Mathematical Notes, Vol. 68, No. 3, ~000

On C*-Algebras Related to Asymptotic Homomorphisms

V. M . M a n u i l o v UDC 517

ABSTRACT. We study the C*-algebra~ related to Mishchenko's version of asymptotic homomorphisms. In particular, we show that their different versions are weakly homotopy equivalent but not isomorphic to each other. We also present continuous versions of these algebras.

KEY WORDS: compact operator, C*-algebra, asymptotic homomorphi-~m, weak homotopy equivalence.

I n t r o d u c t i o n

Let Mk , k E N , denote the k • k matrix C*-a lgebra . To make notat ion uniform, we will wr i t e Mao for the C*-algebra K of compact operators. Le t v = {v(1), v(2), . . . } be a function from N to N U { c r i.e., v is a sequence whose terms v( i ) , i E N, are ei ther integers or infinity. We suppose t h a t t h e sequences v = {v(i)} are monotonically nondecreasing and ei ther stabilize (there exists lim~_,r v( i ) ) or approach infinity as i -+ ~ .

Consider the quot ien t C*-algebra

II ~=I i=I

which is the target C'*-algebra for discrete asymptotic homomorphisms [1, 2], i.e., any discrete asymp- totic homomorphism from a C*-algebra A to matrix algebras or to compact sets gives a genuine .- homomorphism from A to this target algebra. In [3] Mishchenko suggested considering another version of discrete asymptotic homomorphisms, namely, adding the following property: if ~i : A --~ B, i E N, is a discrete asymptotic homomorphism, then it must also satisfy

lira II~,+l(a) - ~ , ( a ) l l = 0 i - -~oo

for any a E A . T h e corresponding target C*-algebras were constructed in [4] as follows. Since any sequence v is nondecreasing, there is a natural inclusion M~(,) C M~(~+I), hence the r ight shif t

0~: ( m l , m 2 , . . . ) ~-~ (0, m l , rn2, . . . ) , where

is well defined on b o t h

m = (T/~l,m2, . . . ) ~ HM~,(i), (1) i= l

c o c ~ c ~

IIM-(,) =d i=I i=l i=I

T h e map (~ is an endomorph i sm of these C*-algebras, and it is easy to see tha t the (~-invariant subset in

II i . ( , ) /0 M.(,) i=l i=l

is a C*-algebra, too. We denote this a-invariant C*-a lgebra by Q(u) . If the sequence v stabilizes,

~ n u(~) = ~ , ~ S N U { ~ } , i - -+r

Translated from Matema~cheskie ZametId, Vol. 68, No. 3, pp. 377-384, September, 2000. Original article submitted September 7, 1999.

326 0001--4346/2000/6834-0326525.00 (~2000 Kluwer Academic/Plenum Publishers

we denote v by n (or by oo if n = oo). In this case we write Q(n) instead of Q(v). The C*-algebra Q(v) can be described as the set B(y) of all sequences (ral, m 2 , . . . ) of matrices rr~ E M~(i) such that

l lm IIm +t - mdl = 0 i-+oo

modulo the sequences converging to zero. The set B(v) is a C*-algebra too, so Q(v) is the quotient C*- algebra of B(v) modulo the ideal I(v) of sequences converging to zero. Moreover, the ideal I(v) is an essential ideal in B(v), so B(v) lies in the multiplier C*-algebra M ( I ( v ) ) .

Note that though the C* -algebras Q(v) consists only of ~-invariant elements of 1-Ii M~(i)/(~i My(1), one should not think that Q(v) are smaller than the latter algebras. For example, consider Q(c~) and let r = r(n) be a sequence of increasing integers with

+ 1) - r ( i ) ) = oo . (2)

Then it is easy to see that the sequences (mr, m 2 , . . . ) with mr(i) = O, i E N, form an ideal Ir of Q ( o o ) and the corresponding quotient C*-algebra Q(oo)/k can be mapped to rIi M o o / ~ i Moo by assigning

( m r , m 2 , . . . ) ~-~ ( m r ( l ) , m r ( 2 ) , �9 �9 �9 ) .

Using (2), one can lift any element of 1-Ii Moo /~ i Moo by linear interpolation, hence this map is ephnor- phic.

Note that the C*-algebra Q(1) is commutative, and its speetznlm is the Higson corona, i.e., the quotient space of the Stone-Cech corona of f in \ l~l modulo the standard action of Z. It is easy to see that the C* -algebra Q(v) is unital if and only if v = n for some finite n . In this case one has O(n) = Q(1) | Mn. Otherwise, Q(v) contalns the tensor product Q(1) | Moo as a proper subalgebra. Indeed, the lifting of Q(1) | Moo to B(v) consists of ,miformly vanishing sequences m = (ml , m 2 , . . . ) 6 B(t 0 . This means that for any such m and for any e > 0 there exists a number j such tha t

[ [ ( m t , r r t 2 , . . . ) -- (pjmtpj,pjm2pj,...)][ < ~,

where pj E Moo denotes the projection onto the first j coordinates. For any v approaching infinity, it is easy to find an element of B(v), which is not uniformly vanishing. To do so consider a sequence {hi} with

lira (ni+t -- n/) ---- 00, i--+oo

and a nonzero element A = (At, A2, . . . ) e B(1) such that An, = 0 for all i e N. Let X[m,n,+,] be the characteristic function for the set In/, n~+t] C N. Let (ki) be another sequence approaching infinity, and let ei E Moo be the projection onto the i th coordinate. Then the element ~]~i X[r~,m+,] Aeu, does not lie in B(1) | Moo.

1. W e a k h o m o t o p y equ iva lence o f Q(v)

The C*-algebras Q(v) are quotient algebras modulo an essential ideal, so it is natural to consider the corresponding weaker notion of homotopy for ,-homomorphisms into corona algebras. Recall that a function g = g(t): [0, 1] -+ B(v) is called continuous wi~h respect to the strict topology [5] if the functions g(t)k(t) and k(t)g(t) are continuous in C([0, 1], I(v)) for any continuous function

k = k(t) "E C([0, 1], I(v)).

We denote by Cstr([0, 1], B(v)) the C*-algebra of all continuous functions with respect to the strict topology. The quotient maps

Cst~([0, 1], S(u)) --+ Cstr([0, 1], B(v))/C([O, 11, I(v)) , B(u) --+ Q(v) are denoted by q.

Let fo, fl: A --~ Q(v) be two .-homomorphisms. They are called homotopic (as maps into quotient algebra) if there exists a family of maps (not necessarily *-homomorphisms) F~ : A --~ B(v) such that Ft(a) is continuous with respect to the strict topology for any a E A, the composition qoFt: A --+ C([0, 1], Q(v)) is a *-homomorphism and q o Fi coincides with fi , i = O, 1.

Since we are dealing with nonseparable C*-algebras, we must consider an even weaker homotopy equivalence. Let A, B be C*-algebras. By [A, B] we denote the set of all . -homomorphisms from A to B.

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Defini t ion 1. We call A and B weakly homotopy equivalent if for any separable C*-algebra C there are natural (with respect to C) maps

~ = r IV, A] -+ [C,B] and r 1 6 2 [C,B] -+ [C,A]

which preserve homotopy of .-homomorphisms and for any f : C --+ A and any g: C --+ B the composi- tions (r o r and (~bo r are homotopic to / and g, respectively.

L e m m a 1. If A and B are weakly homotopy equivalent separable C* -algebras, then they are homotopy equivalent.

Proof. Put / = CA(idA): A -+ B and g = CB(idB): B -+ A. By /* we denote the map [B,X] -~ [.4, X] generated by the . - homomorph i sm/ . Then one has / = /* ( idB) and the naturality of r implies that

CAq~A(idA) = CA/*(idB) = / * r = / * g = g o / ,

so that g o / is homotopic to ida by definition. In the same way one can show that / o g is homotopic to idB. []

Though this weak homotopy equivalence is very weak indeed, it preserves K-theory.

L e m m a 2. If C*-algebras A and B are weakly homotopy equivalent, then K . (A) ~- K . (B) .

Proof. It is easy to see that weak homotopy equivalence of C*-algebras A and B implies the same equivalence of C*-algebras M ~ | and M ~ | The groups K0 and/{1 are defined by homotopy classes of *-homomorphisms into these algebras of the separable C*-algebras C and C0(R), respectively. []

The K-groups of the C*-algebras Q(u) were calculated in [4].

P ropos i t i on 1 (see [4]). For any u one has Ko(Q(t,)) = Z and KI(Q(u)) = O. The group Ko(Q(u)) is generated by the sequence (el, el, e l , . . . ), where el is the projection onto the first coordinate.

Propos i t i on 2. Let Vl and v2 be two sequences approaching infinity, but such that uj ~ oo , j = 1, 2. Then the C*-algebras Q(ul) and Q(t,2) are homotopy, equivalent.

Proof. Let uz be a sequence majorizing both ul and v2. The transitivity of homotopy equivalence shows that we need only prove homotopy equivalence for Q(ul) and Q(u3). As t,l(i) < v3(i) for any i 6 N, there exists a natural inclusion / : Q(ul) -+ Q(v3). To define a map g in the other direction, consider an element m = (ml, m2,. . . ) 6 B(u3), define numbers r = r(i) by / / l ( i - - 1) < u3(r) _< Ul(i), and put (g(m)), = mr. Then g(m) 6 B(t~l) and g(m) 6 I(ul) whenever m 6 I(v3), so the map g: Q(u3) -+ Q(t,1) is well defined. Then the composition g o / : Q(ul) -+ Q(t,1) is reduced to ren-mbering: a sequence m = (ml, m2, . . . ) 6 B(vl) is mapped to the sequence

m l , . . . , . . . ). (3) J �9 �9

81 ~ l l nes 82 ~ m .

We will now construct the homotopy F~ : Q(Ul) -+ C, tr([0, I], B(ul)) connecting g o f with idq(u~). For any a chooee a repr ntati of the form (3). For t [0, I/2] de ne F,(a) by connecting (by a linear path) the sequence (3) with the sequence

F1/2(a) = (ml, ~/%2, " ' " , ~ 2 , m 3 , " ' " , m 3 , . . . ) . (4 ) �9 J '~, ~'

sl+s2"-~l t imes sz ~maes

For t 6 [1/2, 3/4] connect the sequence (4) with the sequence

f 3 /4 ( a ) =( ml ,m2 , m 3 , . . . , m 3 , m 4 , . . . , m 4 , . . . )

81+82-~-s;--I t imes s4 ~mes

etc. Finally put Fi(a) -- (ral, m2 , . . . ) for t -- i . It is easy to see that F~ (a) is continuous with respect to the strict topology (one must check this only at t = 1), that q o Ft(a) does not depend on choice of a representative m, and that Ft(-) is a .-homomorphism modulo C([0, 1], I (u l ) ) , which gives the necessary homotopy. A similar homotopy connects / o g with idQ(~3). []

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P r o p o s i t i o n 3. Let A be a separable C* -algebra and let f : A -+ Q(oo) be a *-homomorphism. Then there exists a finite sequence u such that f (A) C Q(1/) c Q(oo).

Proof . Fix a dense sequence {ai}, i E N, in A and a monotone decreasing sequence of numbers

en > O, n e N. Let m (i) = (m~ i) , taxi),... ) E B(c~) be liftings for f (a i ) . Then the numbers u(n) can be chosen to satisfy

II. < e . (5)

for all 1 <_ i, j <_ n , where P~(n) E Moo are projections onto the first u(n) coordinates. It follows from (5) that for any i the sequences

(ra~ i) , m~(~),... ) and (p~o)m~i)p~(t), p~<~)m~(~)p~(2),... )

give the same element in Q(oo) and the last sequence lies in B(v) . As Q(u) C Q(oo) is a C*-subalgebra, f is continuous, and f(a/) E Q(u), so f (A) C Q(u) too. []

Propositions 2 and 3 imply

T h e o r e m 1. I f 1/ approaches infinity, then all C* .algebras Q(1/) are weakly homotopy equivalent to each other.

2. Growth functions and Q(u)

Now we intend to show that the C*-algebras Q(1/) can be different for different growth of u. We say that 1/has polynomial growth if there exist numbers 0 < C1 < C2 and n >_ 1 such that

ot i" <_ z,(i) <_ c2i lira u(i + 1) - uCi) = O. i i n

It has exponential growth ff there exists a number a > 1 such that for all i one has u(i + 1) _> au(i).

T h e o r e m 2. Let ~1 have polynomial grourth, and let ~2 have exponential growth. 1) There exists a nontrivial Q(1)-valued trace on Q(vl) . 2) There /s no nontrivial trace on Q(u2).

Proof . 1) We define a trace on the C*-algebra B(ut) and then show that it vanishes on the ideal I(v1). Let m - (rnl, ms, . . . ) e B(u l ) , m~ E Mul(i ) . Put

) \ 1/1(1) ' 1/1(2) ' 121(3) ' " " '

where tri is the standard trace on Mi normalized by tr~(li) = i where l i is the identity of Mi. I f llmll = 1, then tr~(0(m~ ) <_ ut(i) , so Ilr(m)tl < 1. One h~s also

1 _< lim +l - m, II + + 1) -+ 0,

hence the image of ~- lies in B(1) . Finally the property ~'(ab) = v(ba) for a, b E B(ul) is obvious. So r is a B(1)-valued trace on B(ut ) . It is also obvious that T(I(ut)) C I (1 ) , so r is well defined on the quotient algebra Q(1/t). It remains to show that r is nontrivial. Let

u1(i)- 1 1 } m~--diag 1, vl(i) ' ' " ' v1(i)

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b e diagonal matr ices in M~,~( 0 . Then

I1-~+~--~11 = I/1(i "]- 1) -- Ul(i) 1 Ul( i+ 1) -- UI(i) l~l(~ -']- 1) < 0 1 ~" "-')" 0 ,

so the sequence m = (m~, m 2 , . . . ) lies in B(~I ) , hence it defines an element in Q(~I) and one has

~ ( ~ ) = \ F~f(~) ' " 2~(~) ' . . . .

Since

t h e trace ~- is nontrivial.

~1(~) + ~ lira = - ~ 0 ,

, 2v~(i) 2

2) For convenience of notation, we assume that a = 2 and that v2(i + 1) ~ 2v2(i). We also assume tha t all M~ are embedded in the C*-algebra of bounded operators on a Hilbert space H wi th a fixed basis ~1,~2, . . . . Let u l , u 2 be isometries on H such tha t Ul(~k) = ~2k-1, u2(~k) = ~2~, k E N. Let m = (ml , m2, . . . ) E B(v2) . Then one can define an element m ~ m E B(v2) as follows. P u t

(m E) m)~ = O, (m (B m),+~ = u~m~u~ + u2m, u~ for i > 1.

Then d im(m ~ m)~+l = 2z/2(i) _~ z/2(i + 1), and it is easy to see that

II(m �9 m)~+l - (-~ �9 m),ll -* 0.

Since [[(mSm)~[[ --~ 0 whenever m E I(v2) , it follows tha t the map m ~+ ( m $ m ) defines a homomorph i sm Q(z/2) --+ Q(v2). In a slm~lar way, one can define a direct sum of 4, 8, etc., snmmands.

Now suppose tha t there exists a trace v on Q(v2) wi th the property 7"(ab) = r(ba) for a , b E Q(v2). Then r extends to a trace on B(v2) and we still denote this extension by r . Let m E B(~2) b e a positive element with [Imll = 1 and ~-(m) # 0. One has m (3 m = m (1) + m (2) , where

$ $ �9 * * m (2) : (0 , U2ml~ t~ , U2m2"U2, u 2 m 3 ~ 2 , . ) . m (1) = ( 0 , U l m l U l , ~ t l m 2 U l , U l m Z U l , . . . ) , . .

Since all n'~ are positive and of dimension v( i ) , the sequences

aj = (0, Uj(ml) 1/2, U j ( m 2 ) 1 /2 , U j ( m 3 ) l / 2 , . . . ) ,

~j = (0, (~/'/,1)1/2~/,;, (~t7,2)1/2~/,;, (~/~3)1/2~/,;,...), j = 1, 2, are well defined as elements of the algebra B(v2) �9 One has

.(~(~)) = ~ ( ~ ) = . ( ~ ) = .(~(m)),

where ~ is the s ~ (1), ~ .(.~U)) = ~(~), as ~ = ~ ( m ) m o d ~ o Z(~2). Therefore, . ( m ~ . ~ ) = 2 . ( m ) . In the same way, we can show tha t for any k one has

Fa(Bm(B . . .E )m E B(u2)

2 ~ t imes

and

2 k times

O n the other hand, it is easy to see that

2 k t imes

hence r ~ r a ~ m ~ . . - ~ m ) < 1 ,

2 ~ t imes

and the contradict ion between (6) and (7) shows tha t v(ra) = 0, ending the proof.

Recall tha t a C*-algebra A is called stable if A ~- A | Moo.

(o)

(7)

[]

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Corol lary 1. If v is of polynomial growth (or smaller), then the C* -algebra Q(v) is not stable. If is of exponential growth, then Q(u) is stable.

Corol lary 2. Let v be of exponential growth. Let v be the ~race constructed in Theorem 2, and let a fi Q(1) | M~o c Q(v). Then v(a) = 0.

Let u be of polynomial growth. Consider a C*-subalgebra I~ C Q(v) generated by all positive elements a w i t h T(a) = 0 . S ince

= T(al/ l/2) < IIbll ( ) = 0

for any b E Q(v), It follows that Ir is an ideal in Q(v). By Lemma 2 and Corollary 2 the K-groups of I~ coincide with those of Q(v) and the inclusion Ir C Q(v) induces an isomorphism of their K-groups. Hence the six-term exact sequence shows that the quotient C*-algebra Q(v) / I , id K-contractible.

3. A cont inuous vers ion for Q(v) Let Cb([0, c~),B) (resp. C0([0, c~), B)) be the C*-algebra of botmded continuous (resp. vanlqhlng at

infinity) functions on the half-line taking values in a C*-algebra B . The quotient algebra

Q b ( S ) -~ Cb([0 , (X~), B)/Co([O, c~), B)

is generally used for describing continuous asymptotic homomorphisms. But this algebra is too big and one must often perform reparametrization in order to obtain slowly varying functions. First consider the case B = Moo. There exists a natural inclusion

Q(e~) -+ Qb(Moo), (8)

given by the natural inclusion N C [0, co) and by linear interpolation from sequences of compact operators to the Moo-valued functions. This inclusion is obviously not epimorphic.

For a function f( t) e Cb([0, ~ ) , Moo) and for an interval [a,b] C [0, ~ ) , put

Vara,b/= sup IIf(t2)--Y(tl)ll tx , tze[a,b]

and in Cb([O, c~), Moo) consider the subset of functions with variation vanishing at infinity

C~b ([0, oo),Moo ) = ~f(t) e Cb(tO, oo),M~) : lira Vara+t,b+t f = 0~. t %-.~ oo )

This subset is a C*-subalgebra and it does not depend on the choice of a and b. Denote by Q~'(Moo) the quotient C*-algebra C~b([0 , c~),M~)/Co([O, oo),Moo).

L e m m a 3. The C*-algebras Q(c~) and Qy(Moo) are isomorphic.

Proof. The image of Q(e~) under the map (8), obviously, lies in Qy(Moo). To get the map in the other direction, one must assign to a function f(t) e C~b ([0, c~), Me~) representing an element of Qy(Moo) the sequence {f(n)} of its values at integer points. One can directly check that these . - homomorphisms are inverse to each other. []

P ropos i t ion 4. The C*-algebras Qb(M~) and Q~"(Moo) are weakly homotopy equivalent.

Proof. It is easy to see that for any .-homomorphism of a separable C*-algebra A into Qb(MeQ) one can fred a reparametrization s = s(t) such that, after passing from t to s, the image of A lies in Q~r(Moo). []

To imitate the continuous version for general v, one must consider telescopes. Let

T(v) = {f( t ) e Cb([O , c~), M~) : f(t) E My(i) for t e [ i , i + 1)}

be the telescope C*-algebra. Put

Co([O, cx~), v) = {f(t) E T(v) : lira IlY(t)ll = O}

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Lemma 4. The C*-algebra~ Q(v) arid C~b([0 , oo), v)/C0([0, oo), v) are isomorphic.

This reseaxch was supported in part by the Russian Foundation for Basic Research under grant No. 99- 01-01201 and by INTAS under grant No. 96-1099.

References

1. A. Connee and N. Higson, "D~ormations, morphismes asymptotiques et K-th~orie bivaxiante," C. R. Acad. Sci. Paris S~r. l, 311, 101-106 (1990).

2. T. A. Loring, "Almost multiplicative maps between C*-algebras," in: Operator Algebras and Quantum Field Theory, Intexnat. Press (1997), pp. 111-122.

3. A. S. Mishchenko and Noor Mohammad, "Asymptotic representations of discrete groups," in: Lie Groups and Lie Algebras. Their Representations, Generali~tions and Applications, Ser. Mathematics and Its Applications, Vol. 433, Kluwer Acad. PubL, Dordrecht (1998), pp. 299-312 . . . .

4. V. M. Manuilov and A. S. Mishchenko, "Asymptotic and Fredholm representations of discrete groups," Mat. Sb. [Russian Ac~l. Sci. Sb. Math.], 189, No. 10, 53--72 (1998).

5. G. K. Pedersen, C* -Algebras and Their Automorphism Groups, Acad. Press, London-New York-Sa~ Francisco (1979).

M. V. LOMONOSOV MOSCOW STATE UNIVERSITY E-mai/: manuilovQmech.math.msu.su

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