of b o u n d e d operators on a Hilbert space L with an

inner p roduc t (., " )L . For each p C L, we have

=

i = 1 L G L G \ i = 1 /

(cf. [32]). Therefore, a~(x) = 0 almost everywhere; therefore, a~(a) = 0 for all x by continui ty and T.,~u = 0. In part icular , u = 0.

Since each operator Ty is an au tomorphism, we obta in

(Tvu, T~v} = f <T.~yu, T:~vv}' dx= / (T~u, Tzv }' d z = (u,v}.

G G

Now we show the equivalence of the two norms, which, in part icular , implies the continui ty of the representat ion. There is a number N > 0 such tha t HT, H' -< N for any x E G. Hence by [32] we have

( s u p ) 2 NuN 2 = N <u,u} NA --N f (Txu, Txu}' dxHd <_ NTxuN' _< N2(NuN') 2. ao \ x E G

On the o ther hand, applying Theorem 2.1.4 and L e m m a 2.7.11, we obtain tha t

-<

G G

<

G G

176

Then (llull9 2 = II <a, u/IIA ~ N211 <~, ~> IIA = N211ull 2- []

R e m a r k 2.7.13. Since 12(P) is a direct summand in 12(A), the previous theorem remains valid for

12(P) and any other countably generated module 34.

R e m a r k 2.7.14. Before averaging, the considered operators, in general, admit ted no adjoints; after averaging, we obtained unitary operators. In relation with this remark, the following problem arises. Is it true that if a given operator represents an element of a compact group, then it admits an adjoint? A negative answer to this problem is contained in Example 2.3.2, since a decomposition into a direct (topological) sum defines a representation of the group Z/2Z.

C o r o l l a r y 2 .7 .15 ([67]). Let A / I = A/ilOA/I2 be a topological decomposition into a direct sum of closed Hilbert modules (not necessarily orthogonal). Then there exists an inner product on the module 3,l equivalent to the initial one, with respect to which the indicated decomposition is orthogonal.

P r o o f . Let us define an operator J : 3,t ~ 34 by the equality

Jx = ~ x, if x E A d l , [ - x , if xEAA2.

It is possible to consider the operator J as a representation of the group Z /2Z on the module A4, and by Theorem 2.7.12 the inner product <x,y}z = <x,y} + (Jx, Jy} is equivalent to the initial one. The

orthogonality of 34 i and AA2 with respect to this inner product is evident. []

In Theorem 2.3.13 of [68], we will show how the averaging theorem 2.7.12 can be generalized from the case of compact groups to the case of amenable groups, but only for Hilbert W*-modules.

3. H i l b e r t M o d u l e s over W * - A l g e b r a s

3.1. W * - a l g e b r a s Detailed information about W*-algebras can be found in [12, 18, 29, 60, 62]. We also recommend the

original papers of Murray and yon Neumann [51], which are still valid. Here we list the basic definitions and necessary facts.

T o p o l o g i e s o n 13(H). Besides the norm topology, we will also consider on the algebra /3(H) of bounded linear operators on a Hilbert space H a number of other locally convex topologies which we define by sets of seminorms on B(H). Let a E B(H), {, {i, r/, rli E H.

1. The ~r-weak topology is defined by the seminorms o o o o (}o

" = i = l i = 1

2. The ~r-strong topology is defined by the seminorms o o o o

p(a) = ~ Ila~ill, ~ II&ll 2 < oc. i--I i=i

3. The o--strong* topology is defined by the seminorms o o (x}

p(a) = Y~(lla~ll + Ila*{ilr) 1/2, ~ II~ill 2 < oc. i = 1 i = l

4. The weak topology is defined by the seminorms p(a) = I(a{, rl) I. 5. The strong topology is defined by the seminorms p(a) = Ila~ll- 6. The strong* topology is defined by tile seminorms p(a) = (lla~ll + Ila*~ll) U2.

177

Oil bounded subsets in 13(H), the ~-weak topology coincides with the weak topology, the o--strong

topology coincides with the strong topology, and the or-strong* topology coincides with the strong*

topology.

A commutant of a subset R C t3(H) is the set R ~ := {a E 13(H) : ar = ra for each r E R}. A

bicommutant of a set R is the set R !! = (R!) !.

T h e o r e m 3.1.1 (von Neumann b icommutan t theorem). Let A c B(H) be an involutive subalgebra. Then the following conditions are equivalent:

(i) the algebra A contains the identity operator and is closed with respect to the er-weak topology; (ii) the algebra A contains the identity operator and is closed with respect to the or-strong topology;

(iii) the algebra A coincides with its bicommutant, A !! = A.

In particular, it follows from this tha t if A C B C 13(H) are two subsets, then A !! C B !!.

D e f i n i t i o n 3.1.2. An involutive subalgebra in 13(H) is called a yon Neumann algebra if it satisfies

the conditions of Theorem 3.1.1.

In the von Neumann algebras, there exists the polar decomposition: any element a E A can be

represented in a unique way as a = uh, where u is a part ial isometry, h is a positive element of the

algebra A, and Ker u = Ker h.

U n i v e r s a l e n v e l o p i n g v o n N e u m a n n a l g e b r a . Let co be a positive linear f lmctional oil a C*-algebra

A and (trio, H~) be a cyclic representat ion of the algebra A on a Hilbert space H~ constructed with the

help of the GNS-constructions. Let us put

: {D ~oEA~_

where A~_ denotes the set of all positive linear functionals on the C*-algebra A. The representation

(re, H) is called universal. The universal representation of tile C*-algebra contains any representation of

this algebra as a subrepresentation.

T h e o r e m 3.1.3. The second dual space A** for a C*-algebra A equipped with the ~r(A**, A*)-topology is homeomorphic to the bicommutant A !! of the algebra A C 13( H) with respect to the universal representation equipped with the c~-weah topology.

The von Neumann algebra A~, where the bicommutant is taken with respect to the universal repre-

sentation, is called a universal enveloping yon Neumann algebra for the C*-algebra A and is denoted by

A**. Any homomorphism of C*-algebras A -+ B admits a natura l extension up to a homomorphism of

the second dual spaces A** -+ B**. If A C 13(Ho) and A --+ B(H) is the universal representation, then

H o C H a n d A ! ! C A ~ A * * .

W * - a l g e b r a s . The notion of W*-algebra allows one to speak about von N e u m a n n algebras without

relation to a concrete Hilbert space where they act.

D e f i n i t i o n 3.1.4. A C*-algebra .4, which, as a Banach space, is dual to some Banach space F,

A = F*, is called a W*-algebra.

The Banach space F is called pre-dual for A.

D e f i n i t i o n 3.1.5. A linear f lmctional F on the yon Neumann algebra A is called normal if for any

increasing net a~ E A, k E A, with the least upper bound a E A, the value ~(a) is the least upper bound

of the set ~(a;~).

178

T h e o r e m 3.1 .6 . Let ,4 be a W*-algebra. Then there exists a unique (up to an isomorphism) pre-dual space for A, which coincides with the space of all normal linear functionals on A.

We will denote the pre-dual space of a W*-algebra A by .4,. By P we shall denote the set of normal

posit ive functionals on A, P C A, . The pre-dual space A , is the linear span of the set P .

3.2. I n n e r p r o d u c t o n d u a l m o d u l e s

We shall call Hilbert modules over W*-algebras Hilber t W*-modules. Some aspects of the theory of Hilbert C*-modules become simpler in the W*-case.

T h e o r e m 3.2.1 ([52]). Let M be a Hilbert A-module. An A-valued inner product (., .} admits an extension to the Banach module A/i', making it a self-dual Hilbert A-module. In particular, the extended inner product satisfies the equality (f, ~) = f (x) for all x E AA, f 6 M ' .

P r o o f . Let f , g 6 f14'. Our task is to define an inner product of ttmse functionals, ( f ,g ) . Let us define for this purpose a map F : P --+ C by the formula F(p) -- ( f~ ,g~)~, where F c P is a normal posi t ive funct ional on A, and let us show tha t the m a p F admits an extension to the set .4, of all normal funct ionals on ,4. For this purpose, the following two technical lemmas will be necessary.

L e m m a 3 . 2 . 2 . Let A1,... ,An 6 C a n d ~ . . . , ~ ff P be such that ~ Ai~i =O. Then ~ AiF(~) =O. i:i i:I

n

P r o o f . Let us consider a normal positive funct ional ~ = ~ ~i. Then ~ _> ~{, i = 1 , . . . , n. If x, y 6 M ,

then, by the assumption, n n n

Z + + : a (x + + = Z y>) = 0; i = 1 i : 1 i = 1

n

therefore, ~ AiV~,~{V~,~{ = 0. Note tha t by Propos i t ion 2.5.7 i = 1

Tt 9% $t 72

A{F(~{) = E A{(f~,{, g~{)~{ = ~ A{(V~,~{f~, V~,,~,~g~)~, = ~ A{,(V,;,~, V~,~,{f~,g;)~ -- O. i : 1 i : l i = l i : 1

This means tha t the map F can be extended to A , . []

L e m m a 3.2 .3 . The map F is bounded.

P r o o f . It is possible to present an arbitrary no rma l functional ~b 6 A , as V) = ~zl - ~z2 + i (pa - ~4), 4

where ~i 6 P and E I1~11 < 2 I1~11. Then i = 1

4 4 4

i:i i:i i:l

as required. []

Let us cont inue the proof of the theorem. We have defined a linear functional on A , which is also deno ted by F. Since ,4 is isomorphic to the space of linear functionals on the pre-dual space A , [62], there exists a unique element (f,g} 6 ..4 such t h a t F(g?) = r for all ~ 6 A, ; in par t icular , ( f~,9;)~ = 9~((f, 9)) for all 9~ 6 P. The sesquil inearty of the defined map (-,-) : M ' x Ad' --+ A follows from the l inearity of the map f ~-~ f~ from M ' to H~ for ~ ~ P. Let us show tha t (-, .) satisfies proper t ies (i)-(iv) of Definit ion 1.2.1.

(i) The inequali ty (f, f} >_ 0 follows from the fact t h a t for all ~ ff P we have ~(( f , f ) ) = (f~, f~)~ _> 0.

179

(ii) Let (f, f ) = 0 for f �9 A4'. Then .f~ = 0 for all q~ ~ P; hence ~( f ( x ) ) = 0 for all x �9 54, whence

f =O. (iii) Since for any ~ �9 P

~ ( ( f , g}) = (f~, g~)~ = (gw, f~)~ = ~((g, f}) = ~((g, .f}*),

we conclude that (f ,g} = (g , /}* .

(iv) Let a �9 A, ~ �9 P. Let us define a funct ional ~ on the algebra A by the equality ~a(b) = g~(a*b), 4 4

b �9 A. Then ~ �9 A . and ~ = ~ A~ i , where A~ �9 C, ~ �9 P. Let us pu t r = ~ + ~ ~z~; then ~b is a i:i i=i

posit ive functional and ~b _> g), q~l, . . . , q~4. It follows from Proposi t ion 2.5.7 tha t

4 4 4

~o(a* (f ,g}) = ~ A ~ i ( ( f , g ) ) = ~ A i ( f ~ , g ~ , ) ~ = ~ A i ( f ~ , Vr162 i:i i=i i:i

But for each x �9 A4

4 4 4

i=1 i = 1 i = I

= ~ ( ( f . a ) ( x ) ) = ( ( f . a ) ~ , x + N ~ ) ~ = ( ( f . a ) ~ , V r v,))~.

Since the subspace M / N ~ is dense in Hr for any ~z E P the following equali ty holds:

4

i f , g}) = = ( ( f - a L , = ( ( f . = g)); i = l

hence a* (f, g} = ( f . a, g}. Pass ing to adjoints, we also obtain (f, g} a = (f , g- a}. The obtained inner p r o d u c t on AJ' is an extension of the inner p roduc t from A4. Indeed, if x, y �9 54,

�9 P , then

~((~,Y)) = ( ' G , G L = (x + N~, v + N~)~ = ~((x ,y}) .

Hence (2, ~} = (x, y}. Fur ther , if f �9 A4', t hen

~( ( f , 2}) = (f~, ~)~o = ~(.f(x));

therefore, (f,~} = f (x) . Let us show tha t A//' is complete with respect to the norm ]]']]M' defined by the constructed inner p roduc t . On 54 ' there also exists the norm ]]'I] defined as the norm of linear maps f rom 54 to A, with respect to which the space 54 ' is complete. Let us prove tha t ]].]]~, = ]].][. Since

_ t 2 f ( x ) * f ( x ) = (.~, f} (f, .~} < H/h~, " (x, x},

we obtain tha t ]]/]] < ]].f]]~,. But , since ]],f~]l < ]].f]] ]]Wii~/e for each qo �9 P , 2 ]lf]I~a, = I](f, f}[] = sup{l]f~l]2~" 9 ~ �9 P, ]1~]] = 1} _< I]fl] 2 ,

and II'll , - - I I l l . Thus, it is proved t h a t A4' is a Hilbert A-module . It remains to verify that it is self-dual. Let

F �9 (A/[')'. The restr ict ion of F to Ad c 54 ' is an element of the modu le 54 ' ; therefore, it is possible to find a functional f �9 54 ' such tha t F(~) = f ( x ) for all x E 54. Let us define a flmctional /7o �9 (54 ' ) ' by the equality

Fo(g) = F(g) - i f , g}, g �9 54'-

I t is obvious that F0(2) = 0 for all x �9 3//. We have to verify tha t Fo(g) = 0 for all g �9 54 ' . Let ~ �9 P . Choose a sequence {y~ + N~} in .A/l/N~, converging to g~. Since F0 is bounded, we canfind a number K such that Fo(h)*Fo(h) <_ K (h, h} for all h �9 A/I'. For all n = 1, 2 , . . . ,

g ~ ( F o ( g ) * f o ( g ) ) = ~ o ( f o ( g - - ~ ) n ) * F o ( 9 - On)) <-- I (g~( (g - - ~]n ,g -- Yn})-

180

But since

= IIg - (Yn + N )ll ,

~((g - ~),., g - ~)n}) ~ 0; therefore

~( Fo(g)* Fo(g) ) = O.

Since equality (1) is t rue for any normal functional ~ E P, we obtain that Fo(g) = O.

(1) []

3.3. Hilbert W*-modules and dual Banach spaces

Propos i t ion 3.3.1. Let J~4,Af be Hilbert C*-rnodules over a W*-algebra .4 and T : J~4 -+ Af be a

bounded operator, T E HomA(A~,Af) . Then there exists a unique extension of the operator T up to an operator T : All' -+ N" .

P r o o f . Let us define an operator T* : H -+ A/I' by the equality (T#y ) ( x ) := (y, Tx) , x E All, y E Af.

Since [[(T#y)(x)[[ <_ []T[[ ][x[[ ]]y[[, the operator T # is bounded, [[T#y[[ _< [[T][ ]]y[[. For any a E ,4,

(T # (y . a) )(x) = (y . a, Tx ) = a* (y, Tx) = ( (T#y) �9 a)(x).

Therefore, the map T # is .4-linear. Let us define a map T : M ' --> N" by the equali ty (T f ) ( y ) = ( f , T # y )

for y E A/', f E J~4'. Since T = (T#) #, the map :F is also a bounded ,4-module map. The equality

(T~)(y) = (Tx~(y) demonst ra tes tha t the operator T is an extension of the opera tor T.

Let us show the uniqueness of this extension. Let S : A//' ~ N" be a bounded .4-module map

coinci_..ding with T on the submodule JM = {2: x E 3//} C A//'. Then their difference V = T - S vanishes

on 3//. Since the module A/I' is self-dual, the operator V has an adjoint opera tor V* : j ~ / __> 3/t'. If g E A/', x E 3//, then

(V*g)(x) = (V*g,~} = (g, V2) = 0,

i.e., V * = 0; therefore V = 0, hence S = T. []

C o r o l l a r y 3.3.2. Let All be a Hilbert ,4-module. Then the map T ~ T defines a monomorphism End~(A/I) C E n d ~ ( M ' ) = EndA(fl4 ' ) . []

Let us show that self-dual Hilbert W*-modules are dual Banach spaces, as well as the C*-algebras of operators acting on them.

Proposi t ion 3.3.3 ([52]). Let jDl be a self-dual Hilbert W*-module. Then .All is a dual Banach space.

P r o o f . Let us introduce the notat ion A//~ for the Hilbert module AJ considered as a Banach space

with multiplication by scalars given by the formula A. x := l,x, x E A/[ ~ Let us consider an algebraic

tensor product -4, | M ~ over the field C, where ,4, is a pre-dual space of normal functionals on "4.

Let us equip the space -4, | M ~ with the maximal cross-norm, and for x E Ad let us define a linear functional 2 on -4, | Ad ~ by the formula

k i = l i = 1

where ~ 1 , . . . , P~ E .4., Y l , . . . , Y~ E Ad ~ This functional is well defined. Since

i = 1 i = 1

it follows from the definition of the maximal cross-norm [3] that ]]:~]] _< ][x]l. Let us show that actually

]]2[] = [[x][. Let { ~ } be a sequence of functionals of the norm 1 in -4. such tha t ]r x ) ) [ -+ ][xt] 2. For

each element of the form ~ | E -4. | ~ we have [['~)n[[ []x[] ---- [Ix]] and [~(r | [[xl[2; therefore,

181

HxH < H2H. Hence it is shown that the map x ~ 2 defines an isometric inclusion 34 C (A. | 34~ *. To prove the s ta tement it is sufficient to demonstrate that the set 3~t = {2 : x E 34} is closed in (A. | with respect to the weak* topology, since this would mean that 34 is isometric to the dual space of some quotient space of A. | 34 ~ Let {2~} be a net in 34 converging to some element F E (A. | AdO) * with respect to the weak* topology. For y E 34, let us define a linear functional on A. by the formula (I)y(r = F ( r | y), where r E A. . The functional (I) is bounded, II~l[ -< IIFII Ilyll; therefore, there exists

a unique element f ( y ) E A satisfying the properties Iif(y)ll <- IIFi] ][Yll and f ( r | y) = r for all r E A . . The map f is linear. Let us show that it is A-linear as well. Let y E 34, a,b E A, and

E A. . Let us define a normal functional r E A, by the equality r = p(a*b). Then it follows from the equalities

p ( f ( y , a)*) = F(%o | (y- a)) = limS:a(%o | (y. a)) = l im~( (y - a, xa}) O~ (2

= limqo(a* (y, xa)) = l im~((y , x a } ) = F (g ) | (:~ Oz

= ~ ( f ( y ) * ) = ~ ( a * f ( y ) * ) = F( ( f ( y )a )* ) ,

which hold for any ~ E A, that f ( y . a) = f ( y ) a . Since the module AJ is self-dual, we can find an element x0 E 3// such tha t f ( y ) = {Xo, y}; therefore, F = ~0, hence 2~4 is closed in (A. | Ado) *. []

Consider the weak* topology on the dual Banach space AJ. Obviously a net {x~} in M converges to an element x E M with respect to this topology iff ~({y, x~}) --~ ~({y, x)) for every p E A, and for

every y E A/t.

Some modification of the previous reasoning allows us to obtain also the following proposition.

Propos i t i on 3.3.4 ([52]). Let A/i be a self-dual Hilbert W*-moduIe. Then the C*-algebra End~4(~4 ) is

a W*-algebra. []

3.4. P r o p e r t i e s of Hi lbert W*-modules The elements of seK-dual Hilbert W*-modules admit the following convenient representation (an

analog of polar decomposition).

Propos i t ion 3.4.1 ([52]). Let AJ be a self-dual Hilbert W*-module . A n y e lement x E Ad can be

represented as x = z . {x ,x} ~/2, where z E 3,4 is such that (z , z} is the projection onto the image o f

(x, x} ~/2. Such a decomposit ion is unique in the sense that i f x = z' . a, where a >_ 0, and if (z', z'} is the projection onto the image of a, then z ~ = z and a = (x, x} 1/2.

Proof . For x E AA, n c N, let us put

--1 a . = ( ( x , x ) + 1/2 x n = x-a.

Since (xn, xn} = ( z , x } ( ( x , x } + l / n ) 1, we have llxntl _< 1. Let y E A4 be a point of accumulation of the sequence {x~} in the weak* topology (which exists due to the compactness of the unit ball). Since

an - (x, x} 1/2 -+ 0 and Xn �9 an = x, we have x -- y . (x, x} 1/2. Let p be the projection onto the image

of Ix, x} 1/2. Then

= =

therefore, x = y . p {x, x} 1/2 and

Hence,

(X,X) ~- (X ,X}I /2p (y ,y ) p (X,X} 1/2 .

182

Since Ilyll 1, we have p - p ( y , y ) p >_ O; therefore,

(x, . )1 /2 (p _ p {y, y) p)1/2 = o,

whence p ( p - p {y, y} p)1/2 = 0, hence p = p {y, y} p. Let us put z = y.p. Then z. (x, x}1/2 = Y'P {x, x} z/2 =

x, (z ,z) = p ( > y ) p = v , and z . p = z. To prove the uniqueness of the decomposition, assume that x = z ' -a , where a > 0, and tha t (z', z'} is

the projection onto the image of a. Then (x, x} = a (z', z'} a = aS; therefore, a = (x, x} 1/2 and (z', z'} = p.

Since (z' - z' �9 p, z ~ - z' �9 p} = 0, we obtain z' = z'p. One also has

( z , x } = = ( z , z ' } 1/2 ,

i.e., (p - (z, z'}) (x, x}1/2 = 0 , whence (p - (z, z'})p = p - (z, z ' . p} = p - (z, z'} = 0. Now it can be easily

seen tha t (z - z', z - z'} = 0, hence z' = z, and this completes the proof. []

Let {Ps} be some set, of projections in a W*-algebra A. For each of them the set J~4~ = p s A C A

has a natural s t ructure of a one-generated project ive Hilbert A-module. Similarly to the definition of the

s t anda rd Hilbert module, we can define the module O A 4 ~ as the set of sequences (ms) , rn~ E Ads c A, s

such tha t the series ~ m ~ m ~ converges wi th respect to the norm in A. The dual Hilbert module s

3//a is called an ultraweak direct sum of the modules A//a. For se l f dual Hilbert W*-modules, we

have the following s t ructural theorem.

T h e o r e m 3.4.2 ([52]). Let All be a se!f-dual Hilbert W*-module over A. Then there exists a set {Pa}

of projections in A such that the module J~4 is isomorphic to the uItraweak direct sum of the modules

p e A . []

P r o p o s i t i o n 3.4.3. Let .IV" C HA be a Hilbert submodule over a W*-algebra A . I f A f ! = 0, then the dual module N" coincides with H A.

P r o o f . Let j " A/" -+ H A be an inclusion of modules. The restriction of functionals f ~-~ f i l l , f E H ' A,

defines a map j ' : H ~ --+ N" dual to j . If f C Hht is such that f i l l = 0, then fAA/, and, by assumption, one has f = 0; therefore, the map j ' is injective. Let us consider the composition of maps

i = j ' o A o j : N'~--~ HA ~-+ H'a --+N".

If n E A f , then i(n) = j ' ( j ( n ) ) = j(n)l H = ~; therefore, the map i coincides with the inclusion map

~" Af ~-+ N". The dual map (after the identification of the first and second dual modules)

( i ' o i ) ' = i' o i " " N " = N " --+ H'a -+ N "

is an isomorphism; therefore, the map i' should be surjective, hence the map j ' is surjective. []

P r o p o s i t i o n 3.4.4. Let A be a W*-algebra and 7~ C HA be an A-submodule without orthogonal complement, i.e. T~ • = 0 in H a . Then T~' = H ~ .

P r o o f . It is sufficient to demonstra te tha t if the orthogonal complement to a submodule Tr in HA

is equal to zero, then the orthogonal complement to /~ in the module H ~ is equal to zero, too. Let us

assume the contrary. Assume that it is possible to find a functional f C H~ such tha t f ( r ) = ( f , r) r 0 oo

* r for some r E 7r But the series ~ f~ i is n o r m convergent in A; therefore, there is a number n such i = 1

tha t f(~)(z) r 0 for f(~) = ( f ~ , . . . , f~, 0 , . . . ) . But since f(~) E HA, we get a contradiction. []

183

3.5. Topological character iza t ion of se l f -dual Hilbert W*-modules Let A be a W*-algebra, 34 be a Hilbert A-module, and P �9 A . be the set of normal states on

A. Let us define (see [22]) two topologies on 34 with the help of sets of seminorms. We denote by ~-~ the topology given by the sys tem of seminorms 9~((', "))z/e, ~ ~ P , and by ~-~ the topology given by the sys tem of seminorms ~((y, .}), y �9 34, 9~ G P- In the case where A = C and 3d is a Hilbert space, the topology ~-~ is ~he norm topology and the topology ~-~ coincides wi th the weak topology; therefore, in general, these two topologies do not coincide.

T h e o r e m 3.5.1 ([22]). Let 34 be a Hilbert W*-module. Then the .following conditions are equivalent:

(i) the module 34 is self-dual; (ii) the unit ball B1(34) is complete with respect to the topology 71;

(iii) the unit ball B1(34) is complete with respect to the topology "r2.

P r o o f . Let us prove the implicat ion (i) ~ (ii). Assume for this purpose tha t the uni t ball B~(AJ) is not complete with respect to the topology ~-1. Let us denote by L the linear span of the complet ion of Bl(Ad) with respect to the topology ~-1. For the extensions of seIninorms from dr4 to L, we use the same notation. By assumpt ion , there exists an element r E L \ M and a net {y~}, oz �9 A, bounded with respect to the norm, such t ha t for any F c P and for any e > 0 there exists some oz �9 A for which

~( ( r - y~, r - y~)) < e for all fl �9 A, ~ _> oz. For arbitrary x EAd, we have

= I ((yg- -< _ y g - -<

for all fl, 7 > oz. Therefore, there exists in the W*-algebra A the l imit (with respect to the v-(A, A. ) - topology)

R(x) = lim (y~,x} �9 A c~

for each x E A4. The inequal i ty

I ((yg, --jlxIIsup(lly li : oz A} ot

shows the continui ty of the m a p R" M -+ ..4, x ~-+ R(x). It is obvious tha t the map R is an A-module map; therefore, R is a funct ional on A/I. By assumption, the modu le M is self-dual; therefore, there exists an element z E A/I such tha t R(x) = (z, x). Then lima (y~, x) = (z, x) (where the limit is taken in the ~r(A, A,) - topology) ; therefore, the net {Ya} converges to the e lement z E M in-the topology T1 and r = z; a contradict ion to our assumption.

Let us prove now the implicat ion (ii) ~ (i). We still denote the extension of the inner product to the dual module A/I' by (.,-}. It is easy to see tha t the ideal (A/t, A/t) E .4 is norm dense in the ideal (AA',3A') E A; therefore, (AA, M ) E A = ( 3 A ' , M ' ) c A. Let us consider at first the case where a W*-algebra .4 is c~-unital. T h e n there exists an exact normal s tate ~p E .4. (see [12, Propos i t ion 2.5.6]). Let {H, 7c, 4} be the cyclic representat ion associated with ~/J. The vector ~ E H is s imul taneously cyclic and separating. The linear space M with the inner product (-,-) = ~p((-, .)) becomes a pre-Hilbert space, and the map ~p(f(.)) : Ad -+ C, where f c M ' , becomes a linear funct ional on M . T h e n one can find an element fr in the complet ion of the space Ad with respect to the norm ~((-, .))~/2 such tha t (fw, x) = ~( f (x ) ) for all x �9 34. This means tha t there exists a sequence (x~), xi �9 3 t , i �9 N, such tha t

0 = l i m ( x ~ - f , , x ~ - f , ) = l i m ~ ( ( 2 ~ - f , ~ - . f } ) = lim ( T r ( ( ~ i - f , S i i - f } ) ) l / 2 ~ 2, i - + o r i - + ~ i - + e c

where .~ denotes the image of the element x under the canonical inclusion 34 C 31~. Since the vector is cyclic and separat ing, there exists the l imit (in the or(A, A . ) - topo logy on A) lim~ ( ~ - f , ~i - f ) = 0 (see [12, Lemmas 2.5.38 and 2.5.39]). Therefore, f E M and the module 34 is self-dual. Let us pass to the general case. If a W*-algebra ,4 is not ~r-unital, then it is possible to choose a directed set of

184

project ions {p~}, a E A, p~ E ,4, such tha t for each a E A the algebra p~Ap~ is a cr-unital W*-algebra

and l imp~ = 1, where the l imit is taken in the or(A, A,)- topology. As was proved earlier, the functional C~

fp~ on a Hilbert p~Ap~-module p~.h4 is an element of the module p~M for all a E A. But then there

exists the limit (in the ~-~ topology) of the net {fp~} which belongs to 3/l and is equal to f , and so the self-duMity of the module M is proved.

It remains to show the equivalence of condi t ions (ii) and (iii). If B~(AJ) is complete with respect to

the topology ~-l, then AJ is self-dual; therefore, it is a dual Banach space with respect to the topology T2 (see Propos i t ion 3.3.3), hence B~(34) is complete wi th respect to the topology ~-2. Let us assume now tha t B~(M) is complete wi th respect to the topology ~-2 and {x~}, c~ E A, is a bounded Cauchy T~-net. For all y E M , /3, 7 E A, and ~ ff P , we have

x ))l -<

As earlier, L denotes the l inear span of ~-~-completion of B~(AJ). There exists the limit in L (with respect to the topology ~-~) lim~ x~ = t ~ L. It follows from inequali ty (2) t ha t {x~} is a Cauchy net wi th respect to the topology 72 as well. Bu t since the topology ~-: is weaker t han the topology ~-~, we have L C 3d; therefore, the l imits l imxi wi th respect to the topologies ~-~ and ~-2 coincide and are equal

i to t ~ 3,l, whence one gets the completeness of AJ with respect to the topology T~. []

4. R e f l e x i v e Hi lber t C * - M o d u l e s

4 .1 . Inner p r o d u c t s o n b i d u a l m o d u l e s

For a given C*-module M over a C*-algebra A, we shall define the bidual Banach right A-module Ad" as a set of bounded A-homomorph i sms f rom the dual module 3,t t into A. It tu rns out tha t an inner

p roduc t on A4 can be ex tended to the bidual module for a C*-algebra A, unlike the dual module , which admi t s an extension of an inner p roduc t only in the case of W*-algebras.

Let x ~ 3 / / , f E A 4 ~. P u t

~(f) := f(x)*. The map x ~-~ ~? is an isometr ic m a p from the A-module A/[ into the A-module Ad ' :

t l ~ ] [ = s u p { i i f ( x ) i i : f E M ' , Itf[i -< l} _< ]]f[[ ]ixi[ -< iixi[,

1 1 11511 _> lI ( :)ll = l i (x ,x} l l = llxlt.

For a funct ional F E M ' , we define a f imctional F E M ~ by the formula

F (x ) := F(~) .

Identifying M and r -- { 2 : x E 3A} c Ad', we obtain t h a t / ~ is the res t r ic t ion of F to M C A d ' . Note t ha t (.~)-= ~ for all x E AA. It is clear tha t the map F ~-+ F is an A-module m a p from AJ" to A& and

/~ < IIF]]. We will soon check tha t this m a p is an isometry.

Let us define an inner p r o d u c t (-,-) : A4" • A4" --+ A by the equali ty

(F, G> := F (G) , F, G e M " . (1)

It can be directly checked t h a t ( F - a, G} = a* (F, G} for a E A. Moreover, for x, y E 3//, one has

therefore, the inner p roduc t defined by equali ty (1) is an extension of the inner p roduc t on M . To check the propert ies of an inner p roduc t , we need the following construction.

185

Consider the right A-module A • 34. In add i t ion to the natural inner p roduc t (-, "}0 defined by the fbrmula {(A, x), (b, Y)}0 = a*b + (x, y}, where a, b c A, x, y E 34, we consider another inner p r o d u c t on

the m o d u l e A • 34. Let us take f E 34' , f ~ 0, and a number t > Nfll, and put

((a,x), (b, y)}Lt := t2a*b + a*f(y) + f(x)*b + (x,y}. (2)

Proper t i es (iii) and (iv) of Definition 1.2.1 obviously hold. Let us check propert ies (i) and (ii). The first

one is valid due to the inequali ty

{(a, x), (a, x)}f, t = t2a*a + a*f(x) + f(x)*a+ (x,x)

1 _> t2a*a + a ' f (x) + f(x)*a + Ilfl[ 2 f (x)*f(x)

> t2a*a + a*f(x) + f(x)*a+ ~ f ( x ) * f ( x )

(3)

(4)

Assume t h a t ((a, x), (a, x))f, t = 0. Then equali ty should be reached at each step in (3)-(4). Subt rac t ing the line (4) from the line (3), we obtain

(ll.fl1-2 - t -2) f (x)*f(x) = 0;

therefore, f (x) = 0, hence t2a*a + (x, x} = 0, and we can conclude t ha t a = 0 and x = 0, so we have checked the validity of proper ty (ii). Thus, the m o d u l e A x A/t with the inner product defined by formula (2), is a Hilbert A-module . We denote the n o r m on this module corresponding to this inner p roduc t by

II.lls,t, and we denote the g i lbe r t module A x Ad equipped with this no rm by (A • d~).f,t. Note tha t

II(0,~)llf,~ = II~ll- For x ,y E 3,t, a ~ A, we have

I I ( / ' a + 2)(y)ll = Ila*f(y)+ (x,y}fl = ((a,x),(O,y)}f,t <_ II(a,x)llf,t" II(0, y)lls,t = IlYll'll(a,x)llf,t. Therefore,

l l ( f . a § ~)11 <-- II(a,x)llf,t. (5)

Let N" c A4 ~ be a submodule containing the module Ad. Then the norm of Proposit ion 4.1.1 ([53]). any .functional ~ E N" satisfies the equality li~b]l = I]~1~]1"

P r o o f . Wi thou t loss of generality, we assume t h a t ]I~bll = 1. Define a funct ional f E A4' by the fornmla

f(x) := ~p(~), x E .s T h e n iiftI < 1. It is necessary to prove the inverse inequali ty fife] _> 1. Take g c Af such t h a t Hg]l < 1, and pu t c = ~b(g) E A. For a E A, x E A/t, we have

]lc~ + f(x)l] = l ie(g- a + .~)il <- ]]g-a + ~1] -< li(~, x)llg,~

(the last inequali ty follows from (5)), hence the m a p

f c : A • 3//--+ A; (a, x) ~-~ ca + f(x)

is a b o u n d e d modular map, Ilfcii(A• --< 1. Therefore,

F~(a,x)*f~(a,x) <_ ((a,x),(a,x)}g,~ (6)

for all a E A, x EAd. From the estimate (6), we get

a'c'ca § a*c*f(x) + f(x)*ca + f (x)*f (x) <_ a*a + a*g(x) + g(x)*a + (x,x}.

Taking a = - 2 g ( x ) , we obtain

4g(x)*c*eg(x) + f(x)*f(x) <_ (x, x) + 2(g(x)*c*f(x) + f(x)*cg(x)).

186

But since

we have

G(x)*c*f(x) + f(x)* cg(x) < g(x)*c* cg(x) + f (x)*f (x) ,

2g(x)*c*cg(x) <_ <x,x) + f (x )* f (x ) <_ (1 + Ilfll 2) <x,x>, 1 hence, Ilg" c*ll _< ~(* § Ilfll2) 1/2, and

1 II~(g-c*)l l = Ilcc*ll = Ilcll 2 _< ~ ( 1 + 11/112) 1/~

The last inequal i ty is valid for all g E AF for which Ilgll < 1, and since 11~11 = 1, the inequality 1 _< @22(1 + Ilfll2)l/2 should be valid too, whence II/ll-> 1. Thus, II/l[ = II l ll = 1. []

Note t ha t in the case where N" = A4' Propos i t ion 4.1.1 means tha t the map F ~ _P is an isometric inclusion 3,i" C 3,t ~.

P r o p o s i t i o n 4 .1 .2 ([53]). For all F E At", one has ( f ,F ) >_ 0 and I I<F,F)I I - - I IFII 2.

P r o o f . Let F E At" , F # 0. Pu t c = F ( F ) , D = IIFII. Let us show at first t ha t D 2 E Sp(c). For

t > D, consider the inner product ( ' , '}~,t on the module A • At . Since .P.a+~2 < II(a,x)ll~,, for all

(a, x) E A • At , the m a p

f r 2 1 5 A; (a,x)~-+ F ( [ ' . a + ' 2 ) = c a + _ P ( x )

is bounded wi th a no rm not exceeding D (we mean here the n o r m defined by the inner product (-, ")~,t)" Therefore,

(ca + F'(x))*(ca + F(x)) <_ D 2 ((a, x), (a,z))~, t .

Inequali ty (7) holds for all t > D, and taking the limit as t -+ D, we obtain

(ca + _F(x))*(ca + !~(z)) <_ D2(D2a*a + ['(x)*a + a*_P(x) + (x,x)).

Taking a = - D - 2 F ( x ) , we get

F(x)*(D-2c - 1)*(D-2c - 1)_F(x) _< D 2 ( - D - 2 F ( z ) * F ( x ) + <x,z>),

hence,

F(x)*((D-2c - 1 ) * ( D - 2 c - 1) + 1)_F(x) _< D 2 <x,x}.

Assume tha t D 2 ~ Sp(c). Then it is possible to find a number 6 > 0 such that

_F(x)*(D-2c - 1)*(D 2 c - 1)F(x) _> 6F(z)*_F(x)

for all x E Ad. Bu t then D 2

~(x)*_~(x) _< - - 1 + 5 <z,z) ,

whence 2 D 2

D 2 = < - - < D 2. - 1 + 5

The contradict ion obta ined shows tha t D 2 E Sp(c). But since

IIoll-- F ( F ) _<ttFI[ F ~ --IIFII ~ - - D 2,

(7)

llcl[ = D 2, hence II(F,F>II = IIFII 2 and II<f , f>l l ~ Sp(<f,Y>). For an arbi t rary element a E A, we have

[[acall = a*F(_r .a) = I I ( F ' a , F ' a ) l l ~ S p ( ( F . a , F . a } ) = Sp(aca).

The following l emma concludes the proof.

187

L e m m a 4.1 .3 ([53]). Let an element c E A be such that the inclusion ]]acalI E Sp(aca) holds for any

a E A, a >_ O. Then c ~ O. []

Proposi t ion 4.1.2 shows tha t the inner p roduc t defined on Jut" satisfies condi t ions (i) and (ii) of Definition 4.1.2. It remains to check condi t ion (iii). Note t h a t

( F + G , F + G ) >_ o,

hence these expressions are self-adjoint. T h e n

((F,C> + (G,F})* = < F , G ) + ( C , F ) ,

( F + iG , F + iG) >_ 0.;

- i ( ( F , C) - (a , F))* = i ( ( f , C) - (G, V));

therefore, (/7, G) = (G, F}*. The module Ad" is a Hilbert A-module since the opera tor no rm on jut" coincides (by Propos i t ion 4.1.2) with the n o r m defined by the inner product . Thus, we have proved the

following theorem.

T h e o r e m 4 .1 .4 ([53]). The map (., .} : ju t" • M " --+ A defined by the formula (F, G) = F(G), F, G E 54" , is an A-valued inner product on Jut". The norm defined by this inner product coincides with

the operator norm on Jut". The map F ~-+ F is an isometric inclusion AA" c Jut'. []

Let us pass now to dual modules of a higher order. Let q) E (Jut")'. Then define a functional

f~ C 54 ' by the formula

F | := x Jut.

Let, further, f E M ' . Define ~ f E (Jut")' by the formula

�9 f ( F ) := (F( f ) )* , F e M " .

The maps �9 ~+ f~ and .f ~ ~ f are A-module morphisms. Consider their composi t ion

54 ' -+ Jut'" --+ Jut', F ~-~ ~ f ~ fq,~. (8)

Since, for any x c Jut, we have

F % ( x ) = ~ f ( ~ ) = (&(f))* = f (x ) ,

the composit ion (8) is an identical map, whence the map M ' --+ AA'" is an isometric inclusion and the m a p J~4"' --+ 3,4' is an epimorphism. Let us show tha t the last m a p is also monomorphic . Apply for this

purpose Proposi t ion 4.1.1 for the case A/" = M " . Let �9 E N" = M ' " . Then the funct ional f~ E AA' is a A

restr ict ion on M of the functional ~, f~ = ~ ] ~ . Assume tha t f~ = 0. Then, by Propos i t ion 4.1.1, we have []#2]] = ]]~]~]] = 0; therefore, the m a p A4'" -+ 54' , �9 ~-+ f~, is monomorphic . Thus, this map is an

isometric isomorphism.

C o r o l l a r y 4.1 .5 . For a Hilbert C*-module A/t, one has ( M " ) ' = 54' and ( M " ) " = J~" . []

So, the series of dual modules A/l, M ' , . . . stabilizes on tile third entry and the inclusions

M c_ 5 4 " = JYt'"' c_ M ' = Jut'"

are isometric. Thus, tim modules Jut and Jut" are Hilbert, unlike the module Jut', which is, generally speaking, only Bausch. Let us i l lustrate by examples tha t all possible variants can be realized:

M = jut" = M ' ; jut # M " = 34 ' ; M = jut" r M ' ; jut # jut" r M ' .

(i) Let A be a unital C*-algebra and let A / / = Ln(A) be a flee A-module with 'n generators. Then

the module 54 is autodual ; therefore, jut = } M " = jut ' . (ii) Let A be a W*-algebra. By T h e o r e m 3.2.1, for any Hilbert A-module M , its dual module 54 ' is

a self-dual Hilbert module, hence jut r Jut" = M ' . (iii) (12a]) Let A = C0(0, 1] be the C*-algebra (without uni t ) of functions on a segment [0, 1] vanishing

at zero, jk4 = A. Then Jut' = C[0, 1], Jut" = Co(0, 1], and Jut = AA" # Jut'.

188

(iv) ([23]) Consider the module Co(0, 1) of functions on the segment [0, 1] vanishing at the end points, over the C*-algebra A = C0(0, 1]. In this case, one has A4' = C[0, 1], A4" = Co(0, 1], that is, 34 r r M'.

Def in i t ion 4.1.6. A Hilbert C*-module A4 is called reflexive if Ad" = 3A.

In the following, we will encounter other examples of reflexive Hilbert modules.

4.2. Ref lex iv i ty of H i l b e r t m o d u l e s over /C +

In this section, we describe the results of [63]. Let/C be the C*-algebra of compact operators acting on a separable Hilbert space H and le t /g+ be the C*-algebra of operators of the form a = A + K, where AE C, K E K ; .

T h e o r e m 4.2.1 ([63]). Any countabty generated Hilbert l~+-module is reflexive.

Proof . According to the stabilization theorem 1.4.1, any countably generated Hilbert module is a direct summand in the standard module /2(/C+); therefore, it is sufficient to prove reflexivity of the module 12(/(;+). Proposition 2.5.5 gives a description of the dual module:

12(1~+)'= { f = (fi) : fi E ~ +, suPN~i=lF[fi < o c } .

L e m m a 4.2.2. If f = (fi) E 12(~+) ', K ~ I~, then f . K = (fiK) E 12(K~+).

Proof . Since the operator K can be approximated by finite-dimensional operators, it is sufficient to prove the lemma in the case where K is finite-dimensional. Note that the operator (f,~K)*(f~K) = K*f,*fiK is a positive operator whose kernel contains K e r K and whose image is contained in ImK*.

Since d i m h n K * and codimKer K are finite, the norm convergence of the series ~ (.f~K)*(f~K) follows i=1

from its weak convergence. []

Let F E 12(/C+) ''. Put F1 = F(~,i)* E 1C +, where {ei} is the standard basis of 12(/C+), ~'i E 12(K+) '. Since 2t4" c Ad' for any Hilbert module 2t4, the sequence _F = (Fi) is an element of the module 12(K+) '.

o ~

Let us prove that the series ~ F/*Fi converges in the C*-algebra/C + to the element F(/T) = (F, F}. i=1

Let K E/~ be a finite-dimensional operator in H. By Lemma 4.2.2,

o o

(F,F) K = (F,F. K ) = E F*FiK; i = 1

therefore,

and for any ~ E H

o o

K* (F,F) K = E K*F~*FiK, i=1

o o

E(K*F~F,K~,~) = ((F, F ) K ~ , K { ) , i=1

where (.,-) is a (scalar) inner product on H. Let r! E BI(H), where BI(H) is the unit ball in H. Choose an element ~ E H and a finite-dimensional operator K ~ so that ~ = K % Then

(N3

E ( Fi~, Fi~) = ( (F, F> ~], rl). i=-1

189

Therefore, for any 7!

L e m m a 4 . 2 . 3 .

o o

~(F~*F~w,~) ~ II(F,F)II" I1~11 ~. i = 1

Let f = (ki) ~ 12(]C+) ', and ki ~ ]C for all i ~ N. Then F ( f ) ~ 1C.

(9)

Proof. Due to the continuity of F and the closedness of the algebra/r it is possible to assume that

all k~ are finite-dimensional operators. Denote by Vi C H the image of the operator k[, dim Vi < co.

Let H = HI G H2 be an orthogonal decomposition of H into a sum of two closed infinite-dimensional

subspaces. Assume at first that each of the subspaces V/ is contained in one of the subspaces HI or//2.

Let F(f) = K+h, K E ~, h G C. Choose a compact operator /r with an image L C //2 such that

dimL = co. Replace by zeroes those terms in the sequence (kl, k2,...) for which V/ C//i and denote /~! ! . . . . . the obtained sequence by ( 1,k2,. )- Then (k~k, k2k,. ) = (k~k,k~2k,..) since the condition kik = 0

is equivalent to the condition Im k* _L Im k. Thus,

F ( k l , k ; . . . ) k = F(k lk , k~k , . . . ) = F(k~k, k 2 k , . . . ) = F ( k ~ , k 2 , . . . ) k = ( K + h)k,

i.e.,

r ( k l , ~;,. . .)1~ = (K + h)l~. Since d i m L = co, F(k[ , k~ , . . . ) = K ' + h with some K' ~ 1~. Interchanging subspaces HI and H2, it is possible to construct a sequence (k[ ~, k" . . . . K " 2 , . . . ) such that F ( k f , k~ ~, ) + h with some K" ~ 1~. Thus,

(Ki , k ~ , . . . ) = ( k l , k ; , . . . ) + (k i ' , k ; ' , . . . ) ,

hence,

F(k~, k2 , . . . ) = K ' + K " + 2h;

therefore, h = 0. In the case of arbi t rary subspaces E , it is possible to find finite-dimensional operators

li, m~, ni such that

I s (I1,12,. . . )El2(]r ( M l , m 2 , . . . ) , ( n l , n 2 , . . . ) E / 2 ( ~ + ) ',

and the image of each of the operators m~ and n* is contained in one of the subspaces H1, H2. []

Put Fi = K~ + hi, (F, F} = K + ,~, where Ki, K E ]C, hi, h E C. Then

F?F~ = K; K~ + h~K; +-X~K~ + Ih~l ~

L e m m a 4.2.4. ~ Ihil 2 = h. i = 1

(N2

P r o o f . At first we show that the series E Ihil 2 converges. Assume the contrary. Put M = II{F,F}II / = 1

N

and find a number N such that ~ Ihil 2 > M + 1. Choose c > 0 to sa t i s~ the estimate i = 1

N 1

E ( 1 + 21hi])E < ~- i = 1

Choose, further, a vector ~ E H with [[~[[ = 1 to satisfy the inequalities

I IK;K~l l _< c, IIKi~ll _< c, IIK;~ll _< c, i =- 1 , . . . , N .

~o N 1

i = 1 i = 1

Then the inequalities

190

oo

contradict (9). So, }-]. IAit 2 < oc. But this means that (A~, A2, . . . ) E 12(1(:+). Then i = 1

K + ~ = (F,F> = F(r~, r ~ , . . . ) = F(K~, K~,... ) + F(~ , ~ , . . . ) oo oo

= F(K1,K2, . . . ) + E F~*ai = F ( K 1 , K 2 , . . . ) + E ( K * + X~)Ai i = 1 i = 1

= F(K1, K 2 , . . . ) + K[A~ + E I A i l 2. i = 1 i = 1

oo (x?

But, since F(K~, K2, . . . ) + ~ K[A~ E 1C, we conclude that ~ la~l = A. i = 1 i----1

[]

L e m m a 4.2.5 ([63]). Let X be a compact Hausdorff space and let fn, f , g~, g be real-valued functions on X, n E N. Assume that the functions f~ and f are continuous, the functions fn + gn and g~ are

o o

nonnegative, the function g is bounded, the series ~ gn uniformly converges to the function g, and the n = l

:N2

series E (f~ + gn) converges pointwise to the J~nction f + g. The., the series E (f~ + gn) uniformly n=l n=l

converges to the function f + g. []

Let X = B1 (H) be the unit ball of H with the weak topology, ~ E X. Put

fn(~) = ( ( K ; K ~ + ~ K ~ + X ~ K ~ ) ~ , ~ ) , g~(~)= IA~I : . II~ll ~, f ( ~ ) = (K~,~), g ( ~ ) = A . II~ll : .

The conditions of Lemma 4.2.5 are satisfied, hence the series ~ (F[F~, ~) uniformly converges on X i = 1

0(3

to the function ({F,F}~,~); therefore, the series E F~Fi converges to (F,F} in the algebra /~+ and, i = 1

therefore, F E 12 (/C +). []

4.3. Ref lex iv i ty of modu les over C(X) In this section, we describe results from [47, 64].

Def in i t ion 4.3.1 ([64]). A compact Hausdorff space X is called an L-space if for an arbitrary sequence .fl, f2,--- of continuous functions on X converging pointwise to some bounded function f the set of continuity points of the function f is dense in X.

Examples of L-spaces are any compact subsets of finite-dimensional Euclidean spaces [1]. Infinite Stonean spaces are not L-spaces.

The definition of L-spaces allows us to give a description of bidual Hilbert modules over algebras of functions on such spaces.

T h e o r e m 4 . 3 . 2 ([47, 64]). Let A = C ( X ) , where X is an L-space. Then any countably generated Hilbert A-module is reflexive.

Proof. According to the stabilization theorem 1.4.1, any countably generated Hilbert module is a direct summand in the standard module HA; therefore, it is sufficient to prove reflexivity of the module HA. Proposition 2.5.5 gives a description of the dual module:

H ~ = { f = ( f i ( t ) ) : f i ( t ) E C ( X ) , suPN ii=l]f~(t),2 < o c } .

191

N

Since the sequence of partial sums ~ If~(t)l 2 is monotone and bounded at each point t C X, the i = 1

corresponding series converges pointwise to a bounded function. By assumption, the set of points of O4)

continuity of the limit function ~ IA(t)l 2 is dense in X. Let us fix a point of continuity to E X. Let i = 1

A(t) be a continuous function on X equal to 1 at the point to. For F E H~, we have

F(fA) = F eifi" ~ + F f - E e i ' f i A , (10) i = 1 i = i

N

where {e/} is the standard basis in HA C H A. The element ~ e/ f /~ belongs to the module HA; therefore, i = 1

F e/f/ . = ZF;f . i = 1 i = 1

where by Eli = F/(t) we denote Let aJ(f) denote the least upper bound of oscillation of the N

function E Ifi(t)l 2. Then, obviously, i = 1

oo N e i f i 2 w( . / )= lim sup E ]fi(t)]2= lira f - E �9

N--+cc t c X N--+oc i = N + I i = 1

Let us choose an arbitrary c > 0 and a function 1(t) such that on the support Supp ~(t) the oscillation aJ(f) is less than c 2. Then find N such that for all N' > N the inequality

f - E eifi .~ < 2e 2 (11) i = 1

holds. It follows from inequalities (10), (11) that at the point to (where 1(to) = 1) one has

N ~

F(f ) ( to) - ~ F[(to)fi(to) < IIFII" 2c i = 1

o o

for all N' > N. Thus, the series ~ F*(t)f i( t) converges to a continuous function F(f ) ( t ) at all points i = 1

O4)

to of continuity of the series ~ If~(t)l 2. Since H~ c H A, the series i = 1

OO

~ F[(t)Fi(t) (12) i = l

is also convergent at each point and coincides with the continuous function (F, F} (t) = F(F)(t) at each point of continuity. Denote by E C X the set of continuity points of the series (12). Let now to be some point of discontinuity of tile series (12). Without loss of generality (multiplying by some continuous function, if necessary), it is possible to assume that there exists a sequence of points t,~ E E converging

OQ

to the point to such that (F,F)( tn) = 1 and y~ IFi(to)l 2 < 1. Choose functions hi(t) ff C(X) to satisfy i = l

the conditions

(i) hi(to) = Fi(to), Ihi(t)l <_ IFi(t)l; o o

(ii) h = (hi) E HA, i.e., the series ~ Ih~(t)l 2 is uniformly convergent. i = 1

192

Define a function A(t) on X by the following conditions:

(i) 0 < A(t) _< 1, )~(t2~)= 0, A(t2~+l) = 1;

(ii) outside the point to the function A(t) is continuous.

Consider the element

F~(t) = hi(t) + ~(t)(Fi(t) - hi(t)).

It can be easily checked tha t the functions fi(t) are continuous on the whole X. Moreover, since IA(t)l <_ IFi(t)l, we have f = (fi) E H~. Then

oo oo

F(f) ( t ) = IF, f} (t) = ~ F[(t)h~(t) + ,~(t) ~ E[(t)(F~(t) - hi(t)). i = 1 i = 1

The series

~ F[(t)hi(t) i = 1

is uniformly convergent since h E HA. On the set E C X, the series cx)

y] hi(t))

(13)

i = 1

converges uniformly to the continuous function {F, F - h} (t) which is continuous on the whole X. Then for t E E one has

{F, f} (t) = (F, h} (t) § A(t)((F, F} (t) - (F, h.) (t)),

and, for each t~,

{F,f} (tn) = {F,h} (tn) + ~(tn)(1 -- {F,h} (t~)). (14)

The functions {F, f} (t) and {F,h)(t) are continuous, and since the series (13) is uniformly convergent,

{F,h}(to)= EIFi ( to ) l 2 < 1. i = 1

But this contradicts the continuity of the funct ion {F, f} (t), since, due to our choice of the function

A(t), the limits of the right- and the leR-hand sides of Eq. (14) are different for even and for odd n. []

4.4. H i lb er t m o d u l e s re la ted to c o n d i t i o n a l e x p e c t a t i o n s o f f inite i n d e x In this section, we describe results from [25].

Let E : A --+ B C A be an exact condit ional expectat ion on a C*-algebra A, i.e., a projection of

norm one onto a C*-algebra B such that the condition E(x*x) = 0, x E A, implies x = 0.

The conditional expectat ion E : A -+ B C_ A is called a conditional expectat ion of algebraically finite n

index if there exists a set of elements {ul, ..., un} C A such tha t for any x E A, x = ~ uiE(u~x). Then i----1

n

the element Ind(E) = ~ uiu~ of the center of the C*-algebra A is called the index of E. Ind(E) does i = l

not depend on the choice of the elements {u~, ..., u~} C A, it is positive, and Ind(E) > 1A [7, 69]. It is

shown in [7] tha t the algebraic finiteness of the index is equivalent to the property of A to be a projective finitely generated C*-Hilbert module over the C*-algebra B.

It is also interesting to consider another class of conditional expectations E : A -+ B C_ A for which

there exists a number K > 1 such that the map ( K . E - idA) is a positive element of the C*-algebra

A. Such conditional expectations are called conditional expectations of .finite index, and they have the following property.

193

P r o p o s i t i o n 4.4.1. Let A be a C*-algebra, and let E : A --+ B C_ A be a conditional expectation .[or which the set of fixed points coincides with B. Then there exists a finite number K >_ I such that the map ( K . E - IdA) is positive iff E is exact and the (right) pre-Hilbert B-module {A, E(<-, "}A)} is complete

with respect to the norm IIE(I.,.}A)II~/2, where <a,b}A = a*b for a,b E A.

P r o o f . If the algebra A is complete with respect to the norm IIE(<.,.}A)II~/2, then there exists a

number K such tha t the inequality KIIE(x*x)l I > IIx*xll holds for any x E A. For a E A, c > 0, put x = a ( c + E(a*a)) -1/2. Note that

(c + E(a*a)) -1/2. E(a*a). (c + E(a*a)) -1/2 <_ 1A,

whence the inequality K . 1A >_ (c + E(a*a)) -1/2. a*a. (c + E(a*a)) -~/2 follows. Multiplying both parts

of it by (c + E(a*a)) 1/2, we conclude tha t K . (c + E(a*a)) >_ a*a for all a E A, c > 0. The inverse

s ta tement obviously follows from the inequality IIE(x)tl >/(-1/21[x]1 , which is valid for any x E A. []

Note that unlike the algebraic finiteness of the index, in the case of conditional expectat ion of a

finite index the Hilbert ,nodule {A, E({. , "/A)} can be infinitely generated.

Define

K(E) = inf{K : ( I< Z - i d A ) positively in A }

Let us call K(E) the characteristic number of the conditional expectat ion E.

Let X be a compact Hausdorff space with an action of a group G. Denote the C*-algebra of

G-invariant continuous functions on X by CO(X), and denote the stabilizer of a point x E X by

= (g c C . g x = z } .

D e f i n i t i o n 4.4.2. A continuous action of a group G on X is called uniformly continuous if for each

x E X and for each of its neighborhoods U~ there exists a neighborhood Vx of the point x such that

g(Vz) C_ Ux for each g E Gz.

Note that continuous actions of compact groups satisfy the conditions of the definition.

D e f i n i t i o n 4.4.3. Assume that a group G acts uniformly continuously on a compact Hausdorff space

X in such a manner tha t the length of each orbit @Gx does not exceed some number k E N. Define a

conditional expectat ion Ec " C(X) --~ EG(C(X)) c_ C(X) by the formula

1 ~ f(gaX), (x E X ) . z G ( f ) ( x ) = # ( c / G ) o,G/G

L m m a 4.4.4 ([25]). The conditional expectation EG is well defined.

T h e o r e m 4.4.5 ([25]). Assume that a group G acts uniformly continuously on a locally compact Hausdorff space X so that k := max{@(Gx) : x E X} < +oc . Th, en the characteristic number of the conditional expectation Ec satisfies the equality

K(EG) = k := max # ( G x ) . x c X

P r o o f . Let x E X be an arbitrary point and let kx = @Gx. Then

K E a ( f ) x = k 1 kx" E f(gaX)>_.f(x), gaEG/G~

where f is an arb i t rary nonnegative function in C(X). Assmne tha t K(Ec) < k and choose a point x such that k~ > K(EG). Then it is possible to choose a neighborhood U~ of the point x, small enough

to satisfy the condit ion giU.~ N gjUx = 0 (i ~ j) for the set {gl = 1, g2, . . . ,g ,~} = G/Gx. Let f be a

194

continuous nonnegative function with the support contained inside Ux. Then

K(EG)EG(f)x = K(EG) r(gax) < f (x ) . gaEG/Gx

The contradiction with the definition of K(EG) completes the proof. []

T h e o r e m 4.4.6. Let X be a compact Hausdorff space and let G be a group uniformly continuously acting on X . If all orbits of the action of G have the same finite number of points, then the conditional expectation

1 E ( f ) ( x ) - # (Gx) ~ f(gix)

gic(c/cx)~

is well defined on C(X), and the Hilbe t CG(X)-module {C(X),E((-, /)} is finitely generated and pro- jective.

P r o o f . The idea of the proof is contained in [69], but beforehand two technical lemmas are required.

L e m m a 4.4.7. Let X be a compact Hausdorff space with a uniformly continuous action of a group G,

and let all orbits contain an equal finite number of points. Then for any point x E X and for any element g E Gx, one can find an open neighborhood Ux of a point x on which g acts identically.

P r o o f . Let us denote the length of orbits ~(Gx) by n. Let x l , . . . , X n E X be the orbit of the

point x and let hi C G be such elements that hix = xi. Choose go E Gx and assume tha t each

neighborhood U~ of the point x contains some point y E Ux such that goy ~ y. Fix neighborhoods U~

of the points xi satisfying the condition Ux~ N U~j = 0 for i ~ j . Then we can find a neighborhood

V~ c_ U~ of the point x such tha t hi(Vx) C_ U:~ i . Since the group G acts nnifornfly continuously, it

is possible to find a neighborhood Wx C_ Vx of the point x such that g(W~) C_ Vx for each g E G~.

If y E Wx and goY ~ y, then the orbit Gy of this point includes not less than n + 1 different points

{hiy E Uxi : i = 1 , . . . , n} U {y, goY E Vx}. The contradiction obtained proves the lemma. []

L e m m a 4.4.8. Under the assumption of Lemma 4.4.7, for any point x E X one can find a neighborhood V~ of this point such that the action of the subgroup G~ on V, is the identity mapping.

P r o o f . We should show that a neighborhood U, of Lemma 4.4.7 can be chosen for all g E G~ simultaneously. For each g E Gx, put

Ux(g) = {y ~ X : gy = y}.

Suppose the contrary, i.e., tha t the set ['1 Ux(g) does not contain any neighborhood of the point x. This gEGx

means tha t any neighborhood Ux of the point x contains some point z such tha t for some g~ E Gx we have

gzZ ~ z. Consider a neighborhood Vx of the point x and neighborhoods {Unix} for fixed representatives

{hi ~ e} E G of cosets in G/G~ such that their intersections are empty and h~(V~) C Uhi~. As in the

proof of Lemma 4.4.7, we find a neighborhood Wx C_ Vx of the point x such tha t g(Wx) C_ Vx for each

g E Gx. Put Ux = [J G(VVx) c_ Vx. It is a G~-invariant open neighborhood of the point x ~ X. The gCG~

assumption gzZ ~ z for some z E Ux, g~ C Gx means tha t the orbit of the point z consists of not less than n + 1 points. []

Let U~ C V~ be a neighborhood of the point x such tha t the action of G~ on V~ is an identity

mapping. Then one can find a function f~ E C(X) such that snppf~ c V~ and fxiu~ = 1. For each g E G, one has either (gV~) N Vx = 0 or gVx - Vx; therefore,

{ og(f~) 2 if gx = x, c~g(f,~), fx = 0 if gx ~ x,

195

where a denotes the action of G on functions, Ctg(f)(x) = f (g- lx) . Let { U ~ , . . . ,U~,} be a finite

covering of the space X by sets of the above form. P u t

k

v = E f~ 2 1, Ui = v-1/2(fx~)l/2 E C(X). i = 1

Note tha t if we take one element gi in each coset G/G~, then by Lemma 4.4.4 the map

n

1 Z f(g x ) E G ( f ) ( x ) - #(Gx) i=l

is well defined for all x E X, f E C(X), and it is a condit ional expectation on C(X). Moreover, for each

function f E C(X), one has

z =f, i : 1 i----1 j : l

hence the set { u l , . . . ,u~} is a basis of the Hilbert Ca(X)-module {C(X),EG({.,.})}. Therefore, this

Hilbert module is finitely generated and projective. []

Theo rem 4.4.6 generalizes the results of [69] and shows that if all orbits consist of an equal finite

number of points, then the corresponding conditional expectat ion is of algebraically finite index. From

the finite generatedness and the projeetivity, it follows tha t the Hilbert module A = {C(X), E(( . , .})} is

auto-dual. In the case of a finite index (when K(EG) < oc and the pre-Hilbert module A is complete),

we cannot expect that this module is self-dual. However, sometimes this module is reflexive, i.e., A" = A,

where A' is the dual Banach c G ( X ) - m o d u l e of bounded cG(X) -homomorph i sms from A into c a ( x ) .

T h e o r e m 4.4.9 ([25]). Let the group a acts uniformly continuously on a compact Hausdorff space X . Assume that all orbits consist of not more than n points, and the number of point for which the length of their orbit is less than n is finite. Then the Hilbert CG(X)-module {C(X) , EG({-, "})} is reflexive.

P r o o f . Describe at first the dual Banach Ca(X)-module A'. Let x l , . . . ,x,~ be the points with orbits

shorter t han n,. It is possible to choose open neighborhoods U1, .. �9 , U,~ of these points in such a way tha t

each neighborhood Ui is invariant with respect to tile act ion of the subgroup Gx~ and if for some h E G

one has hxi = Xy, then hUi = Uj. Denote by Y the G-invariant compact set X\(U1U...UU,~). Let F C A'

be a cG(X) -va lued functional on the module A. Consider its restriction on the Hilbert CG(Y)-module

{C(Y),EG({.,.})}. For a function g E C(Y), we take its extension ~ E C(X) and define Fly by the

equality FIy(g ) = F(~) lz . This definition does not depend on the choice of an extension ~. If Y' D Y

is also a compact G-invariant subspace not containing the points x l , . . . ,Xm, then (Fly,)ly = Fly. Since the orbit of each point of the set Y has constant length, by Theorem 4.4.6 the CG(Y)-module

{ C(Y), EG( (.,. }) } is finitely generated and projective, therefore auto-dual. Denote by C ( X \ { x l, �9 �9 �9 , x ~ }) the set of continuous functions on the noncompact space X \ { x l , . . . , x,~}. Tile restriction on this space

defines a map

A' > C(X \ { x ~ , . . . , x , d ) . (15)

It is easy to verify that the map (15) is a monomorphism.

Let us s tudy local properties of functionals on A' close to the points xl , .. �9 , x~. Let Xo be one of

these points. It has a neighborhood Ux o such that if gxo = Xo, then gUxo = Uxo. The group Gx o contains a normal subgroup Go of the elements which do not move points from the neighborhood U~ o. Choose a representative g{ in each coset G/Gxo. Then outside the point x0 the action of the functional F E A'

196

can be wri t ten as n

1 E F*(gix) . f (gix) , (16) F( f ) ( x ) = n i = 1

and this action can be continuously extended to the point x0. Let x0 = x ~ x l , . . . , x k-~ be the orbit of the point Xo. Then it is possible to write the sum (16) in the form

Passing to the limit (which exists by assumption), we obtain

x----~ Xo n j=O i:gixo=xJ

hence the limit (for f = 1 e C(X))

l lira F(g x) n x - + x o

i:gixo~X j

exists for any x E X \ {x~, . . . ,x,~}. Recall that the function F(x) is defined only outside of the point x0. If we want the action F on the Hilbert Cc(Y)-module {C(Y), Ea((-, .})} to be of the form (16) on the whole X, it is necessary to define the function F(x) at the point x0 by the equality

1 lira E F(gix) (17) F( o) =

i:giXo~xJ

To complete the proof we need the following lemma.

L e m m a 4.4 .10. The module A' is isomorphic to the module of all bounded functions F(x) on X which are continuous on X \ {Xl , . . . ,Xm}, and satisfy the condition (17).

P r o o f . We need to show that the image of the monomorphism (15) consists of bounded functions. Suppose the contrary. Then there exists a point 2 such that ]F(.~)t > n . [[F[[, where [[F[[ is the norm of F in the dual Banach Ca(X) -modu le A'. Moreover, it is possible to choose a neighborhood U~ of the point 2 so tha t U~ n giU~. = fi for those elements of the group G for which gi2 r 2. Consider a function f E C ( X ) such that f(2) = 1 and supp f C U~. Then it follows from equality (16) that

F(f)(2~) = 1 r* (2 ) . f(c?), ?7

and the inequality

-llF*( )I �9 If( )l = -1 IF( )l > IIFII , lF('f)(x)[ = n n

giving a contradiction. []

Now, having the above description of the dual module A', it is possible to describe the bidual module A ' . Since there exists the canonical inclusion A" C A', and the immr product on A can be in a natural way extended to an inner product on A" (see Theorem 4.1.4), making it a Hilbert module, it is sufficient to verify on which functions from A' it is possible to extend the Cc(X)-va lued inner product. Consider a function F from A' N A ' . Adding to it (if necessary) a continuous flmction from A, we can assume that F(Xo) = 0. Then the Cc(X)-va lued inner product of F by itself is an element of Ca(X) , having the form

1 ~ [F(gcr)i 2 (18) (F,F} (x) = E(IF(x)] 2) = ',-t i:giXO~XJ

197

for all x E X. But since (F, F} (x0) = 0, it follows from the assumption F E A" that

lim F(g~x) = 0 X ~ + X 0

for each summand of equality (18). Therefore, we obtain from (17) that the function F is continuous at the point x0, hence the module A is reflexive. []

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