Solid yon Neumann algebras

Acta Math., 192 (2004), 111-117

@ 2004 by institut Mittag-Leflter. All rights reserved

Solid yon N e u m a n n algebras

by

NARUTAKA OZAWA University of Tokyo

Komaba, Japan

Dedicated to Professor Masamichi Takesaki on the occasion of his 70th birthday

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

Recall t h a t a von N e u m a n n algebra is said to be diffuse if it does not contain a minimal projection. We say t h a t a yon N e u m a n n algebra 3,l is solid if for any diffuse von N e u m a n n subalgebra A in 34, the relative c o m m u t a n t . A ' N M is injective. A solid yon N e u m a n n algebra is necessarily finite. We prove the following t h e o r e m which answers a question of Ge [G] whether the free group factors are solid.

THEOREM 1. The group von N e u m a n n algebra s of a hyperbolic group F is solid.

Recall t h a t a factor 34 is said to be prime if A d ~ A d l ~ 3 4 2 implies t h a t either 341 or 342 is finite-dimensional. T h e existence of such factors was proved by P o p a [P2]

who showed t h a t the group factors of uncountable free groups are prime. T h e case for countable free groups had remained open for some time, but was settled by Ce [G]. This was generalized by ~tefan [~t] to their subfactors of finite index. Our t h e o r e m gives a further generalization. Indeed, combined with a result of P o p a [P3] (Proposition 7 in this paper), we obtain the following proposition.

PROPOSITION 2. A subfactor of a solid factor is again solid, and a solid factor is non-F and prime unless it is injective.

Notably, this provides infinitely m a n y prime I I l - f a c t o r s (with the p r o p e r t y (T)).

Indeed, t h a n k s to a t h e o r e m of Cowling and Haagerup [CH], for lattices Fn in Sp(1, n), we have s whenever mT~n. However, we are unaware of any non-injective solid factor with the Haagerup p r o p e r t y other t h a n the free group factor(s). This proposition also distinguishes the factor ( s 1])*L;F~ from the free group factor s which answers a question of Shlyakhtenko.

The author was supported by the JSPS Postdoctoral Fellowships for Research Abroad.

2. P r e l i m i n a r y r e s u l t s o n r e d u c e d g r o u p C * - a l g e b r a s

For a discrete group F, we denote by A (resp. Q) the left (resp. right) regular representa- tion on 12P, and let C~F (resp. CoF ) be the C*-algebra generated by A(F) (resp. Q(F)) in B(12F) and/2F=(C~,F)" be its weak closure. The C*-algebra C~F (resp. the von Neu- mann algebra/2F) is called the reduced group C*-algebra (resp. the group yon N e u m a n n algebra).

The study of C*-norms on tensor products was initiated in the 1950s by Turumaru, but the first substantial result was obtained by Takesaki [T] who showed that the minimal tensor norm is the smallest among the possible C*-norms on a tensor product of C*- algebras. He also introduced the notion of nuclearity and found t h a t the reduced group

*F

C*-algebra C~ 2 of the free group F2 on two generators is not nuclear. Namely, the ,-homomorphism

C~,F2| ~ E ai| > a i x ~ ~ B(12F2)

i = 1 i = 1

is not continuous with respect to the minimal tensor norm. Yet, Akemann and Ostrand [AO] proved the remarkable theorem t h a t it is continuous if one composes it with the quotient map 7r from B(12F2) onto the Calkin algebra B(12F2)/K(12F2). By a completely different argument, Skandalis (Th~or~me 4.4 in [Ski) proved the same for all discrete sub- groups in connected simple Lie groups of rank one, and Higson and Guentner (Lemma 5.2 in [HG]) for all hyperbolic groups. In summary, we have the following result.

THEOREM 3. Let F be a hyperbolic group or a discrete subgroup in a connected simple Lie group of rank one. Then, the *-homomorphism

i = 1

is continuous with respect to the minimal tensor norm on

C~r|

The crucial ingredient in the proof was the amenability of the action of F on a suitable boundary which is 'small at infinity'. For the information on amenable actions, we refer the reader to the book [AR] of Anantharaman-Delaroehe and Renault. Since F acts amenably on a compact set, C~,F is embeddable into a nuclear C*-algebra and thus has the property (C) of Archbold and B a t t y (Theorem 3.6 in [AB]). Although we do not need this fact, we mention that the property (C) is equivalent to exactness by a deep theorem of Kirehberg IN]. By Effros and Haagerup's theorem (Theorem 5.1 in lEg]), the property (C) implies the local reflexivity. In summary, the following lemma is true.

S O L I D V O N N E U M A N N A L G E B R A S 113 LEMMA 4. Let F be as above. Then, C~F is locally reflexive, i.e., for any finite- dimensional operator system Ec(C~F)**, there is a net of unital completely positive maps Oi: E-+C~F which converges to idE in the point-weak* topology.

3. P r o o f o f t h e t h e o r e m

We recall the following principle [Ch]: If ~: A - + B is a unital completely positive m a p and its restriction to a C*-subalgebra A o C A is multiplicative, t h e n we have q~(axb)=

q2(a)~(x)q~(b) for any a, beAo and x ~ A .

Let A r c f14 be finite von N e u m a n n algebras with a faithful trace T on A/[. Then, there is a normal conditional expectation c2r from ~4 onto Af, which is defined by the relation

~-(cx(a)x)=T(ax) for aEAA and x e A f . This implies t h a t a v o n N e u m a n n subalgebra of a finite injective von N e u m a n n algebra is again iujective. Moreover, ear is unique in the sense t h a t any trace-preserving conditional expectation from A/I onto A/" coincides with c]r Indeed, for any aEA/[ and x C N , we have

T(c' (a)x) = T(c'(ax)) = T(ax) = T(eN(a)x).

We say t h a t a yon N e u m a n n subalgebra A/[ in B(7{) satisfies the condition (AO) if there are unital ultraweakly dense C*-subalgebras B c J t 4 and C c ~ 4 t such t h a t B is locally reflexive and the * - h o m o m o r p h i s m

u : B | 1 7 4 >~r aixi c B ( n ) / K ( 7 - / )

i = ]

is continuous with respect to the minimal tensor n o r m on B |

We have seen in w t h a t the group von N e u m a n n algebra s satisfies the condition (AO) whenever F is a hyperbolic group or a discrete subgroup in a connected simple Lie group of rank one.

LEMMA 5. Let B C ~4 and C C M ~ be unital ultraweakly dense C*-subalgebras with B locally reflexive, and let ArC M be avon Neumann subalgebra with a normal conditional expectation cH onto A/'. Assume that the unital completely positive map

n n

OPjV: B | ~ E a~| ~ E c.~'(ai)x~ C B(7-/)

i = l i = 1

is continuous with respect to the minimal tensor norm on B | Then Af is injective.

Proof. Since B| CcB(7-l)| and B(7-/) is injective, 6pr162 extends to a unital completely positive m a p ~: B(7-/)| C--+B(7-/). Then, 9 is automatically a C-bimodule map. P u t ~ p ( a ) = ~ ( a | for a e B ( H ) . Then, for every aeB(7-t) and x e C , we have

xr = ~ ( l | 1 7 4 = ~ ( a | = ~ ( a | 1 7 4 = r

Hence, r maps B('~) into C ' = M . It follows that ~=aN~b: B ( 7 / ) - + H is a unital com- pletely positive map such that ~IB=gr Let I be the set of all triples (s ~', ~) where g C A f and ~ C A f . are finite subsets and r is arbitrary. The set I is directed by the order relation ( g l , ~ l , c l ) E ( E 2 , f f 2 , r if and only if E1CE2, ~'1C~'2 and r Let i = (g, 9 r, 6)E1 and let E C N" be the finite-dimensional operator system generated by g.

We note that E C M = p B * * and r where pEB** is the central pro- jection supporting the identity representation of B on 7-/. Since B is locally reflexive (cf. Lemma 4), there is a unital completely positive map 0i: E - ~ B such that for a E g and f E f f , we have I(ei(a), r - ( a , r162 Take a unital completely positive ex- tension ~)~: B(?-/)--+B(~/) of 0i, and let a i = ~ 0 i : B(N)--+Af. It follows that for aGE and f E Y ,

I(~(a), f ) - ( a , f) l = I(r f) ^- (a, f) l = I(e~(a), r162 ^- (a, c ~ ( f ) ) l < r Therefore, any cluster point, in the point-ultraweak topology, of the net {(ri}i~z is a

conditional expectation from B(?-l) onto Af. []

We now prove Theorem 1, or more precisely the following result.

THEOREM 6. A finite yon Neumann algebra M satisfying the condition (AO) is solid.

Proof. Let A be a diffuse von Neumann subalgebra in M . Passing to a subalgebra if necessary, we may assume that A is abelian and prove the injectivity of N ' = A ~ N M . It suffices to show that ~.~c in Lemma 5 is continuous o n ]~@min C . Since A is diffuse, it is generated by a unitary u E A such that l i m k - ~ u k = 0 ultraweakly. Indeed, every diffuse abelian von Neumann algebra with separable predual is .-isomorphic to L ~ [0, 1], and the unitary element u(t)=e2~itcL~[O, 1] has the required property (where i here, but not elsewhere, is the imaginary unit). Let ~Pn(a)=n -1 ~-~.~=1 ukau-k for acB(?-l), and let ~:B(~)--+B(7-I) be its cluster point in the point-ultraweak topology. It is not hard to see that ~ 1 ~ is a trace-preserving conditional expectation onto X and hence 9 1M=cr162 It follows that for a n y Ein__l a i @ x i C . ~ @ . / ~ t, we have

q2 aixi = r162 =aPW ai| .

i=1 \ i = 1

Since l i m ~ _ ~ u k = 0 ultraweakly, we have K ( ? - l ) c k e r ~ . This implies that ~ = ~ r for some unital completely positive map ~:B(N)/K(?-/)-+B(7-I). Since ~ in the condi- tion (AO) is continuous o n B@minC , so is (I)Af=02/]. []

The following proposition was communicated to us by Popa [P3]. The author is grateful to him for allowing us to present it here.

S O L I D V O N N E U M A N N A L G E B R A S 115 PROPOSITION 7. Assume that the type II1 factor A4 (with separable predual) con- tains a non-injective yon Neumann subalgebra Afo such that AfgNfl4 ~ is a diffuse yon Neumann algebra, where All ~ is an ultrapower algebra of All. Then there exists a non- injective yon Neumann subalgebra .hflC A4 such that Af~ AA/[ is diffuse.

Proof. Replacing it with a subalgebra if necessary, we may assume that the non- injective yon Neumann subalgebra No is generated by a finite set {xl, x2, ..., Xm}. By Connes' characterizations of injectivity (Theorem 5.1 in [Co]), it follows that there exists c > 0 such that if { 1, . . . , x ~ } C A / [ X / are so that ]]x~-xil]2<~c then {x~}i generates a non- injective yon Neumann subalgebra in ~4. Indeed, if there existed injective von Neumann algebras B k C A 4 such that lima I]xi-cgk(x~)[]2=0 for all i, then for any ~j~=l a j | Af0| we would have

~< lim inf eB~ aj

aj B(n) k - - ~ B(n)

n n

= l i m i n f ~ - ~ r 1 7 4 <~ a j | ,

k -~. oo

which would imply C* (N'0, Ad')~N'0| Ad' and the injectivity of Af0 (cf. the proof of L e m m a 5).

Since JV'g A 2t4 ~ is diffuse, it follows by induction that there exists a sequence of mu- tually commuting, ~--independent two-dimensional abelian ,-subalgebras An C A/l, with minimal projections of trace 89 such that

< E for all i.

But if we let B n = A I V A 2 V . . . V A , , then we also have

for all i.

2n+1

Since ~B-nMOSA:~+lnM=C~;~+lnM, if we let A = V ~ B n and take into account that c A , n ~ =limn-~oo r (see e.g. [P1]), then by triangle inequalities we get

]]x~-CA,nM(Xi)]]2 <~c for all i.

Thus, if w e take All to be the y o n N e u m a n n algebra generated by

then All satisfies the required conditions. []

4. R e m a r k

We note the possibility t h a t , for a hyperbolic group F, the . - h o m o m o r p h i s m

i=1 \ i=1 "

m a y be continuous with respect to the minimal tensor norm. If this is the case, then it would follow t h a t a yon N e u m a n n subalgebra A r C / : F is injective if and only if

c* (At, c ; r ) n K(t2F) = {0},

which would reprove our results (modulo T h e o r e m 2.1 in [Co]).

Acknowledgment. T h e author would like to t h a n k Professor Sorin P o p a for showing him Proposition 7, and Professor Nigel Higson for providing him with the references

[HG]

and [Sk]. This research was carried out while the author was visiting the University of California at Los Angeles under the support of the Japanese Society for the P r o m o t i o n of Science Postdoctoral Fellowships for Research Abroad. He gratefully acknowledges the kind hospitality of the UCLA.

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NARUTAKA OZAWA

Department of Mathematical Sciences University of Tokyo

Komaba 153-8914 Japan

narutaka@ms.u-tokyo.ac.jp Received February 19, 2003