A class of II1 factors without property P but with zero second cohomology
B. E. J o n N s o ~
University of Newcastle upon Tyne, England
I f ~ is a y o n N e u m a n n algebra, or i n d e e d a n y B a n a c h algebra, c)~2(~l, 9/) is t h e q u o t i e n t of t h e space o f c o n t i n u o u s bilinear m a p s S : 9~ • 9~ - + ~I such t h a t
bS(a, b, c) ~- aS(b, c) -- S(ab, c) -[- S(a, bc) -- S(a, b)c = 0 (a, b, c E 9.1) b y t h e subspace of t h o s e m a p s of t h e f o r m
S(a, b) = aR(b) -- t~(ab) + R(a)b = ((~R)(a, b)
for some c o n t i n u o u s linear m a p R : 9~ --> 9~. T h e b a c k g r o u n d to t h e p r e s e n t p a p e r is t h e t h r e e p a p e r s [6], [7] a n d [5], in w h i c h it is s h o w n t h a t c2/~(% ~l) = 0 for t y p e I y o n N e u m a n n algebras a n d for h y p e r f i n i t e y o n N e u m a n n algebras. I n this p a p e r we c o n s t r u c t some n o n h y p e r f i n i t e I I 1 f a c t o r s w h i c h h a v e this p r o p e r t y . Besides t h e t h r e e p a p e r s a b o v e we shall also use ideas f r o m [4].
LEMMA 1. Let G be a group of permutations of a set X , x o c X and H = {g : : g E G, gx o ---- x0}. S u p p o s e H is amenable and G is 3-fold transitive on X . T h e n
l ) = o.
f ~ ( X ) is t h e space of b o u n d e d f u n c t i o n s on X a n d if f E F ~ 1 7 6 g E G we define gf b y (gf)(x) ~ - f ( g - l x ) (x q X). C 1 is ~he set of c o n s t a n t f u n c t i o n s in ~ ( X ) a n d is closed u n d e r m u l t i p l i c a t i o n b y e l e m e n t s o f G so t h a t if F E f ~ 1 7 6 1, g F is well defined. S a y i n g 9 0 ( G , / ~ ( X ) / f J 1) = 0 m e a n s t h a t w h e n e v e r q~ is a m a p G - + ~ ~(X)/C 1 w i t h
Ilq~(g)li < K g E G
q)(gg') = qS(g) + gqb(g') g, g' E G,
t h a t is, if q~ is a b o u n d e d crossed h o m o m o r p h i s m , t h e n t h e r e is F C ~ ~176 1 w i t h
154 ~. ~. z o ~ s o ~ +(g) = g ~ - ~ g c r
t h a t is ~b is a p r i n c i p a l crossed h o m o m o r p h i s m . S a y i n g t h a t G is 3-fold transitive m e a n s t h a t if {x 1, x~, x3}, {x~, x' 2, x~} are t w o sets of 3 d i s t i n c t p o i n t s o f X t h e n t h e r e is g C G w i t h gxl = x'i i = 1, 2, 3. A m e n a b i l i t y o f g r o u p s is discussed in [3, w
P r o o f . I f X is f i n i t e t h e r e s u l t is a c o n s e q u e n c e of [4, T h e o r e m 3.4]. Accord- ingly we a s s u m e X has at least 3 points. L e t ~b be a b o u n d e d crossed h o m o m o r - p h i s m as above. As H is a m e n a b l e a n d as f ~ 1 7 6 1 is t h e d u a l of ~ - { a ; a E ~ ( X ) , (a, 1 ) ~ 0} a n d F ~ g F is t h e a d j o i n t o f t h e m a p a ~ a g w h e r e ag(x) = a(gx) on ~00, t h e r e is F in ~ ( X ) / C 1 w i t h ~ b ( h ) ~ h F - F for all h in H [4, T h e o r e m 2.5]. R e p l a c i n g ~b b y T : g ~ q~(g) - - g F -~ F we see t h a t T is a b o u n d e d crossed h o m o m o r p h i s m a n d T is p r i n c i p a l i f a n d o n l y i f ~b is.
T is zero on H so T ( g h ) = T ( g ) , g E G , h E H . T h u s T is c o n s t a n t on t h e left cosets o f H , w h i c h are in one t o one c o r r e s p o n d e n c e w i t h t h e p o i n t s o f X , a n d so can b e c o n s i d e r e d as a f u n c t i o n T ' on X w h i c h is zero a t x 0. I f ~ 0 = { f : f E S ~ ( X ) , f(Xo) = 0} t h e n t h e q u o t i e n t m a p q o n t o ~ | 1 is one to one on
+~ 0 a n d i f we define
(g o f ) ( x ) = f ( g - ~ x ) - - f ( g - l x o ) f C ~0, g E G, x e X
t h e n g o f E ~ 0 a n d q(g o f ) = gq(f). T h u s we can a s s u m e t h a t T ' t a k e s values in +~ r a t h e r t h a n F ~ ( X ) / C 1 a n d we h a v e
T ' ( g g ' ) = ~ ' ( g ) + g o ~g'(g').
L e t O(x, y) ~ ( T ' ( x ) ) ( y ) (x, y E X). T h e n O is a b o u n d e d c o m p l e x v a l u e d func- t i o n on X • X w h i c h is zero i f e i t h e r v a r i a b l e is x 0. T h e crossed h o m o m o r p h i s m p r o p e r t y for T shows
O(x, g-~y) - - O(x, g-~Xo) - - O(gx, y) + O(gx o, y) ~-- 0 g E G, x, y C X .
I f x, y E X ~ { x 0 } t h e n t h e r e is g E G w i t h gx o - - x o, g x ~ - - y a n d t h e a b o v e e q u a t i o n yields O(x, x) ~ O(y, y). I f g x o ~-- Xo, z ~ g - l y we get O(x, z) ---- O(gx, gz) so t h a t , because G is 3-fold t r a n s i t i v e on X , O is c o n s t a n t off t h e diagonal of (X ~ {x0} ) • (X ~ {x0}). I f ~ is t h e v a l u e o f O on t h e d i a g o n a l a n d fi t h e v a l u e o f f t h e diagonal t h e n w r i t i n g g-~y ~ z a n d choosing g w i t h gx ~ - - x o we h a v e
O(x, z) - - O(x, x) § O(gxo, gz) = 0
so t h a t if x, x o, z are d i s t i n c t t h e n ~ - - fi ~- ~ = 0. D e f i n i n g ~(x) = --fi if x :fi Xo,
~(xo) --~ 0 we easily c h e c k t h a t O(gxo ' y) : F ( g 4 y ) _ q)(g-~xo ) _ ~(y) for all g E G , y E X . ~ E ~ ~ a n d this e q u a t i o n can be r e w r i t t e n T ' ( g x o ) = g o q ~ - - f r o m w h i c h we see T ( g ) = gq(~) - q(q~).
T ~ E o ~ 2. L e t (Z, ~) be a locally compact, g-compact m e a s u r e space a n d G a group o f h o m e o m o r p h i s m s o f Z such that ~ o g is absolutely c o n t i n u o u s w i t h respect
A C L A S S O F I I 1 F A C T O R S W I T H O U T ] ? I ~ O P E R T Y P B U T W I T H Z E R O S E C O N D C O H O ~ O L O G Y ] 5 5
to ~ for all g in G. Let K be an amenable normal subgroup of G and H an amenable subgroup containing K . We suppose that
(i) ~ ; N c ' Z U g = ( 0 ) i f g # e (ii) K is ergodic on Z
(iii) G is 3-fold transitive on the left coset space G / H
where the notation is that of [1, p. 134]. Let 95 be the yon ~Veumann algebra con- structed f r o m Z, ~, G [1, p. 133--135]. T h e n 9E~(~N, c~) = 0.
Proof. W e shall use the n o t a t i o n of [1, Ch. 1, w in particular t h a t o f E x . 1 p. 137.
L e t 910 be t h e n o r m closed *subalgebra of ~ ( ~ ) g e n e r a t e d b y t h e operators { q ) ( T ) ; T E t Z } , {Uk;/c C K}, 91 the w e a k closure of 92[ 0 , go the n o r m closed algebra g e n e r a t e d b y {q)'(T) ; T E t Z } a n d {U~',; h E H } a n d ~3 t h e w e a k closure of go. I t is easy to see t h a t t h e s u b g r o u p cU of the u n i t a r y group of 910 generated b y the { r U - 1 = U * E a } a n d { U k ; / c E K } is an extension of the abelian group {qs(U): U - l = U* E t;Z} b y a group isomorphic with K and so ~Zd is a m e n a b l e [3, T h e o r e m s 17.5 a n d 17.14]. T h u s 91 0 is s t r o n g l y a m e n a b l e [4, Pro- position 7.8]. Similarly ~3o is strongly amenable. I f M is a t r a n s l a t i o n i n v a r i a n t m e a n on ell t h e n defining
( P X ~ , ~ ) = M ( U * X U ~ , v ) ~ , ~ E ~, X e?-~(~O),
U E IZ
where the right h a n d side indicates t h e v a l u e of M at t h e function U ~+ ( U * X U~, ~]), we define a projection P : ~E-(~) --> 910 = 91' with ! P ( X B ) = P ( X ) B , P ( B X ) =
= B P ( X ) for B E 91', X E 5~(~)). There is a similar projection Q onto ~3'.
B y [5, L e m m a 5.4] to show 9~2(c75, ~r~) = 0 it is enough to show t h a t if S : qvo• N _+ ~/r~ is s e p a r a t e l y u l t r a w e a k l y continuous, dS = 0 a n d S(a, b ) = 0 if either a or b lies in 910 (and so too if a or b lies in 91) t h e n N = dR 0 for some n o r m continuous m a p Ro : ~N - ~ ~-r--~. Using [7, T h e o r e m 2.4] we see t h a t there is a n o r m continuous m a p /~ : ~ --~ ~ ( ~ ) with S = dR and b y [5, L e m m a 5.5]
with ql~ = ~ ( ~ ) we can t a k e /~ to b e u l t r a w e a k l y continuous. As R(ab) =
= aR(b) -~ l~(a)b for all a in 91 using t h e definition of a m e n a b l e algebra [4, w we see t h a t t h e r e is x C ~ ( ~ ) with J ~ ( a ) = a x - - x a for all a in 910 and so, b y n l t r a w e a k continuity, for all a in 91. Replacing _R b y a ~+ t~(a) - - ax -~ xa if necessary we can assume R is zero on 9I. R e p l a c i n g /~ b y QR if necessary we can assume in addition t h a t R m a p s 9~ into ~3'. W e h a v e 0 = S(a, b) =
= aR(b) - - R(ab) a E 91, b E 9~. Similarly JR(ba) = R(b)a a E 91, b E ~ .
The set of generators of 910 is m a p p e d onto itself u n d e r t h e a u t o m o r p h i s m X ~ + U* X~/g of ? ~ ( 5 ) so U* 9 l U g = 91 for all g in G. H e n c e _R(Ug)U* A U g =
156 B . E . g o d . s o N
= R(A~Ts) = A/~(~Ts) for all A C 91, g E G, so t h a t R(~;g)U* E 9/'. Also R(~Tg)Ug*
E ~ ' because /~(~?g) a n d ~?g are.
is a direct s u m of copies of ~ ~) so a n y element L of ~_Z(~) can be repre- s e n t e d as a G • G m a t r i x w i t h entries from ~ ( ~ ) ) . W e shall investigate t h e special form this m a t r i x t a k e s w h e n L E 9/' fl ~ ' . As Lq~(T) = r we h a v e Ls,~ T =
= TL~,= for all T in ~ , s, u in G. As ~7~ is m a x i m a l abelian this shows Ls, ~ E ~Z..
A similar calculation s t a r t i n g f r o m Lq~'(T) = qb'(T)L shows Ls, u E U~ ~ U* ----
= ~ Us~-, so L ~ , , E ~ N ~ U s ~ _ , = { 0 } i f s v e u . Thus Ls0~= ~,~Y, where for each s in G, Y, E ~ . The e q u a t i o n LUk = (TkL shows Yk~ = UkY,,U*
for / c E K , u E G . The e q u a t i o n LLT~= l~'hL shows Y , h = Y~ for all u E G , h E H . T h u s Y~ = Uk Yk~Uk : Uk Y~-,k~Uk = * * UkY~Uk, u E G , * k E K . As K is ergodic on Z this implies Y ~ = y ~ I o for some y ~ E C . T h u s if L E 9 / ' A !3' t h e n
L~,t = (5~,ty~Io
for some complex v a l u e d f u n c t i o n y on G which is c o n s t a n t on t h e left cosets of H . Clearly y is bounded. W r i t i n g J L for y we see t h a t J is a linear i s o m e t r y of 9~'N ~ ' o n t o ~ ( X ) where X is t h e space of left eosets of H in G. More- over UgLU* e 9/' gl ~ ' a n d J(UgL~]*) = g J L where t h e p r o d u c t of g E G, J L C f ~ ( X ) is as d e f i n e d in L e m m a 1. A n o t h e r calculation shows t h a t J ( ~ N 9/') =
= e l .
r u t 0(g) = The e q u a t i o n S ( @ = where
S ( U v Ug,)Ug, Ug*-*E~ a n d R(Ug,,)Ug* C 9 ~ ' ~ 1 ~ ' for all g " E G shows t h a t bR(~7, Ug,)U~ U* e ~ f'l 9/' f r o m which we see gq~o(g') -- #(gg') + #(g) E C 1.
Thus q~b 0 is a b o u n d e d crossed h o m o m o r p h i s m f r o m G into / ~ 1. L e t z ~ / * ( X ) w i t h q#o(g) = gq(z) -- q(z) (using t h e L e m m a ) a n d let L 0 E 9~' N ~ ' w i t h J L o = z . W e h a v e
-- U~L oU~ + Lo) E C l
so t h a t -- U~LoU ~ + L o E C I ~ ~ ~/~. - * T h u s defining R0(B) = R(B) -- -- ( B L o --/LoB ) for all B in ~ we see t h a t R o is a n u l t r a w e a k l y continuous m a p from ~ into !3'.
Because /Lo E 9/' a n d R ( A B ) = A R ( B ) , B ( B A ) = 2R(B)A a n d _R(A) ---- 0 if A ~ 9/, B ~ ~ , R o has t h e same properties. I n a d d i t i o n if g ~ G t h e n /?o(fig) ----
= (R(Us)U* -- ~7~Log~* + L0)U ~ e ~ . T h u s i f T E ~ , g E G t h e n Bo(q~(T)Ug ) :
= Og(T)Ro(~7) E ~?~. As R o is u l t r a w e a k l y continuous a n d t h e u l t r a w e a k l y closed linear span of t h e 09@) Ug is ~2~ we see t h a t R 0 ( ~ ) _~ ~2~. F o r all B x, B~ in
w e h a v e
S(B~, B~) -~ B~R(B~) -- R(BxB~) + R(B~)B+= = B~Ro(B~) -- Ro(B~B~) + Ro(B~)B v
A C L A S S O F I I 1 : F A C T O R S W I T H O U T P R O : P E R T Y P B U T W I T H Z E R O S E C O N D C O t I O M O L O G Y ] 5 7
T h u s t o p r o v i d e o u r e x a m p l e we h a v e o n l y t o show t h a t t h e h y p o t h e s e s can be satisfied in some s i t u a t i o n in w h i c h ~ is a t y p e I][ 1 f a c t o r w i t h o u t p r o p e r t y P [8, D e f i n i t i o n 1]. T o f a c i l i t a t e this we simplify c o n d i t i o n * [1, p. 135].
L]~MMX 3. Let Z be a locally compact a-compact metrizable space, ~ a positive Radon measure on Z, s a homeomorphism of Z. Then the following condition * is satisfied i f and only i f v((z : z E Z, sz : z}) = O.
(*) For each measurable set Z' in Z with v(Z') r 0 there is a measurable subset Z" of Z' with r(Z")=/= 0 and Z " n 8 Z " = 0 .
Proof. I f F = { z : z E Z , z = sz} has v(F) = O, Z' c Z, v(Z') > 0 is a m e t r i c on Z c o m p a r a b l e w i t h t h e t o p o l o g y t h e n
a n d d
Z~ = {z : z e Z', d(z, sz) > n -~}
defines a m o n o t o n i c increasing sequence o f m e a s u r a b l e subsets o f Z w i t h u n i o n Z ~ F w h e r e v ( Z ~ F ) = v ( Z ) > 0. T h u s for some n , v ( Z , ) > O. T a k i n g a c o m p a c t subset K of Z , w i t h v(K) > 0 a n d a ball B c e n t r e z 0 o f radius (2n) -1 w i t h ~ ( B n K ) > 0 we p u t Z " = B f ] K . T h e n if z E Z " we h a v e d(z0, s z ) >
> d(z, sz) -- d(z 0, z) > (2n) -1 showing sz ~ Z". T h e converse is obvious.
= Q~
Example 4. I n T h e o r e m 2 let Z Z 2 , t h a t is t h e p r o d u c t of a c o u n t a b l e n u m b e r o f copies o f t h e g r o u p o f integers rood 2, t h e f a c t o r s being i n d e x e d b y p a i r s o f r a t i o n a l n u m b e r s , w i t h t h e u s u a l p r o d u c t t o p o l o g y a n d let v be H a a r m e a s u r e on Z w i t h v(Z) = 1. T h u s Z is a c o m p a c t m e t r i z a b l e g r o u p . L e t Z 0 = ( z : z E Z , z e : 0 for all b u t a f i n i t e n u m b e r o f p E Q2} a n d let K be t h e set of all m a p p i n g s o f Z o n t o itself o f t h e f o r m z ~ + z ~ - z 0. K is t h e n a n abelian g r o u p o f h o m e o m o r p h i s m s p r e s e r v i n g v and, in p a r t i c u l a r , is a m e n a b l e [3, T h e o r e m 17.5]. I f F E L ~ ( v ) has F ( l c z ) : F ( z ) for a l m o s t all z in Z for each k E K t h e n we h a v e f f ( z -- zo)F(z)&(z ) -- ff(z)F(z)dv(z) ( f E C(Z), z 0 E Z0). As
y~ff(z-
y)F(z)dv(z) is c o n t i n u o u s a n d Z o is dense in Z this shows F v is an i n v a r i a n t i n t e g r a l on C(Z) a n d h e n c e F is a c o n s t a n t . T h u s K is ergodic on Z.F o r a n y one t o one m a p ~ o f Q~ o n t o itself a ' : z ~ - ~ { z ~ ( p ) ; p E Q~} is a n a u t o m o r p h i s m o f t h e topological g r o u p Z a n d so is a h o m e o m o r p h i s m p r e s e r v i n g v.
G is t h e g r o u p o f h o m e o m o r p h i s m s of Z g e n e r a t e d b y K a n d t h e a ' w i t h a E SL(2, Q) w h e r e t h e m a t r i x g r o u p acts on Q2 in t h e usual way. E v e r y e l e m e n t of G can be w r i t t e n /ca', /~ E K , ~ ESL(2, Q) in e x a c t l y one w a y . I f a E SL(2, Q), a n d a is n o t t h e i d e n t i t y t h e n ~ has an i n f i n i t e n u m b e r of n o n f i x e d p o i n t s in its a c t i o n on Q2 so we can f i n d an i n f i n i t e subset E of Q2 w i t h E f l a ( E ) : O. I f / c E K is t h e h o m e o m o r p h i s m z ~-> y - ~ z t h e n t h e e q u a t i o n ]ca'z = z is e q u i v a l e n t t o t h e s y s t e m zp = z~(p) -~- yp (p C Q2) so t h a t if E o is a subset of E c o n t a i n i n g e x a c t l y n e l e m e n t s t h e set o f f i x e d p o i n t s of /ca' is a subset o f
158 B . E . JOHNSON
{z : z ~ Z, % = z~(~) § y~, p ~ Eo}
where this l a t t e r set has v measure 2 -~. Thus t h e set of f i x e d points of k~x' has measure zero. I f k E K t h e n k has no f i x e d points unless k is t h e i d e n t i t y . T h u s b y L e m m a 3 we see t h a t G satisfies condition *. G is ergodie on Z because K is.
Thus [1, p. 135] condition (i) of Theorem 3 holds.
L e t H be t h e subgroup of G containing K a n d those h o m e o m o r p h i s m s ~' where c ~ E / " I = { c ~ : ~ S L ( 2 , O), ~ = 0} a n d H~ t h e group g e n e r a t e d b y K a n d t h e c~' w i t h
a E r 2 = {c~ : ~ E SL(2, O), oh1 = 1, ~21 = 0}.
F 2 is n o r m a l in 1"1 a n d //1 a n d H are t h e inverse images of /"2 a n d F 1 u n d e r t h e isomorphism s:/c~'~-> c~ of G/K onto SL(2, Q) so t h a t K is n o r m a l in H 1 , / / 1 is n o r m a l in H a n d K, H1/K a n d H / H 1 are abelian. Thus [3, Theorems 17.5 a n d 17.14] H is amenable. I n t h e usual w a y SL(2, Q) acts on the r a t i o n a l projective line a n d /"1 is t h e subgroup leaving (0, 1) fixed. U n d e r z t h e action of G on G/H is m a p p e d onto this action a n d it is well k n o w n t h a t the action of SL(2, Q) on t h e projective line is 3-fold transitive. Thus condition (iii) of T h e o r e m 2 is satisfied. B y [1, p. 135] c~ is a I I 1 factor.
To complete our example we copy t h e a r g u m e n t in [8, L e m m a 7] to show t h a t does n o t have p r o p e r t y P . As G contains a group isomorphic w i t h SL(2, Z) which in t u r n contains a free group on two generators ([2, p. 26]), G is n o t amen- able [3, T h e o r e m 17.16]. H o w e v e r ~N is spatially isomorphic w i t h ~L~' [1, p. 137, Ex. 1] so t h a t if ~d has p r o p e r t y P so has ~?-3' a n d in this ease there is a s t a t e on ~ ( ~ ) ) w i t h T ( U * A U ) = ~(A) w h e n e v e r A E ~ s a n d U is u n i t a r y in
~;~' [8, Corollary 6]. I f F C f ~ ( G ) a n d A F is the element of ~-2(~) defined b y (AF),, , = F(s)Ia if s = t, (AF),, t = 0 otherwise, t h e n denote z(AF) b y M(F).
M is t h e n a state on / ~ ( G ) a n d M(gF) = z(AgF) : z(~J'gAf~7"g*) = z(AF) = M(F) so t h a t M is a n i n v a r i a n t m e a n for f ~ ( G ) .
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8. SCHWARTZ, J. T., Two finite, non-hyperfinite, non-isomorphic factors, Comm. Pure Appl.
Math. 14 (1963), 19--26.
Received September 11, 1972 ]3. E. Johnson
U n i v e r s i t y of Newcastle Newcastle upon Tyne E n g l a n d
