Cohomology of operator algebras II. Extended cobounding and the hyperfinite case
RICHARD V. KADISO~ and JoI~N 1~. ]~I:NGROSE University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A.
and
University of Newcastle upon Tyne, England
1. Introduction
We continue the programme begun in [1] of studying the (topological) eohomo- logy of operator algebras. In t h a t article, we proved that cohomology of a t y p e I roll lXTeumann algebra with coefficients in the algebra vanishes [1: Theorem 4.4].
Employing t h a t theorem and the various preparatory results on centre adjustment of eocycles, we prove (Theorem 2.4), in this paper, t h a t each cocycle on a (general) yon N e u m a n n algebra with coefficients in the algebra cobounds a cochain with coefficients in the algebra of all bounded operators on the ttilbert space on which the yon l~eumann algebra acts. This theorem is, then, used to prove (Theorem 3.1) t h a t cohomology with coefficients in the algebra vanishes for hyperfinite yon I~eumann algebras.
The argument proving Theorem 2.4 is structurally the same as t h a t appearing in [4; Theorem 4]. I t is made more difficult b y the fact t h a t higher-dimensional
(norm-continuous) cocycles do not satisfy automatic weak continuity conditions (as do derivations [4; L e m m a 3]). This same difficulty rules out certain direct approaches to dealing with the hyperfinite case.
We wish to express our thanks for the hospitality of the Centre de Physique Th~orique, C.N.I~.S., Marseille and the Mathematical Institutes of the Universities of Copenhagen and Aarhus, Denmark during various stages of the research in this article. Both authors acknowledge with gratitude the partial support of the NSF, and the first-named author t h a t of the Guggenheim Foundation.
56 AI~!~IV F6~ ]~IATEI~IATIK. Vol. 9 No. 1
2. Cobounding in ~(c)/)
In this section we show (Theorem 2.4) t h a t each n-cocycle with coefficients in our yon •eumann algebra ~ is the coboundary of a cochain with coefficients in
~ ( P d ) . This is precisely analogous to the step in the proof of the Derivation Theorem establishing t h a t derivations are ))spatial>> (cobound an operator ~7~ in ~ ( 9 ~ ) -- see [4: Theorem 4]). The present proof is similar in structure to the proof for deri- vations. We extend our cocycle to the algebra generated b y a maximal abelian subalgebra of the c o m m u t a n t and our yon l~eumann algebra, after establishing suitable boundedness conditions. Unlike the derivation case, the extension of the cocycle from this algebra to its weak-operator closure, a yon Neumann algebra of t y p e I, is not a result of automatic ultraweak continuity (as is the case with deri- vations [4; L e m m a 3]). I t is easy to convince oneself that (norm-continuous) higher- dimensional cocycles are not necessarily ultraweakly continuous b y passing to the coboundary of a suitable (norm-continuous) eochain which is not ultraweakly continuous.
In fact t h a t (Theorem 2.1), nonetheless, each cocycle on a (concretely repre- sented) C*-algebra with coefficients in its weak-operator closure can be adjusted b y a coboundary so t h a t the resulting cocyele has an extension to the weak-operator closure (which is, again, a coeycle) replaces the missing automatic ultraweak conti- nuity. This fact is established with the aid of [1; Theorem 3.4] (the centre adjustment of coeycles) and the properties of the universal representation.
In the argument which follows, we will have occasion to extend a multilinear mapping from a C*-algebra to its weak-operator closure. One would hope, of course, t h a t when the mapping is ultraweakly continuous (separately) there is an ultra- weakly continuous extension. This has been proved in another connection and will appear elsewhere. I t involves other techniques; and, since we will be dealing, here, with the universal representation, we have chosen (and are able) to use the simpler (more awkward) procedure of successive extensions.
TI~.ORE~ 2.1. I f q~ is a faithful representation of the C*-algebra 91 and e e Z~(~(~) , ~(~)-), there is a ~ in Z~(q~(91)-, q~(9~)-) whose restriction ~ to q~(9~) is cohomologues to ~ (in Z~(qJ(~), ~0(9/)-)).
Proof. F r o m the properties [3: pp. 181--182] of the universal representation of 9i, there is a central projection P in ~(9I)- and an isomorphism ~ of ~(91)-P o n t o ~(9/)- which extends the mapping ~f(A)P ~ ~(A). We denote b y ~, the isomorphism induced b y ~ of the cohomology of ~p(?l)P and ~p(~)-P having coefficients in ~(9I)-P with that of ~0(9~) and ~0(91)-, respectively, having coeffi- cients in ~(9i)-. Let ~0 (in Z~(y~(91)P, ~(Pj[)-P)) be a;~(e). I f there exist ~o (in C~-I(y~(~)P, y~(Pj[)-P)) and 3 0 (in Z~(~(91)-P, ~(~/)-P)) such t h a t ~ 0 - A~0 :
~o]~(~)P, let ~ (in Z:'(~(91)-, ~(~)-)) and ~ (in C~-~(~0(91), ~(91)-)) be a.(~0) and a.(~0), respectively. Since, also, ~ ~ a,(~0), it follows t h a t ~ -- z~ ---- u proving the theorem. I t is now sufficient to consider the case in which ~ is the
C O H O ~ O L O G Y O F O P E R A T O R A L G E B R A S I I . 57 faithful representation A ---> ~ , ( A ) P of ~ (one could even assume t h a t 9/[ is given, acting on a H i l b e r t space 9~, in its universal representation, a n d q ( A ) = A P for A in g[).
W i t h ~ ( A ) = l v ( A ) P a n d ~ replaced b y ~0, let ~1 in Z2(~v(N), ~v(91)-P) be fl, l(Qo), where fl is t h e isomorphism ~v(A)--->~,(A)P of ~v(9I) o n t o ~v(~)P (the i d e n t i t y isomorphism of ~o(91)-P being used in defining fi.).
Since ~x is bounded, using t h e properties of t h e universal r e p r e s e n t a t i o n [3;
pp. 181--182], A --~ ~(A , A 2 , 9 9 9 , An) has a n u l t r a w e a k l y continuous extension f r o m ~v(~) to ~v(gi)-. The resulting m a p p i n g A 1 . . . A - - ~ n ( A 1 , . . . , An) is multilinear from ~v(91)- • ~v(91) • . . . • ~o(~) to ~v(N)-P. Continuing, w i t h successive extensions, we construct, for each k ( = 1 , . . . , n), a m a p p i n g ~lk which is multi- linear from ~v(91)- • . . . • ~v(N)- • ~v(91) • . . . • ~v(91) (where t h e first k factors are ~v(91)- a n d t h e last n - k are ~v(~)) into ~v(~)-P. Moreover, ~lk is u l t r a - w e a k l y continuous in its k t h a r g u m e n t a n d extends a n d has t h e same b o u n d as
~lk-1- F o r n o t a t i o n a l convenience, let ~x0 be ~1 a n d gl be ~ln ( = e1~+1). T h e n
~ e c2(~(~)-, ~(~)-P).
B y showing, inductively, t h a t
(A~,k)(A~ . . . . , An+~) = 0 ; A~ . . . A,. ~ ~(9.1)- , A~+~ . . . A,~+~ e ~(9.I) , (P~)
we will establish (P,+~) t h a t ~1 ~ Z](~v(~)-, ~v(gI)-P). Since ~x e Z2(~v(~ ), ~v(91)-P), (P0) follows. Suppose (P~_~) is given. W i t h A o . . . A~ in ~v(gl)- a n d A~+ 1 . . . . ,A=+ 1 in ~o(gI), we h a v e
(z] elk)(/1 . . . A~+~) ~- AI~Ik(A 2 , . . . , An+~)
n + l
-- ~ (--)J~,,(A 1 . . . A j _ 2 , A j _ I A j , A j + I , . . . , A ~ + I ) (1) j=2
+ (--)n+~ol~(A~ . . . An)An+~ 9
B y construction of ~,~ a n d ~ _ ~ , A~--> e~(A~ . . . A~ . . . A~+~) =
~o~_~(A~ . . . A~ . . . A~+~) is u l t r a w e a k l y continuous, as is each of t h e other t e r m s on t h e right side of (1) in its a r g u m e n t A~. Thus A~---+ ( A ~ ) ( A ~ . . . A~, . . . , A,+t) is u l t r a w e a k l y continuous on ~o(0/)-. Now, (AO~)(A~ . . . A,,+~) vanishes w h e n A~ ~ ~0(9~), b y inductive hypothesis, since (from (1)) it coincides w i t h ( A ~ _ ~ ) ( A ~ , . . . , A,,+~), in this case. The u l t r a w e a k d e n s i t y of y~(9~) in ~o(gI)- combined with t h e foregoing, yields (P~).
F r o m [1; T h e o r e m 3.4], there is a cochain ~1 in C2-~(~o(N) -, ~0(9s such t h a t ~ - A ~ ( ~ u vanishes w h e n a n y of its a r g u m e n t s lies in t h e centre of
~(~l)-. L e t ~o (in C2-~(y~(9~)P, ~(~I)-P)) be fi, (~I~p(9~)). F r o m [1; L e m m a 3.2],
~ ~ N Z y ( w ( ~ ) - , F(~I)-P); so t h a t
v~(A~P . . . . , A n P ) = u . . . A n ) P = 71(A~ . . . A~), for A ~ , . . . , An in ~o(91) (recalling t h a t ~(A~ . . . A,) ~ o ( 9 I ) - P ) . Thus
58 hgliIg FOR 3IATEMATIK. VO1. 9 No. 1
It follows t h a t e0 - - A~0 = ~01~~ P , w i t h ~0 t a k e n as 7~1[~0(~)-P (e Z2(~0(9/)-P,
~(~)-P)).
T h e l e m m a w h i c h follows describes a c a n o n i c a l e x t e n s i o n of a c e n t r e n o r m a l i s e d n - c o c h a i n on a y o n N e u m a n n a l g e b r a t o a c e n t r e n o r m a l i s e d n - c o c h a i n on a C*- a l g e b r a w i t h t y p e I y o n N e u m a n n a l g e b r a closure.
LEM~A 2.2. I f ~ is a yon N e u m a n n algebra acting on a Hilbert space c)C , d is a m a x i m a l abelian *-subalgebra of ~ti', a n d S is the C*-algebra generated by
a n d ~4, then each ~ i n N C ~ ( ~ , ~ ) extends u n i q u e l y to a n element ~ of N C ~ ( 5 , S ) . T h e m a p p i n g , ~--->~, is linear a n d isometric; a n d A~ = A~.
Proof. W i t h ~ t h e l a t t i c e o f p r o j e c t i o n s in d , a n d 50 t h e s e t o f all o p e r a t o r s of t h e f o r m E 1 T 1 H- 9 9 9 H- E , , T , , , w h e r e E 1 . . . E ~ E ~ a n d T 1 . . . T~ E ~ , S o is a n o r m dense * - s u b a l g e b r a of 3 . G i v e n S 1 . . . S , in 50, we m a y s u p p o s e t h a t
S i = Ei, 1Tj,~ @ . . . + Ej, m(j)Tj,,,(j) (j = 1 . . . n) (2) I f a C N C ~ ( 5 , 5 ) t h e n , since e a c h Ei,k lies in t h e c e n t r e ~ / o f 5,
-,0) m(~)
a(S~, . . . , S~) = ~ . . . ~ a(El.k0)Tl,k(1) . . . En,k(,)T,,k(,~)) k(1)=1 k(~)=l
re(l) re(n)
= ~ . . 9 ~. E l , k(x). 9 9 E,,,k(,)a(T1, k(1), 9 9 9 , T~, k(,)) 9
~(~)= 1 ~(~)= z
I n p a r t i c u l a r , if ~ satisfies t h e conclusions of t h e l e m m a , t h e n
re(l) re(n)
~($1 . . . S n ) = ~ . . . ~ E x , k 0 ) - . . E , , ~ ( , ) ~ ( T I , k ( , ) , - . - , T , , k ( , ) ) . (3)
k(1) = 1 k ( n ) = 1
T h i s e q u a t i o n d e t e r m i n e s ~(S1 . . . S~) u n i q u e l y w h e n e v e r S1 . . . . , S , E 50;
a n d t h e u n i q u e n e s s o f } (if it exists) n o w follows f r o m its n o r m c o n t i n u i t y , t o g e t h e r w i t h n o r m d e n s i t y o f S 0 in 5 .
I n o r d e r to p r o v e t h e e x i s t e n c e o f }, we s h o w f i r s t t h a t , for $1 . . . S , in 50, t h e r i g h t h a n d side o f (3) d e p e n d s o n l y on $1 , 9 9 9 , S , ( b u t n o t on t h e p a r t i c u l a r w a y in w h i c h t h e s e o p e r a t o r s a r e r e p r e s e n t e d in t h e f o r m (2)). F o r this, it is s u f f i c i e n t t o s h o w t h a t t h e r i g h t h a n d side o f (3) is zero i f Ei, 1 T j , 1 - @ . . . -~ Ei, m ( j ) T j , ,.(1) ~-- 0 for s o m e j , 1 _ < j < n. B y [2: L e m m a 3.1.1], t h i s l a s t c o n d i t i o n e n t a i l s t h e e x i s t e n c e o f o p e r a t o r s C~,~ ( r , s = 1 . . . re(j)) in t h e c e n t r e ~ o f q~ s u c h t h a t
m(.i)
C~,,Tj,~ -= 0 (S = 1 . . . m ( j ) ) , (4)
r ~ l
-*(1)
E j , , C r . , ~-- E j , , (r ---- 1 . . . r e ( j ) ) . (5)
s = l
C O H O ~ O L O G Y OF O P E R A T O ~ A L G E B R A S I I . 59 Since q E N C ' ~ ( ~ , c ~ ) , it follows f r o m (4) a n d (5) t h a t
-,(.i)
E i,k(j)Q(Tl,k(1) , 9 9 9 , T j , k(j), 9 9 9 , Tn,k(n)) k(j)=l
m(j) m(j)
= ~. ~, E i , , C k ( j ) , , e ( T l , k ( 1 ) , . . . , T j , k ( j ) , . . . , T,~,k(,)) k(j)=l s=1
~(i) ~(~)
= ~, Ej,s ~,
e ( T l , k ( 1 ) , 9 9 9 , C k ( j ) , , T j , k ( j ) , . . . , Tn,k(,,)) = O .s= 1 k(j) = 1
B y o p e r a t i n g on t h e left h a n d side of t h i s c h a i n o f e q u a t i o n s w i t h E I , k(1) 9 . ' Ej_1,k(,._i)Ej+l,k(j+x) . . 9 En, k(,), s u m m i n g o v e r e a c h o f t h e v a r i a b l e s k(1) , . . . . k ( j - - 1), k ( j ~- 1 ) , . . . , k ( n ) , a n d u s i n g t h e c o m m u t a t i v i t y o f t h e p r o j e c t i o n s E .... we d e d u c e t h a t
re(l) re(n)
9 "" ~ E l , k ( 1 ) . . 9 E n , k ( n ) ~ ( T 1 , k(1) . . . T n , k(n)) = O . k ( 1 ) = l k ( n ) = l
T h i s shows t h a t , if $ 1 , 9 9 9 Sn in S 0 a r e r e p r e s e n t e d as in (2), t h e n t h e e q u a t i o n
re(l) re(n)
~0(S1 . . . . , ~qn) ~-- ~. 9 "" ~ E l , k(1)" 9 9 En,
k(n)~(T1,
k(1) . . . . , Tn, k(n)) (6)k(1) = 1 k(n) = 1
defines, u n a m b i g u o u s l y , a n e l e m e n t r . . . . , S~) o f 30. I t is a p p a r e n t t h a t ~0 is a m u l t i l i n e a r m a p p i n g o f (30) ~ i n t o ~0; a n d ~0 e x t e n d s Q since
~ 0 ( R 1 . . . . , R n ) = ~ ( R 1 . / . . . . , R ~ . I ) = Q(R1 . . . . , R=) w h e n R 1 , . . . . . R~ E ~ . O u r n e x t o b j e c t i v e is to p r o v e t h a t ~0 is b o u n d e d . F o r this, s u p p o s ~ t h a t S 1 , . . . , Sn in 30 are r e p r e s e n t e d as in (2). T h e r e is a n o r t h o g o n a l f a m i l y { F 1 , 9 9 9 F~} of p r o j e c t i o n s in ~ s u c h t h a t e a c h Ej, k o c c u r r i n g in (2) is t h e s u m of a collection o f F j s . E a c h Sj c a n b e e x p r e s s e d in t h e f o r m
= F 1 R j , 1 -~ . . . -~ F m R j , m , (7)
w i t h t h e Rj, k's in ~ ; a n d , since F~ . . . . ,F,~ a r e p a i r w i s e o r t h o g o n a l ,
rn
~0($1 . . . . , S , ) = ~ F k ~ ( R x , k . . . , R , , k) (8) W i t h Qk t h e c e n t r a l c a r r i e r of Fk in ~ ' , we c a n r e p l a c e Fk b y FkQk in (8).
Since ~ E N C ~ ( ~ , ~,~), we o b t a i n
C o ( S 1 , . . . , S,~) = ~ F k e ( Q k R ~ , k , . 9 QkRn, k) 9 (9)
k = l
Since F 1 , . . . , F k are o r t h o g o n a l p r o j e c t i o n s w h i c h c o m m u t e w i t h ~ ( a n d t h u s w i t h e a c h v a l u e o f ~), while t h e m a p p i n g Q k R - + F k R is a * - i s o m o r p h i s m f r o m
~?~Q~ o n t o ~)~F~ ( a n d is t h e r e f o r e i s o m e t r i c ) , it follows f r o m (9) a n d (7) t h a t
60 At~KIV l~6R MATE/CIATIK. Vol. 9 No. 1
Ii~0(s~ . . . S~)ll = m a x ]IFk~(QkR~, k . . . QkR=, k)Ii < m a x He][ [IQkR~, ~ll. 9 9 I[QkR=, kl[
l ~ k < _ r n l < k < m
= m a x II~ll IIFkR~, kl[. 9 9 IIFkR~, k]l = m a x II.oll IlFkSilI . . . lYks,][ < I]QII llSx[I 9 9 9 IlSoll 9
l < k < m l<k<rn
Thus Qo is a b o u n d e d multilinear m a p p i n g of (50) = into 5o, a n d II~01l < II~lt- The reverse i n e q u a l i t y is a p p a r e n t ; so [IQoll : II~ll. Since go is n o r m dense in 5 , eo e x t e n d s b y c o n t i n u i t y to a n element ~ of C2(5 ~ , 5), a n d II~ll--l[e0!l = II~ll; t h e linearity of t h e m a p p i n g e--~ ~ is evident.
W e show n e x t t h a t ~ r NC'~(S, 3); t h a t is, ~ ( $ 1 , . . . , E - I , E E , E+~ . . . . , S~) = E ~ ( S ~ , . . . , S ~ ) for all S 1 , . . . , S , in ~ a n d E in t h e centre ~ of 5 ~. Since
is t h e n o r m closed linear span of ~ a n d 5 o is n o r m dense in ~, it is sufficient, b y t h e c o n t i n u i t y a n d m u l t i l i n e a r i t y of ~, to establish this last e q u a t i o n for t h e case in which S ~ , . . . , S n e S o a n d E e ~ (whence, ~ can be replaced b y Co).
W e m a y suppose t h a t S~ . . . Sn are represented as in (7); t h e corresponding expression for E S i is t h e n EF~Ri, 1 - ~ - . . . - ~ EF,~Rj, m, a n d t h e appropriate e q u a t i o n of t h e f o r m (6) yields
eo(S~ . . . S~_~ , E S i , Sj+t . . . S,,) = ~ E F ~ ( R 1 , ~ . . . R,,, ~) = E~o(S ~ . . . S,~).
k = l
I t remains to prove t h a t A~ = (A~i. W i t h So . . . . , S n in 50 represented as in (7), t h e corresponding expression for Sj_~Sj is F , R i _ I , ~Rj,~ q- ... q- FmRi_~, ~Ri, m.
Since ~ e x t e n d s ~0, a n d F~ . . . F= are pairwise orthogonal projections which c o m m u t e w i t h c~ (and hence w i t h each value of ~), it follows from (8) t h a t ( ~ ) ( S o . . . . , s~) = ~'oeo(S1 . . . s , ) + ~ ( -
1)Jeo(so,..., sj_:, Sj_lSj, Sj§ ... so)
j--1
+ ( - ~F+loo(So . . . . , S~_l)S~
= ~ Fk{R0. ko(Rl~ . ~ . . . . , R . , .
~)
k=l
~- ~ (-- 1)Jo(Ro,
k , " " " , Rj--2, k , J~k--1, k ' R j , k , " R j - r l , k . . . R n , k ) j--1-[- ( - - 1 ) n + l ~ ( R o , k . . . . , R n - l , k l R n , k } : ~ Fk(d~)(Ro, k , 9 9 Rn,~)
k - - 1
= (~e)(s0 . . . sn)
(where the last step results f r o m t h e equation, corresponding to (8), for t h e element AQ o f _YCy+l(~?~, c?~)). The m u l t i l i n e a r forms z]~ a n d (zl~) on ~n+l are b o u n d e d , a n d t a k e t h e same values on t h e dense subspace (50)"+1; so, b y c o n t i n u i t y , A~ = ( ~ ) .
C O H O M O L O G Y O F O P E R A T O R A L G E B R A S I I . 61 COrOlLARY 2.3. With ~ , ~4 and S satisfying the conditions of Lemma 2.2, each ~ in N Z ~ ( ~ , c-~) extends uniquely to an element ~ of N Z ~ ( J , 5).
Proof. I f Q E NZ2(C2~, oN) then, b y L e m m a 2.2, ~ extends uniquely to an element ~ of NC~(3 ,~), and A ~ = ( A ~ ) = 0 = 0. Thus ~ E N Z 2 ( J , 5 ) .
The possibility of extending a cocycle as in L e m m a 2.2, ~)adjusting)) it and then extending it to the (type I) weakoperator closure (of 3) combined with the fact [1; Theorem 4.4] t h a t n-cocycles on t y p e I yon N e u m a n n algebras cobound allows us to prove:
T~EoIcE~ 2.4. I f cN is a v o n Neumann algebra acting on a Hilbert space 9~
and o C Z 2 ( C ~ , ~ ) , there is a ~ in C~(Q~, ~ ( 9 ( ) ) such that ~ = AS.
Proof. F r o m [1; Theorem 3.4]; there is a ~ in C2-~(c~ , ~ ) such t h a t -- A ~ ( ~ ~) E N Z ~ ( ~ , ~ ) . With ~ a maximal abelian *-subalgebra of ~ ' and J the C*-algebra generated b y q~ and ~4, there is, b y Corollary 2.3, a (unique) ~2 in
NZ'2(g
, J ) extending ~ . Theorem 2.1 provides a S~ in C ~ - ~ ( ~ , 3 - ) such t h a t ~ - A~2 ( = ~a) has an extension ~a in Z 2 ( $ - , ~ - ) . Since 5" contains both c/~ and ~ , and ~4 is maximal abelian in ~?~';, ~ ' ~ Q4'fl r ~4. Thus ~-, having an abelian commutant, is of t y p e I.
F r o m [1; Theorem 4.4], there is a ~a in C~-~(5 "- , g - ) such t h a t ~a = A~-~. Let
~3 be ~1~74; so t h a t ~3 E C2-~(c~, S-) and A~]3 = ~lc~. Let ~ be ~21~; so t h a t ~ ~ C2-~(c~, ~-). With ~2 t a k e n as O~ 1 ~74 , a2 E Z 2 ( ~ , 3) C Z2(c~ , ~-).
Then % -- zJ~]: -~ A~]3- BUt ~1 = ~21C~%) = 0"2 = zJ(V2 @ ~]3) ~- ~ -- A~I" Now S ~ E C 2 - ~ ( ~ , ~ ) c C 2 - ~ ( ~ , X - ) ; so t h a t 0 = A ( ~ + ~ + S ~ ) , with ~z@~a-t-S1 in C~-i((?~,S-). Choosing Vz + ~a + S~ as S completes the proof.
3. The hyperfinite case
I n this section, we prove that the cohomology of a hyperfinite yon N e u m a n n algebra with coefficients in t h a t algebra vanishes. The result t h a t cohomology of u.h.f. C*-algebras with coefficients in a dual module vanishes is established by meaning techniques.
T I t E O R ~ 3.1. I f ~'~ is a hyperfinite yon Neumann algebra H2(~4 , ~ ) = O.
Proof. Suppose, first, t h a t ~ ' is hyperfinite. F r o m [5; L e m m a 5], there is a bounded projection z of ~ ( g g ) , the algebra of all bounded operators on the tIilbert space 9 i on which ~ acts, onto c/~, having (among others) the property t h a t ~ ( A T B ) = Aze(T)B, for A and B in c~. With ~ in Z2(c?,~, c~), Theorem 2.4 tells us t h a t there i s a S in C~-I(,~ , ,~/~(c)~)) such t h a t ~---At. I f ~ is ~ o S then ~ C C:-l(c/~, c~), and, with A s , . . . , A , in ~?~,
62 ARKIV FOR MATESIATIK. Vol. 9 No. 1
( A ~ ) ( A , , . . . , A , ) = A I ~ ( A 2 , . . . , A ~ ) - ~ ( - ) i ~ ( A ~ . . . . , A s : , A j A j _ x , A j + I , . . . , A n ) j = 2
+ (--)"~(A1 . . . . , An i)An
= :~[A~(A1 . . . , AD] = ~[e(A1 . . . A,)] = ~(A 1 , . . . , A~) . T h u s H~(~?~ , ~ ) - - - - 0 w h e n c~, is h y p e r f i n i t e .
F r o m [6; T h e o r e m 12.2], each y o n N e u m a n n a l g e b r a can be r e p r e s e n t e d in a ))standard>) f o r m in which, in p a r t i c u l a r , it is * a n t i - i s o m o r p h i c t o its c o m m u t a n t . Since H ~ ( ~ , ~2() is i n d e p e n d e n t o f t h e r e p r e s e n t a t i o n o f ~ as a y o n N e u m a n n algebra, we m a y a s s u m e t h a t ~?( is r e p r e s e n t e d in this s t a n d a r d f o r m . T h e n ~ ' is h y p e r f i n i t e ; and, f r o m t h e preceding, H : ' ( ~ , ~ ) = 0.
R e m a r k 3.2. T h e r e are o t h e r r o u t e s we c o u l d t a k e t o a p r o o f o f t h e p r e c e d i n g t h e o r e m w h i c h w o u l d a v o i d [6]; b u t t h e y r e f e r to t h e )>internal)> p r o p e r t i e s of t h e h y p e r f i n i t e ~ a n d h a v e t o be a r g u e d m o r e closely. T h e use of zr gives some i n f o r m a t i o n n o t n o t e d in t h e s t a t e m e n t of T h e o r e m 3.1; viz. H:(C?~, ~() = 0 if
~ ' is h y p e r f i n i t e . I t is quite possible, t h o u g h , t h a t this occurs o n l y w h e n ~ is h y p e r f i n i t e .
I n t h e n e x t result, t h e dual m o d u l e m i g h t be, for e x a m p l e , t h e y o n N e u m a n n algebra closure o f t h e C*-algebra in a g i v e n r e p r e s e n t a t i o n .
THEOREM 3.3. I f a C*-algebra 91 is subgroup cg o f its u n i t a r y group, a n d //2(91, ~ ) = 0.
the n o r m closed linear s p a n o f a n a m e n a b l e cH~ is a two-sided d u a l 91-module, then
Proof. F r o m , [1; L e m m a 3.3], t h e r e is a m e a n fi f r o m I~(~V, c)~'l) i n t o c}11.
W i t h ~ in Z2(91, cff~), Ax . . . An_ 1 in 9I a n d V in q/, define $(A1 . . . A , _ I ) to be fi(~), w h e r e ~(V) = V * ~ ( V , A 1 . . . . , A n _ l ) . Since ~ d e p e n d s m u l t i l i n e a r l y on t h e p a r a m e t e r s A 1 , . . . , A , _ 1 a n d fi is linear, ~ is multilinear. M o r e o v e r
I I ~ ( A 1 , - . . , An_0][ = ])~(~)1[ --< I!~[I --< Ii~l]" HAIII . . . IIAn-l[I ; so t h a t ll~Ii --< I[~1], a n d ~ e C2-1(91, off/).
W e p r o v e t h a t Q = A ~ . W i t h W in ~V, since ( A Q ) ( V , W , A 1 , . . . , A n . 1 ) - ~ 0 (in t h e n o t a t i o n o f [1; L e m m a 3.3]),
~ w ( V ) = ( V W ) % ( V W , A , , . . . , & - O
= W * V * [ V ~ ( W , A 1 . . . A~_I) @ ~ ( V , W A 1 , A 2 . . . A ~ _ I )
- - ~ ( V , W , A I A 2 , Aa . . . A , _ I ) - / . . .
@ ( - - 1 ) ' e ( V , W , A m .. . . . A , _ 3 , A n _ 2 A , _ i ) @ ( - - 1)~+I~(V, W , A ~ , . . . , A ~ _ 2 ) A n _ I ] 9 Thus, f r o m t h e p r o p e r t i e s o f fi [1; L e m m a 3.3],
~(A 1 . . . A~_I) = / i ( ~ ) = fi(~w) = W*[~(W, A 1 , . . . , An_l) -~- ~e(WA1, A 2 , . . . , A~_I) - - $ ( W , A 1 A 2 , A 3 , 9 . . , A~ 1) + 9 9 9 -~ ( - - 1)~( W , A~ . . . . , A , _ 3 , A , - 2 A n 1)
@ ( - - I ) " + I ~ ( w , A 1 , . . . , An_2)A._I]
= W * [ ~ ( W , A 1 . . . . , An_l) @ W ~ ( A 1 . . . . , An_i) - - ( A ~ ) ( W , A 1 , . . . , An_l) ] ;
O O H O M O L O G Y O F O P E R & T O ~ A L G E B R A S II. 63
and e(W, At, . . . , A,_I) = (A~)(W, A I , . . . , A , _ 0 , (10) for W in D/. As 9~ is t h e n o r m closed l i n e a r s p a n o f ~V a n d b o t h e a n d A~
a r e m u l t i l i n e a r a n d b o u n d e d ; it follows f r o m (10) t h a t e ~ ~ . T h u s H ~ ( ~ , ct]~) = 0.
COROLLARY 3.4. I f 9~ is an abelian or a u.h.f. C*-algebra, Hy(9~, c)/]/) _-- 0 for each two-sided dual ~l-module c~?.
References
I. K A D I S O N , i~. u a n d R I ~ G R O S E , J. I~., C o h o m o l o g y of operator algebras I. T y p e I y o n l ~ e u m a n n algebras. Aeta Math., 126 (1971).
2. K A D I S O N R, V., U n i t a r y invariants for representations of operator algebras. A n n . of Math., 66 (1957), 304--379.
3. -->)-- Transformations of states in operator theory a n d dynamics. Topology, 3 (suppl.2) (1965), 177--198.
4. --~)-- Derivations of operator algebras. Ann. of Math., 83 (1966), 280--293.
5. SChWArTZ, J. T., Two finite, non-hyperfinite, non-isomorphic factors. Comm. Pure Appl.
Math., 16 (1963), 19--26.
6. TAKESAKI, M., Tomita's theory of modular Hilbert algebras and its applications. Lecture Notes in Math. 128, Springer, Berlin, 1970.
Received October 10, 1970. R i c h a r d V. K a d i s o n
University of P e n n s y l v a n i a D e p a r t m e n t of M a t h e m a t i c s Philadelphia, Penn. 19104 USA
J o h n R. Ringrose University of Newcastle Newcastle u p o n T y n e E n g l a n d
