Integral Equations and Operator Theory Vol. 9 (1986)

0378-620X/86/050679-1551.50+0.20/0 Q 1986 Birkh[user Verlag, Basel

SPECTRAL PROJECTIONS OF L 1 OPERATORS

IN TYPE IIIk VON NEUMANN ALGEBRAS

Victor Kaftal and Richard Mercer

The LP spaces of a type III~ factor are represented in

terms of the discrete decomposition of the factor. The LP

operators are characterized through the notion of wandering

projections, and those projections that are spectral projections

of positive LP operators are completely characterized. Several special cases are considered, including a description of those

operators that are bounded parts of a positive LP operator.

1: In~rgduction

If M is a semifinite factor with trace T, then a

projection P in M is finite if and only if for some operator A in

L1 (M,T) +, P = ~(~,~) (A), the spectral projection of A on the

interval (S,~) . If M is a type III factor then the only finite

projection is zero but there is a large class of spectral

projections of L 1 operators (belonging not to M but to the crossed

product of M by the modular action of a fixed weight). Can these

projections be used to introduce a notion of finiteness in M? In

general, can some notion of finiteness in M be inherited from the

semifiniteness of this crossed product? This last question was investigated for the case of type III~ and type III 0 factors in

[2,3]. A discrete decomposition {N, 8,~} of M was used instead of

a continuous one, and a class J, analogous to the class of compact

operators in semifinite algebras, was defined as the hereditary

C*-algebra generated by the finite projections of N (identified

680 Kaltal and Mercer

with its image in M) . A natural question thus arises: if we

represent LP(M) as LP 0 , an isomorphic space of unbounded

measurable operators affiliated with N (as indicated by Haagerup

in [i]), and we take the hereditary C*-algebra generated by their

spectral projections (on intervals (8,~) for s > 0), do we again

obtain J? The answer is yes (Proposition 3.13), indicating that J

may provide a reasonable extension of the notion of finiteness to type III% factors.

Our main objective in this paper is to study the spectral

projections of LP 0 operators, which is of independent interest as

well as shedding light on the question posed in the previous

paragraph. This is done in Section 3. In fact, we provide a

complete characterization of those projections in N which are the

spectral projections of LP 0 operators (Theorem 3.5) and analyze

several special cases (Propositions 3.10-3.12). In preparation

for these results we provide a useful characterization of LP 0

operators (Proposition 3.3). These results provide insight into

the structure of a discrete decomposition {N,8, Z}. In Section 2, we

define the discrete LP 0 spaces and discuss their relationship with

the usual LP spaces in preparation for Section 3.

2; Preliminaries on Discrete LP Spaces

Let M be avon Neumann algebra, and let ~ be a fixed

faithful semifinite normal (f.s.n.) weight on M. Let N =

9{ (M,(~, ~) be the crossed product of M by the modular action

associated with 9. Then N is a semifinite algebra and there is a

relatively invariant trace Z, i.e. a f.s.n, trace scaled by the

dual action @ of G~: ~~ = e-St, s ~ R. Let ~ be the set of

T-measurable operators, the closed densely defined operators A

affiliated with N such that <(~(~,~)(IAi)) < ~ for some ~ > 0.

Then Haagerup defined the LP spaces associated with M for

1 _< p _< oo as

(I) LP(M) = {A ( ~ 1 8s(A) = e-S/PA, s ~ ~}

L~176 = {A ~ ~ i @s (A) = A, s ~ ~}

Kaftal and Mercer 681

N @ is defined to be {A ~ N I @s(A) = A, s ~ ~}. Recall that

L~(M) = N 8 = ~(M) where ~ is the canonical embedding of M into N,

whereas for p < ~ all nonzero elements of LP(M) are unbounded

operators. Recall also that for all positive ~ ~ M, the Radon-

Nikodym derivative A~ of the dual weight ~ by the trace r belongs

to LI(M) (here ~ = ~(A~.) ), and the map ~ ~ A~ extends to an

isomorphism between M, and LI(M). The positive linear functional

tr on LI(M) given by tr A~ = ~(i) defines a norm on LP(M) :

(2) i OAf Ip = (tr(IAIP)) I/p = (Z(X(l, ~) (JAJP))) I/p

Also, tr implements the duality between LP(M) and Lq(M) when i/p

+ i/q = i. Recall finally that the LP spaces on M associated with

different f.s.n, weights (or equivalently, with different

continuous decompositions of M) are isomorphic as Banach spaces.

For details see [i] and [i0].

Assume now that M is a separably operating type III%

factor; then instead of a continuous decomposition we can use a

discrete one and we can express the LP spaces in terms of it.

This has been stated without proof by Haagerup in [i, Section 3].

For the reader's convenience we are going to present here a proof

of the equivalence of the two formulations following a slightly

different approach.

Assume that the f.s.n, weight ~ is periodic with period T

= -in ~, so O~T = i. By Takesaki's analysis [9, Theorem 10.6],

there is a type II~ factor N O operating on the Hilbert space H0,

an automorphism 80 of NO, and a f.s.n, trace Z0 on N O such that

(3)

(4)

(5)

(6)

(7)

Z0~80 = %~0

9~(N0,80,Z) = ~(N, 8, S>_-- M

N = N 0 • L~(0,T) (which we identify with L~((0,T) ,N O ) )

(A) = [o T e-tz 0 (A(t)) dt for A 6 N +

@r(A) (t) =80n+l(A(t-s+T)) for t ~ [0,s)

@0n(A(t-s)) for t ( [s,T)

where r = s + nT, s ~ [0,T) and A ~ N.

As a consequence of (6) we obtain

(8) ~(B| = T~0 (B) for all B 6 ~0 +,

multiplication by e t on L2(0,T) .

where Q is

682 Kaftal and Mercer

DEFINITION 2.1: ("discrete" LP spaces [I, Section 3])

LP 0 = {B 6 ~0 I 80(B) = kl/PB}, 1 < p < ~.

L~0 = {B ~ ~0 I 80 (B) = B}.

i iBI Ip = ~0(IB1P X(~,I] (IBLP))I/P for B ~ LP0, 1 < p < ~.

It is easy to see that for p < ~ the LP 0 are linear

subspaces of n O spanned by their positive parts and that the only

bounded operator in LP 0 is 0. We shall obtain from the next

proposition that lIB1 ip < ~ for all B ~ LP 0, that 11"i Ip is a norm

and LP 0 is a Banach space isomorphic to LP(M) . For p = ~ we have

(9) L~0 = M 0 = {B ~ N O i 80(B) = B}

as in the case of L~(M) . An operator B with polar decomposition

UIBI belongs to LP 0 if and only if U 6 M 0 and IBJP ~ LI0 Since

80 is integrable [7, Prop.23.9], it is dominant [7, 20.10, 23.9]

and thus M 0 is isomorphic to ~(N0,80,Z) andhence to M and L~(M) .

This can also be obtained directly by the same technique used in

Proposition 2.3 to follow. LFA~MA 2.2: (i) Let A be a positive operator affiliated with

N such that 00 (A) = ~A. Then ~0 (~(E) (A)) = 2C(A-IE) (A) for any

Borel set E c ~. (ii)If B 6 (LIo)+, then ~O(2~(An, o=) (B)) =

l l-n (I-A) -I~ 0 (Z(A, I] (B)).

PROOF: (i) A = [~y dX(-~, Y) (A), so 80(A) =

By the uniqueness of the spectral resolution, 00(X(-~,~) (A)) =

%(_~,~-iy) (A) for each y, and this result may be extended to Borel

sets. (ii) By part (i) we have 80n(%(k,l] (B)) = %(~-n+l,~-n] (B)

for all n. Thus 80n(~(~,l] (B)) are mutually orthogonal projections

and therefore ~0 (%(~n,~) (B)) = ~0 (%(kn-k,~n-k-l] (B)) =

~l-n ~%k~0(%(~,i ] (B)) = ~l-n(~-l)-l~0(%(~,l ] (B)) . H

This Lemma implies that ~0(%(~, ~) (B)) < ~ for all ~ < ~,

and also that ~0(B%(~,I] (B)) _< ~0(%(I,I] (B)) < ~, hence

i iBI Ii < ~- Thus I IBt i p < ~ for all B 6 LP 0 .

PROPOSITION 2.3: For 1 <_ p < ~ the map ~p: LP 0 -~ LP (M)

given by ~p(B) = B~pI/P is an isomorphism of Banach spaces.

PROOF: Let B ~ LP 0 and A = ~p(B) . Then clearly A is

closed densely defined and affiliated with N. Using (7) we can

Kaftal and Mercer 683

check that js(A) = e-s/PA. By (2), I IAI IpP = Z(%(I,~) (IAIP)) =

[Ze-t~0((%(l, ~) (IAIP))(t))dt = [Ze-t~0 (% (l, ~) (etIBlP))dt =

[~e-t~0(%(e-t, ~) (IBlP))dt = I(~,1]~0(%(s, ~) (IBlP))ds =

(I-I)~0(%(~,~) (IBIP)) - [(X,I]Z0(%(%,s] (IBLP))ds. By Lemma 2.2,

(i-~)~0(%(I,~) ( I BI p)) = ~0(%(%,i] (IBIP)) �9 Integrating by parts we

then have I IAI IpP = Ill, l] s d~0(%(l,s] (IBIP)) =

~0(IIX, 1] s d%(~,s] (IBIP)) [6, 12.6] = ~0(;(I, 11 s d%(-~,s] (IBIP)) =

~0(IBI p %(~,i] (IBIP)) = I IBI Ip p �9

By the remark following Lemma 2.2, I IBI Ip p < ~, so rAl p and A are

�9 Thus A = ~p(B) ~ LP(M) and I I~p(B) I Ip = I IBI Ip.

It is clear that ~p is linear and injective. To show

that fp is onto, let A E LP(M) . Then A E ~, so by (5) we have A =

leA(t) dt with A(t) ~ ~0 almost everywhere. Since 8s(A) = e-S/PA,

we have 8(A) (t) = e-s/P A(t) for all s and almost every t. In

particular for s ~ [0,T) we have by (7)

(I0) 80(K(t-s+T)) = e-s/P A(t) for almost every t in [0,s) .

A(t-s) = e-s/P A(t) for almost every t in Is,T) .

Let E = { (s,t) I s,t E [0,T), s < t and A(t-s) = e-S/P A(t) }.

Assume for the moment that E is a measurable set. Then by the

above, for almost every t ~ [0,T), A(t-s) = e-s/P A(t) for almost

every s E [0,t) . Given t,u E [0,T), there exist r E [0,t), s

[0,u) such that t-r = u-s and A(t-r) = e-r/P A(t) , A(u-s) =

e-s/P A(u) . Then A(t) = e(r-s)/P A(u) = e(t-u)/P A(u) = et/P B,

where B = e-u/P A(u) ~ ~0. Therefore

(ii) A(t) = et/P B for almost every t ~ [0,T) .

Substituting into (I0) we have e(t-s+T)/P 80(B) = e-s/P et/P B,

hence 80(B) = e-T/P B = ~i/p B Therefore B ~ LP 0 and A =

B| 1/p

The proof of the measurability of E will only be

sketched. As {A(t) } is known to be a measurable field of closed

densely defined operators, it is not difficult to show that both

sides of (i0) are measurable fields of closed densely defined

operators on { (s,t) I s,t ~ [0,T), s _< t}. As measurable fields of

684 Kaftal and Mercer

closed operators on H 0 can be realized as measurable fields of

closed subspaces of H0~H 0 using the graphs of the operators as

in [4], classical direct integral theory can be used to show that

equality in (i0) takes place on a measurable set. !

Recall that LI(M) =-M, and hence L10 ~ M, We will now

display this isomorphism explicitly. Let P0 = .~80 n and P =

[;8 s ds be the canonical f.s.n, operator valued weights from N O

to M0 and from N to K(M) ^ respectively [7, Prop. 19.8, 23.9].

LEMMA 2.4: P(B•I) = T(Po(B)(~I) for all B ~ NO + .

PROOF: Since P, P0 are normal and semifinite, it is

enough to verify the claim for all B ~ NO + such that P0(B) is

bounded. Let x 6 H0, y 6 L2(0,T) . Then the claim follows from

this calculation: <P(B~I) x| x| =

x| x+y> : [: x> ,y<t) ,2 d t ) d s

: [ / l y ( t ) 2 (I[nT,(n+I)T)<Os(B| ds) dt =

[~ly(t) I 2 (n~= { I[0,t' <o0n+l(B)x'x> ds + [[t,T)<e0n(B)x,x>ds } ) dt

= [oTly(t) 2 {t<P0 (B) x, x> + (T-t) <P0 (B) x, x> } dt =

T<P0 (B) x, x> <y,y> = <T(P0(B)|174 |

Recall that the dual weight ~ of a semifinite normal

weight y on M is given by ~= ~oZ-Iop Let V be the isomorphism

of M onto M 0 induced by the isomorphism of M onto K(M) = N @, i.e.

(12) V-I(B) = K-I(B| for B ~ M 0 .

Then we can define a dual weight ~ on N O by

(13) ~ = ~oV-IoP0

Let BI~ be the Radon-Nikodym derivative of ~ by Z0 , i.e.

(14) ~ = r (By.) .

We then have PROPOSITION 2.5: For all positive VI ~ M. we have B~I

L10 Moreover All[ = B~/J2 .

PROOF: By [I0, Theorem II . 7] A~ ~ L 1 (M) , hence by

Proposition 2.3 there is a B ~ LI0 such that A~ = B| Then for

all C s NO + we have Z0(BC) = T-IT(BC~) (by (8)) = T-Ir C|

Kaltal and Mercer 685

= T-16(C| = T-I~oK-IoP(C~I) = ~oK-I(P0(C)| (by Lemma 2.4) =

~oV-IoP0(C) (by (12)) = 6(C) = Z0(B~C). Therefore B = By. |

COROLLARY 2.6: [I, Proposition 3.2] (i) (LIo) + = {Bv! ~ 6 M, +}

(ii) If ~ 6 M, +, then ~(I) = 30(BvX(A,I] (B~))

REMARKS: All the proofs in this section and the next

hold without change for algebras M for which Theorem 10.6 of [9]

holds, i.e. if M has a continuous decomposition {N,8,T} where N is

type II~ and the action @ is ergodic on the center of N, but @T is

not ergodic on the center of N for some T > 0. This holds if M is

a properly infinite continuous algebra with a periodic f.s.n.

weight. In particular this is the case if M = ~(N,8,Z) where N

is a semifinite algebra with f.s.n, trace ~ and automorphism @ such

that ~o8=~ for some 0 < ~ < !.

It is not difficult to verify that LP 0 is independent, up

to Banach space isomorphism, of the discrete decomposition

{N0,@0,~ 0} chosen.

3: Spectral Pro~ections of L" 0DeratQr~

Let N be a semifinite algebra with f.s.n, trace z and

automorphism 8 such that Zo8 = iZ for some 0 < ~ < 1 We will

say that P is a finite trace projection if Z(P) < ~. (If N is a

factor this is of course the same as being finite, but in general

it is not.) The S-span P8 of a projection P is defined to be

the supremum ~@n(p) . p@ clearly belongs to the fixed point

algebra N 8. A projection P in N is called wandering if PSn(P) = 0

for n ~ 0. From [3, Proposition 4.2, Corollary 4.3, Remark 4.5] we have the following :

PROPOSITION 3.1: (i) Let Q be a nonzero finite trace

projection in N. Then there is a nonzero wandering projection

P E {Sn(Q)} ~ with P S Q. (ii) Let Q be a projection in N ~. Then

Q = P@ for some finite trace wandering projection P in N.

By the usual maximality argument Proposition 3.1 yields

686 Kaltal and Mercer

COROLLARY 3.2: If Q is a projection in N then Q can be

written as the orthogonal sum of finite trace wandering

projections.

Define LP 0 = {A ~ ~ I 8(A) = II/PA} and I IAi Ip =

~(IAIP~(~,I] (IAIP)) I/p. By Proposition 2.3 and the remarks after

Corollary 2.6, LP 0 is then isomorphic as a Banach space to LP(M)

where M = ~(N,~,Z).

PROPOSITION 3.3: There is a one-to-one correspondence

between elements A 6 (LI o) + and elements A 0 ~ N + such that

APo -< AO -< Po for some finite trace wandering projection PO 6 N.

This correspondence is given by the formulas

15) A = An @-n (Ao) " A 0 = AX(A,I] (A)

PROOF: Let A e (LI0)+, and let P0 = Z(k, I] (A). Then P0

is finite trace, and as 8n(P0 ) = X(~-n+l,l -n] (A) by LEMMA 2.2,

8n(P0)P0 = 0 for n ~ 0. Therefore P0 is wandering. If we define

A 0 = AP 0 then clearly IP 0 -< A 0 < P0, and A = ~. A X(~-n+l,~ -n] (A)

= ~ As-n(P0) . But As-n(P0) = 8-n(~nA)8-n(P 0) = ~n@-n(AP0) =

~nB-n(A0) . Hence A = ~ ~n 8-n(A0) .

Conversely, suppose A 0 is as given.Then 8-n(A0 ) -< 8-n(P0 )

for all n and since all the 8-n (P0) are orthogonal, A =

~ns-n(A0) defines an unbounded self-adjoint operator which is

clearly affiliated with N. As 8(A) = ~ ~n 8-n+l (A0) = n=-~

~ ~n@-n(A0) = ~A, we need only show that A is ~-measurable to

conclude that A E LI0, and hence we need to show T(~[I,~) (A)) < ~.

But X[I, ~) (A) = ,~X[I, ~) (Ins-n(A0) ) = ~Z X[l-n, ~) (8-n(A0) ) =

~. 8-n(x[~-n,~) (A0) ) = ~ @n(x[~n,l] (A0)) = zsn(x[kn,l] (A0)) n= -~ n = - , ~ , n = o

(I IA011 _< i) < ~8n(P0 ) (since A 0 -< P0) �9 Hence ~(%[i,~) (A)) -< n=0

(.~oz(en(PO)) = (.~0Xn)z(po) < o~. I

The following Corollary results from considering LP 0

instead of LIo and using the polar decomposition to remove the

Kaftal and Mercer 687

restriction to positive elements (See Definition 2.1 and the

remarks following.)

COROLLARY 3.4: There is a one-to-one correspondence

between elements A 6 LP 0 and elements A 0 6 N such that

AI/PPo ~ /Ao/ ~ PO for some finite trace wandering projection PO

e N. This correspondence is given by

(15) A = ~ %n/p @-n(AO) AO = AX(AI/P,I] (/A/) n=-~ �9 �9

and furthermore //A//p = ~(/Ao/)P)I/P

Members of LP 0 of the form D = ~ In/P @-n(P 0) for some

finite trace wandering projection P0 will be called diagonal

elements. If A 6 (LP0)+ is of the form ~ kn/P @-n(A0) with

II/PP0 S A0 S P0 and D is the associated diagonal element as

above, then ~i/p D S A S D

In this section we use the terms - ~ and ~-purely

~nfinite to refer to subalgebras of N on which the fixed trace �9 is

respectively semifinite or purely infinite.

Let A ~ (LI0) +, and let A be the abelian algebra

generated by A. By Lemma 2.2, A is @-invariant. Notice that the

~-measurability of A implies that A is ~-semifinite In fact

if Q is a nonzero projection in A with Q = X(E) (A) for some

Borel set E c ~ then there is an s > 0 such that P =

%(EN[s (A) ~ A is nonzero. By Lemma 2.2, ~(P) < ~. Therefore A

is ~-semifinite. Notice that Q must also have the property that

@n(Q)Q = Q@n(Q) for all n, as @n(Q) belongs to A for each n. A

projection with this property is called 9-compatible. In general,

a family of projections {Pi} c N is called @-compatible if @m(P i)

commutes with @n(pj) for all m,n,i,j.

We are now in a position to characterize those

projections in N which are spectral projections of LI0 operators.

THEOREM 3.5: Let Q be a projection in N. Then the

following are equivalent :

(i) There exists A 6 (LIo)+ and a Borel set Ec@ such that Q =

X (E) (A) .

(ii) Q is the orthogonal sum of @-compatible finite trace

wandering projections.

688 Kaital and Mercer

(iii) There exists a @-invariant abelian ~-semifinite subalgebra A

of N containing the projection Q. PROOF: (i)~(iii) was shown in the preceding discussion.

(iii)~(i) : We know that .~ 8nQ ~ A 8, so by Proposition 3.1(ii)

applied to A there is a finite trace wandering projection P0 ~ A

such that .~ 8UP 0 = .~ 8he . Let en = (8-nQ)P0 �9 Then .~ 8nQn

= Z Q 8UP0 = Q, and each Qn -< P0" Since AP 0 is abelian we

can find an operator A 0 with spectrum in (~,I] which generates

AP 0. Therefore there exist Borel sets E n c (~,i] such that Qn =

Z(En) (A0) . Let E = -0 ~-nEn and A = ~-nsnA 0. Then A 6 L10 by

P'roposition 3.3 and Z(E) (A) = ~ ~((E) [}~-nsnA0] = Z %(~%nE) [SnA0 ]

= .~ 8nz(%nE)(A0) . As %hE N (k,l] = En, this last sum equals

.~ 8 n %(E n) (A 0) = .~ 8nQn = Q.

(ii)~ (iii) : If Q = Z Qn is an orthogonal sum of 8-compatible

finite trace wandering projections, then let A = {Sk(Qn ) }" A is

then abelian, atomic and 8-invariant, and �9 is semifinite on A

because z(Sk(Qn ) ) = %k~(Q n) < ~ for all k,n

(iii)~(ii) : Follows directly from Corollary 3.2 applied to A. |

In Theorem 3.5 one might ask whether condition (iii)

could be replaced by the simpler

(iv) Q is 8-compatible, i.e. QSn(Q) = 8n(Q)Q for all n.

In the case that z(Q) < oo, the answer is clearly yes because then

(iv) implies (iii) . But in general the answer is no, because of

the following example.

EXAMPLE 3. 6: Suppose that N O is a IIoo factor with

f.s.n, trace ~0 and an automorphism 80 such that ~0080 = k~ 0.

Suppose that N 1 is a Ioo factor with trace Zl, automorphism 81 , and

maximal abelian subalgebra A 1 such that A 1 is 81-invariant and ~I

is purely infinite on A I. In this case we have ZlOSI = Zl- Let Q1

be a projection in A 1 such that {81n(Ql) }" = A I. For example,

let x = L2 ([0,I]), N 1 = B(K), ~I the usual trace, A 1 =

L~176 and 81(a) = u*au, where u is a unitary operator induced

Kaftal and Mercer 689

by an irrational translation on [0, I] . Q1 can then be any

nontrivial projection in A I. Now let N = N0| 8 = 80| ~ =

~0| and Q = I| I. Then @ is an automorphism, ~ is a f.s.n.

trace, ~o8 = (~0|174 = (~0o80)| = ~0| = ~ and

N is a IIoo factor. Thus Q is 8-compatible projection in N, i.e.

Q satisfies (iv). If A is any @-invariant abelian subalgebra of N

containing Q then A must contain I| hence A c (I| N N

= N0| I. By the following Lemma 3.7, �9 is purely infinite on

N0| 1 and hence also on A. Thus Q does not satisfy (iii) .

LEMMA 3.7: Suppose that ~j is a faithful normal trace

on a von Neumann algebra Nj, j = 0,1. Then if ~I is purely

infinite on NI, ~Oe~l is purely infinite on No@N 1 .

PROOF: Recall that ~0| = sup {90| i 9j ~ (Nj) *+,

9j -< Tj, J = 0,i} [7, Sections 8.2, 8.3]. Also 90| = 910#(90)

for a conditional expectation #: N0| 1 -~ N 1 [5, p.100] . Thus

for all nonzero T ~ (N0| I) + we have (90| (T) = 91o~(90 ) (T) ~ 0

for some 9j ~ (Nj),+, 9j < ~j, j = 0, i, due to the faithfulness of

Z0| But then ~(90) (T) ~ 0, so z0e~I(T) _> sup {1~(~(90) (T)) ] ~;

(NI) *+, �9 -< ~i} = ~i(r (T)) = ~. !

A projection in N thatsatisfies the conditions of Theorem

3.5 will be called an LI0 projection (although the projection

itself is of course not in LI0) . We now investigate some special

cases of these projections.

LEMMA 3.8: Suppose e is ergodic on a @-invariant abelian

subalgebra A c N .

(i) If a projection P 6 A is wandering, then it is minimal in A.

(ii) If A is atomic, then for any minimal nonzero projection

P 6 A, A = {@n (p) }~,.

(iii) If A is T-semifinite, then A is atomic and minimal

projections are wandering.

PROOF : (i) If P is wandering and P1 is a nonzero

projection of A strictly dominated by P, then (PI)@ ~ A is a

@-invariant projection strictly dominated by P@, contradicting the

assumption that 8 is ergodic on A. (ii) If P and P1 are nonzero

minimal projections in A such that P1 is not any of the @n(p),

then P1 is orthogonal to P@ in contradiction to the ergodicity of

690 Kaftal and Mercer

8. (iii) Since every projection in A dominates a finite trace

wandering projection by Proposition 3.1(i) and these are minimal

by part (i), A is atomic. If P is a minimal projection in A, then

so is 8n(P) for each n. As each of these has finite trace and

~(Sn(P)) = ~nT(P), 8n(P) ~ P for n ~ 0. Since A is abelian, we

must have @n(p)p = 0 for all n ~ 0. |

PROPOSITION 3.9: For a projection Q in N, the following

are equivalent:

i) There is a @-invariant atomic abelian subalgebra A

containing Q.

ii) Q = ~{Z ~n (Pi) : n 6 S i} for some collection {Pi} of finite

trace wandering projections with (Pi)@ orthogonal and S i oZ.

PROOF: (ii)~(i) : Just take A = {@n(Pi) }~'. Note that A

is then ~-semifinite. (i)~(ii) : First assume that 8 is ergodic on

A. Then by Lemma 3.8, A = {Sn(P0) } 11 for some wandering projection

P0 If P0 is of infinite trace, let P0 = ~Pi where {Pi} is an

orthogonal collection of finite trace projections in N. Then it is easy to check that {Pi} is a collection of wandering

projections with (Pi)8 orthogonal. Thus A 1 = {Sn(Pi) } " is an

atomic abelian algebra generated by finite trace wandering

projections. Since Q s A c AI, (ii) follows easily. In the

general case, decompose A with respect to A @. Since A is assumed

to be atomic, this results in A being written as a direct sum of

@-invariant atomic algebras on each of which 8 is ergodic. The

result then follows from the ergodic case by combining the

resulting collections of projections. !

Note that if the algebra A in condition (i) is not

~-semifinite, the arguments of the proof show that A can be

embedded in a ~-semifinite A 1 also satisfying (i) . A projection Q

satisfying the conditions of this Proposition is therefore an L10

projection.

One may try to determine whether Q is an L10 projection

on the basis of the structure of A 0 = {@n(Q) } ~'. If A 0 is atomic,

the above Proposition and remarks show that Q is an L1 0

projection. However, Q may take the form of condition (ii) above

and yet have A 0 continuous.

Kaftal and Mercer 691

The following Propositions discuss some of the special cases of the relationship between the projection Q and the operators in

L10 of which it can be a spectral projection.

PROPOSITION 3.10: Let Q be a projection in N. Then

the following are equivalent:

(i) Q = $6(B)(A) for some Borel set B and some diagonal operator

A ~ L10 .

(ii) Q = z{@n (Po) : n 6 S} for some finite trace wandering

projection PO and some S c Z.

(iii) There exists a @-invariant atomic abelian subalgebra A c N

containing Q such that @ is ergodic on Q@A

PROOF: (i)~ (ii) : If A = ~ ~n 8-n(P0) is a diagonal

element of (LI0)+, then X(B)(A) = Z{8-n(P0):~n ~ B}. (ii)~(iii) :

Clearly Q8 = Z~ 8n(P0) . Let A = {Sn(P0 ) }" Then A is atomic

with minimal projections {Sn(P0 ) } and 1 - QS" Therefore the only

invariant projections in A are Q8 and 1 - QS. (iii)~(ii) : By

Proposition 3.1, Q8 = .~ 8n(P0) for some finite trace wandering

projection P0 ~ A , and P0 must be minimal by Lemma 3.8(i) . Since

8 is ergodic on QSA, {Sn(P0) } are the only minimal projections in

QSA , and so Q = z{Sn(P0) : n ~ S} for some S c z !

PROPOSITION 3.11: Let Q be a finite trace projection in

N. Then

(i) Q is wandering if and only if Q = Z(A,I] (A) for some

A 6 (L I0) +

(ii) Q 2 8(e) if and only if Q = X[~, oo) (A) for some A 6 (LIo)+

The proof is omitted as it is a straightforward

application of methods used elsewhere in this section.

PROPOSITION 3.12: Let A 6 N + with / /A// = 1 and let

A 0 = ~(A) - ~A. Let PO be the range projection of A O. Then the

following conditions are equivalent:

(i) There is an operator B 6 (LIo)+ such that A = B ~(0,1] (B) .

(ii) A 0 > 120 , T(Po) < oo and PO A = O.

692 Kaftal and Mercer

9ROOF: (i)~(ii) : B = ~ ~n@-n(B I) where kP 1 < B 1 _< P1

and Pl is a finite trace wandering projection (Proposition 3.3).

Then %(0,i] (B) = ~0-n(Pl) and so A = BZ(0,1] (B) = ~ ~n@-n(Bl) . n=o n=o

Thus A 0 = ~nS-n+l (BI) - ~kn+ls-n(Bl) = e(Bl) . Therefore n=o n=o

P0 = 0(PI) and hence ~(P0) < ~- As 8(BI) >_ ~0(PI) , it follows that

A 0 _> IP 0 and, since P1 is wandering, P0 A = 0(P I) ~ %ne-n(Bl) = 0.

(ii) ~ (i) : Since ~A and A 0 are selfadjoint and have orthogonal

range projections, r 10(A) i I = max(1 HA01E,l) . Therefore i [A011 = 1

and so kP 0 < A 0 < P0" Moreover, if P is the range projection of

A, then @(P) = P + P0 is the range projection of @(A) . It is then

easy to see that P0 is wandering. By routine calculations as in

[3], we see that @(P) >_ ~0-n(P 0) and Q = @(P) - ~@-n(P 0) ~ N @, n=0 n=o

On the other hand, 0 = QPoA0 = QA 0 = 0(QA) - ~QA, so QA 6 LI0.

Since QA is bounded, by the remark after Definition 2.1 we have

QA = 0, hence Q = 0 as Q -< P. Thus P ~ ~ @-k(P0) . k=1

Using @(A) = kA + A 0 and iterating we have 8n(A) = ~nA +

~o ~n-l-k @k(A0) " For all 0 < @ -< n-l, eJ(P0) is orthogonal to P,

so 8J (P0)@n(A) = kn@J (P0)A + n~ kn_l_k@j (p0)@k(A0) = xn_l_j@j (A0) . K;O

So we have, for all k _> i, @-k(P0) A = ~k-i e-k(A0) " Thus A = PA

= ~. @-k (p0) A ~- ~ Xk-l@-k (A0) = ~ Xk @-k (e-i (A0)) . Since

kS-i (P0) < 8-i (A0) <_ @-l (?0) and 8 -I (P0) is a finite trace

wandering projection we may define B = ~ ~ke-k(@-l(A0)) e LP 0, ' k=-~

and then A = BZ(0,1] (B) . I

PROPOSXTION 3.13: Let P be a nonzero finite trace

projection in N. Then there exist nonzerg finite trace LIo

projections QO and Q1 such that QO -< P <- QI" PROOF: By Proposition 3.1{i), P dominates a finite trace

wandering projection Q0 which is trivially an L10 projection. Let

Q1 = ~@n (p) . Then P -< Q1 and as @(QI) -< QI, Q1 is an LI0

projection by Proposition 3.11(ii) . Hence ~(QI) -< ~ ~(en(P)) =

~Ln~(p) < ~. I

Kaftal and Mercer 693

References

[i] Haagerup, U. : LP-spaces associated with an arbitrary von Neumann algebra. (preprint)

[2] Halpern, H. , Kaftal, V. : Compact operators in type IIIl and

type III 0 factors, I. (To appear in Mathematische Annalen)

[3] Halpern, H. , Kaftal, V. : The relative Dixmier property and

classes of compact operators in type IIIl factors. (preprint)

[4] Lance, E. C. : Direct integrals of left Hilbert algebras, Math. Ann. 216 (1975), 11-28.

[5] Sakai, S. : C*-Algebras and W*-Algebras. Springer, New York. (1971)

[6] Segal, I. : A noncommutative extension of abstract integration, Ann. Math., 57 (1953), 401-457.

[7] Stratila, S. : Modular Theory in Operator Algebras, Abacus Press, Tunbridge Wells, England. (1981)

[8] Takesaki, M. : Theory of Operator Algebras I, Springer, New York. (1979)

[9] Takesaki, M. : Duality for crossed products and the structure of von Neumann algebras of type llI, Acta Math., 131 (1973), 249-308.

[10]Terp, M. : LP spaces associated with yon Neumann algebras. Copenhagen Univ. Rapp. #3 (1981).

Victor Kaftal, Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221 USA

Richard Mercer, Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435 USA

Submitted: November 16, 1984 Revised: May 30, 1985