On Maximal Injective Subalgebras of Factors

Advances in Mathematics  1536 advances in mathematics 118, 3470 (1996) article no. 0017

On Maximal Injective Subalgebras of Factors Liming Ge* Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395 Received March 1, 1995

1. Introduction There are three closely related themes in this article. 1. Under what conditions does a subalgebra of a tensor product ``split?'' 2. What are the maximal injective subalgebras of an algebra? 3. What are the maximal injective subfactors of a factor? The splitting property referred to in Question 1 states that, in certain situations, a subalgebra of the tensor product A1 A2 of algebras A1 and A2 ``splits'' as a tensor product of subalgebras of A1 and A2 . More specifically, we shall consider unital algebras A1 and A2 over C and subalgebras of A1  A2 that contain A1 (that is, A1  C1). If A1 is M n(C), the algebra of all n_n matrices over the complex numbers C, and A1 is another complex matrix algebra, with the tensor product taken over C, the splitting results from a simple algebraic calculation. If A1 and A2 are infinite-dimensional algebras over fields of arbitrary characteristic, with varying assumptions on their structure, the situation is less clear. When A1 and A2 have topological and analytic structure, and the tensor product A1  A2 is formed to reflect that structure, the splitting question becomes a deeper and more intricate one. One of the principal results of this article involves von Neumann algebras R1 and R2 and their von  R2 . Specifically, we prove the Neumann-algebra tensor product R1  following result (see also [Ge]). Theorem A. If M is a factor of finite type, R is a finite von Neumann algebra, and S is a von Neumann subalgebra of M  R that contains M (=M  CI), there is a von Neumann subalgebra R0 of R such that S=M  R0 . * Research supported by a Sloan Dissertation Fellowship. Current address: Department of Mathematics, MIT, Cambridge, MA 02139.

34 0001-870896 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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In the preceding statement, we use ``finite'' in the sense of Murray and von Neumann [M-N1]: M is either a factor of type II 1 or of type I n (isomorphic to M n(C)). To illustrate the depth and importance of a theorem of the nature Theorem A (without a restriction on the type), we note that it is a stronger result than the celebrated ``commutation theorem'' of Tomita (T$ denotes the set of operators in B(H) that commute with all operators in T, the commutant of T) (cf. [Ta1]) (R  S)$=R$  S$, answering the difficult and longstanding problem posed by Dixmier [Di1]. To see this, suppose we have our splitting property for subalgebras of R$  B(H) containing R$  CI. (For simplicity, we assume that R is a factor.) It is immediate that (R  S)$ contains R$  S$ and, hence, R$  CI. On the other hand, it is easy to show that (R  CI )$= R$  B(H). Since R   CIR  S, we have that (R  S)$(R   CI )$R$  B(H). From our ``splitting'' assumption (R  S$)=R$  T, for some T. Thus CI  T commutes with CI   S, and TS$. At the same time, CI  S$R  S$(R  S)$=R$  T, whence S$T. Thus T=S$, and (R   S)$=R$  T=R$   S$. In work in progress with R. Kadison, we are studying the possibility of removing the restriction on types in the tensor product splitting result. With regard to Questions 2 and 3, Connes [Co2] shows that a factor M acting on a separable Hilbert space H is hyperfinite (the weak-operator of an ascending union of finite-dimensional C*-algebras) if and only if there is a conditional expectation (equivalently [To1], an idempotent mapping of norm 1) sending B(H), the algebra of all bounded operators on H, onto M. Von Neumann algebras that are the range of such an expectation are said to be injective. In [F-K], Fuglede and Kadison established the existence of maximal hyperfinite subfactors of a II 1 factor. (They showed that, when such a maximal hyperfinite subfactor is proper, it is properly contained in its second ``relative commutant.'' This answered a ``normality'' question posed by Murray and von Neumann.) Fuglede and Kadison ask if each maximal hyperfinite subfactor of a II 1 factor has a trivial relative commutant (that is, only the scalars in the factor commute with the subfactor). In an outstanding series of papers, Popa [Po1, Po2, Po3] answers this last question, in the negative, as well as some other longstanding questions

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raised by Kadison. At the same time, Popa raises other questions involving maximal injective subalgebras and subfactors. Broadly then, Questions 2 and 3, above, acquire special importance in studying the structure of nonhyperfinite II 1 factors. We shall use our tensor product splitting result, together with ultraproduct and free product techniques to answer several of Popa's questions. In Section 3, we show that the tensor product of the hyperfinite II 1 factor R with a maximal injective subalgebra B of a finite von Neumann algebra S is maximal injective in R  S (cf. [Po2; Problem 4.5(1)]). In Section 4, we show that each non-atomic finite injective von Neumann algebra appears as (is isomorphic to) a maximal injective subalgebra of some II 1 factor. As a consequence, we prove that the lists of equivalence classes (up to * isomorphism) of maximal injective subalgebras of free group factors are all the same. In Section 5, using some results in Section 4, we construct some examples of maximal injective subfactors of some II 1 factors: in particular, one of a maximal injective subfactor of a II 1 factor with a non-injective relative commutant, and one of a maximal injective subfactor of a II 1 factor with two-dimensional relative commutant such that the von Neumann subalgebra generated by the subfactor and its relative commutant is not maximal injective. (See [Po2; the end of Section 1]). The next section contains some preliminary results and preparatory discussion of conditional-expectation, ultraproduct, free-product and slicemapping techniques. For the general theory of operator algebras (C*- and von Neumann algebras), we refer to [K-R I-IV]. The author wishes to express his deep gratitude to Prof. R. V. Kadison for many valuable and stimulating conversations.

2. Preliminaries There are four topics in this section: conditional expectations, free products, ultra-products and central sequences, and slice maps. (2.1) Conditional Expectations. Tomiyama [To1] shows that an idempotent 8 of norm 1 from a C*-algebra A onto a C*-subalgebra B is a conditional expectation: 1. 8 is linear and positive, 8(A)0 when A0; 2. 8 is a B-bimodule mapping on A, 8(B 1 AB 2 )=B 1 8(A) B 2 for all B 1 , B 2 in B and all A in A. Suppose R0 is the ultraweak closure of an ascending union of finitedimensional C*-algebras A n acting on a Hilbert space H. Each A n has a finite group Un of unitary elements that generates it linearly. In [Sc],

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Schwartz constructs an idempotent linear mapping of B(H) onto the commutant R$0 of R0 (a Schwartz projection) by averaging over the groups Un , successively, and passing to a Banach limit. Certain features of the Schwartz-projection construction associated with a free ultrafilter will be needed at several places later; we describe the construction in detail (in a formulation proposed by Kadison). where a n is the With T in B(H), let 8 n(T ) be (1a n )  Un # Un U n TU *, n order of Un . Since Un is a group, U n 8 n(T )=8 n(T ) U n for each U n in Un . As Un generates A n linearly, A n 8 m(T )=8 m(T ) A n for each A n in A n , when mn. Moreover, &8 n(T )&&T &. Let x and y be vectors in H and p be a point in ;(N)"N (a free ultrafilter). The function n  ( 8 n(T ) x, y), bounded by &T& &x& &y&, extends (uniquely) to a continuous function .(T, x, y) on ;(N). Routine calculation shows that .(T, x, y)( p) is a conjugate-bilinear form in x, y satisfying |.(T, x, y)( p)| &T & &x& &y&.

(1)

From the Riesz theorem, there is a (unique) 8(T ) in B(H) such that .(T, x, y)( p)=( 8(T ) x, y) for each x and y in H. From (1), &8(T )& &T &. Again, routine calculation shows that 8 is linear in T. Moreover, with A n in A n , A n 8 m(T )=8 m(T ) A n when mn. Since N"[1, ..., n] is dense in ;(N)"[1, ..., n], .(T, x, y) is continuous on ;(N), p  N, and .(T, A n x, y) &.(T, x, A * n y) vanishes on N"[1, ..., n], we have that ( 8(T ) A n x, y) &( 8(T ) x, A * n y) =0.

(2)

As (2) holds for all x and y in H, 8(T ) A n =A n 8(T ). Hence 8(T ) # A$n , and 8(T ) # R$0 . With T$ in R$0 , 8 n(T $)=T $ for all n, whence 8(T $)=T $. Thus 8 has range R$0 . With co R 0(T ) the convex hull of [UTU*: U a unitary in R0 ], we show that 8(T ) is in co R 0(T ) &, its ultraweak closure, for each T in B(H). More specifically, we prove that 8(T ) is in the ultraweak closure of [8 n(T ): n # N], which will establish our assertion since each 8 n(T ) # co A n(T ) co R 0(T ). Given x 1 , ..., x m , y 1 , ..., y m in H and a positive =, the continuity of .(T, x j , y j ) on ;(N) assures us that there is a neighborhood N j of p such that |.(T, x j , y j )( p)&.(T, x j , y j )( p$)| <= when p$ # N j . Let N be  m j=1 N j . Since N is dense in ;(N), there is an n in N & N. For this n, we have that |.(T, x j , y j )( p)&.(T, x j , y j )(n)| = |( [8(T )&. n(T )] x j , y j ) | <=

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for each j in [1, ..., m]. Thus 8(T ) is in the ultraweak closure of [8 n(T ): n # N]. (We use, here, the fact that the weak-operator and ultraweak topologies coincide on bounded subsets of B(H)cf. [K-R II; Remark 7.4.4].) Definition 2.1. A conditional expectation 9 of a von Neumann algebra S onto a subalgebra is proper when 9(T ) # co S (T ) & for each T in S. A family [A m ] of finite-dimensional C*-algebras A m whose union is ultraweakly dense in a von Neumann algebra R0 is a tower for R0 . We denote the conditional expectation constructed from a tower t for R0 and a point p in ;(N)"N by ``8 t, p .'' The results of the preceding arguments are summarized in the following: Proposition 2.2. Each 8 t, p is a conditional expectation of B(H) onto R$0 proper on each von Neumann algebra S containing R0 (onto S & R$0 ). In the next result, we describe a conditional expectation 8 of a finite von Neumann algebra M onto a von Neumann subalgebra N and establish some properties of it that will be needed. The SakaiRadonNikodym theorem [Sa2] can be used to define the conditional expectation 8, or a slightly more elaborate argument using the simpler [K-R II; Proposition 7.3.5] in conjunction with [K-R II; Theorem 7.2.15] will produce 8. Since the existence of 8 is well known by now, we shall not give the details of its construction. Theorem 2.3. If N is a von Neumann subalgebra of a finite von Neumann algebra M with (normal faithful ) trace {, there is a conditional expectation 8 of M onto N, whose trace { | N we denote by ``{ 0 '', such that {(TA)={ 0(8(T ) A)

(T # M, A # N).

(3)

Moreover, 8 is the unique conditional expectation of M onto N such that {(T )={ 0(8(T ))

(T # M).

(4)

If M admits a proper conditional expectation 9 of M onto N, then 9=8. Proof. As remarked, we shall assume the existence of the conditional expectation 8, satisfying (3), as given. Setting A equal to I in (3), we see that 8 satisfies (4). Suppose (4) is valid for 8 0 in place of 8, where 8 0 is a conditional expectation of M onto N. Then { 0(8(T ) A)={(TA)={ 0(8 0(TA))={ 0(8 0(T ) A)

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for each T in M and A in N. Hence { 0([8(T )&8 0(T )] A)=0

(T # M, A # N).

In particular, this last equality holds when A=[8(T )&8 0(T )]*. Since { 0 is faithful on N, 8(T )=8 0(T ). Hence 8, satisfying (4), is unique. Suppose 9 is a proper conditional expectation of M onto N. Then 9(T ) # co M (T ) &. By ultraweak continuity of { on M and that {(S)={(T ) when S # co M (T ), {(R)={(T ) for each R in co M (T ) &. In particular, Q.E.D { 0(9(T ))={(T ) for each T in M. Thus 9=8. If M is a factor in Theorem 2.3, then { (non-zero) is automatically normal and faithful. Whenever we refer to a conditional expectation of a type II 1 von Neumann algebra M onto a von Neumann subalgebra N, we always mean the unique conditional expectation that lifts the trace. In this case the conditional expectation is ultraweakly continuous. We denote it by 8 N when M is understood. (2.2) Free Products. Now, we recall some basic facts about free products of finite von Neumann algebras (see [Ch] and [V-D-N] for details). Suppose that M and N are finite von Neumann algebras with faithful normal traces { 1 and { 2 respectively. (In this paper, we always assume that traces are normalized, i.e. they take value 1 at the identity.) Let M Va N be the algebraic free product of unital algebras M and N. Equipped with the adjoint operation (M 1 N 1 M 2 N 2 } } } M n N n )*=N n*M n* } } } N 2*M * * M 1*, 2 N1 M Va N is a *-algebra. The conditions {(I )=1

and

{(A 1 A 2 } } } A n )=0,

where A j is in one of M or N and A j+1 is in the other, for j in [1, ..., n&1], and each A j has trace ({ 1 or { 2 , as is appropriate) 0, define, uniquely, by linear extension a trace function { on M Va N, the free product trace on M Va N. (We denote by I the multiplicative identity of M, N and M Va N.) Using { to define an inner product on M Va N and completing, we construct a Hilbert space, L 2(M V N, {) on which M Va N may be represented, faithfully, by ``left-multiplication.'' The weak-operator closure of the image of the representation is a von Neumann algebra M V N to which { extends (uniquely) as a trace (denoted, again, by {). Thus M V N is a finite von Neumann algebra, the free product of the finite von Neumann algebras M and N (with respect to the traces { 1 on M and { 2 on N).

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We define the free product F V G of two subsets F and G of finite von Neumann algebras M and N, respectively, to be the set of all operators in M V N that have the form A 1 A 2 } } } A n , where each A j is in one of F or G and A j+1 is in the other. Suppose M and N are von Neumann subalgebras of a finite von Neumann algebra R with a trace {. We say that M and N are free subalgebras of R (with respect to {) if the identity mappings of M and N into R extend to a * isomorphism of M V N (with respect to { | M and { | N) onto the subalgebra of R generated by M and N. This notion of freeness is equivalent to that in [V-D-N], which is defined in terms of the trace. (That is, M and N are free von Neumann subalgebras of R with respect to a trace { on R, if and only if for M j # M, N j # N, {(M 1 N 1 M 2 N 2 } } } M n N n )=0, when {(M j )={(N j )=0 for all j.) Two subsets of R are said to be * free if the von Neumann subalgebras they generate are free. Two von Neumann subalgebras M and N of R are independent with respect to a trace { on R if {(MN )={(M ) {(N ) for all M # M, N # N. This is equivalent to the orthogonality of the Hilbert subspaces L 2(M, {)  CI and L 2(N, {)  CI in L 2(R, {) where L 2(R, {) is the completion of R under the inner product ( A, B) ={(B*A) for A, B # R. The norm induced by this inner product is called the trace norm, denoted by & & 2 . Free von Neumann subalgebras are independent. Further results on independence appear in [Po3]. In particular, Lemma 2.5 of [Po3] is used frequently in our Section 4. For completeness, we state and prove it as follows.

Lemma 2.4. If B is a von Neumann subalgebra of a von Neumann algebra M of type II 1 (with trace {) and U is a unitary element in M such that for any =>0 there is a finite dimensional von Neumann subalgebra A = of B such that {(E)<= for all minimal projections E in A = , and UA = U* and B are independent (as subalgebras of M), then U is trace-orthogonal to B and to B$ & M. Proof. From our assumption, for any positive =, there is a finite-dimensional von Neumann subalgebra A = of B such that {(E )<= for all minimal projections E in A = , and U A = U* and B are independent (as subalgebras of M). Then, for each unitary operator V in B or B$ & M, VUA = U*V* and VBV* (=B) are independent. Thus {(EVUEU*V*)={(E )_ {(VUEU*V*) for any projection E in A = . Let E 1 , ..., E n be a maximal family of mutually orthogonal minimal projections in A = with sum I. Let A be the maximal abelian subalgebra of A = generated by E 1 , ..., E n . Then 8 A$ & M (A)= j E j AE j for each A in M, where 8 A$ & M is the conditional

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expectation from M onto A$ & M. Then, for any unitary operator V in B or B$ & M,

"

2

|{(VU)| 2 &8 A$ & M (VU)& 22 = : E j VUE j

" =: &E VUE & j

2

j

j

2 2

j

=: {(E j VUE j U*V*E j )=: {(E j ) {(VUE j U*V*) j

j

= : {(E j )==. j

As this inequality holds for each positive =, {(VU)=0 for each unitary operator V in B and B$ & M. This implies that {(VA)=0 for any A in B and B$ & M. Thus U is trace-orthogonal to B and B$ & M. Q.E.D The following lemma generalizes some related results in [Ka] and [Po3], and is related to the above lemma. Lemma 2.5. Suppose R is a non-atomic finite von Neumann algebra with separable predual and G is a discrete group. Let LG be the von Neumann algebra generated by the left regular representation of G (so that LG is a finite von Neumann algebra). If A is a non-atomic abelian von Neumann subalgebra of R, then A$ & (R V LG ) is contained in R. Proof. Since A is non-atomic abelian, A is isomorphic to LZ (see [K-R II; 9.4]). As isomorphisms are normal and {, the trace on R V LG , is normal, there is a unitary operator U in A such that [U n: n # Z] forms an orthonormal basis for L 2(A, { | A). Extend [U n: n # Z] to a orthonormal basis of L 2(R, { | R) and denote it by F (since R is dense in L 2(R, { | R) as a linear subspace and L 2(R, { | R) is separable, by assumption, we can suppose that F is contained in R (cf. [Di3])). We know that all unitary operators L g , g # G, form a natural orthonormal basis of L 2(LG , { | LG ). The free product, denoted by O, of F"[I ] and [L g : g # G, g{e], (as the set free product defined above) together with I forms an orthonormal basis for L 2(R V LG , {). If A is in A$ & (R V LG ), then A is also an element of L 2(R V LG , {). Write A= ! # F * ! !+ ' # (O"F) * ' ', where * ! , * ' # C. For each unitary operator V in R V LG , the mapping B  VBV* (B # R V LG ) extends to a unitary operator on L 2(R V LG , {). Since U # R, B  UBU* maps R (in L 2(R V LG , {)) onto itself hand, hence, its orthogonal complement onto itself. Thus U n( ! # F * ! !) U &n is R and U n( ' # (O"F) * ' ') U &n is orthogonal to R. Since U nAU &n =A, we have that U n( ' # (O"F) * ' ') U &n=  ' # (O"F) * ' ' for each n # Z. Since elements of O can be viewed also as elements in R V LG , U n'U &n makes sense in R V LG for ' # O.

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We prove, now, that U n'U &n is weak-operator convergent to 0 as n tends to infinity for each ' in O"F. In effect, we must show that {(`*U n'U &n!) tends to 0 as n tends to infinity for all ! and ` in O _ [I ]. Suppose '=E 1 L h1 E 2 } } } L hk&1 E k ,

!=F 1 L g1 F 2 } } } L gm&1 F m ,

`=H 1 L g$1 H 2 } } } L g$l&1 H l where E j , F j , H j # F, I  [E 2 , ..., E k&1 , F 2 , ..., F m&1 , H 2 , ..., H l&1 ] and h j , g j , g$j # G "[e]. That ', !, ` are expressible as indicated is a consequence of the definitions of O and F. Note, in particular, that E 1 , E k , F 1 , F m , and H 1 , H l are permitted to be I (which is in F), so that we include the possibility that ', !, or ` have expressions that begin or end with some L g ({I ). Note, too, that k>1, that is, there is some L hj ({I ) in the expression for ' since ' is assumed to be in O but not in F. All of the preceding information is needed in the computation that follows as well as the fact that R and LG are free relative to { in R V LG . We have that n {(`*U n'U &n!)={(H l*L* g$l&1 } } } H * g$1 H * 2 L* 1 U E 1 L h1 E 2

} } } L hk&1 E k U &nF 1 L g1 F 2 } } } L gm&1 F m ) n n ={(H *L* 1 g$l&1 } } } H * g$1(H * 2 L* 1 U E 1 &{(H * 1 U E 1) I

+{(H 1* U nE 1 ) I ) L h1 E 2 } } } L hk&1(E k U &nF 1 &{(E k U &nF 1 ) I +{(E k U &nF 1 ) I ) L g1 F 2 } } } L gm&1 F m ). Now, with the noted exceptions where they may be I, each of the E j , F j , H j in the preceding equality has trace { equal to 0 since each is (trace) orthogonal to I and each of the L g , L g$ , and L h has trace equal to 0 (since none is I, by choice). Moreover, as noted, L h1 {I (and L h1 may well be L g$1 ). On expanding the right-hand side of the preceding equality (using the distributive law, taking account of the information just noted and freeness of L G and R relative to {), it becomes the sum &n {(H *L* F 1 &{(E k U &nF 1 ) I ) l g$l&1 } } } H * g$1 L h1 E 2 } } } L hk&1(E k U 2 L* n _L g1 F 2 } } } L gm&1 F m ) {(H * 1 U E1)

* L* * U nE 1 &{(H 1* U nE 1 ) I ) +{(H l*L* g$l&1 } } } H 2 g$1(H 1 _L h1 E 2 } } } L hk&1 L g1 F 2 } } } L gm&1 F m ) {(E k U &nF 1 ) * L* +{(H l*L* g$l&1 } } } H 2 g$1 L h1 E 2 } } } L hk&1 L g1 F 2 } } } L gm&1 F m ) n &n F 1 ). _{(H * 1 U E 1 ) {(E k U

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n Since {(H * U n ) is the inner product of H 1 E * 1 U E 1 )={((H 1 E *)* 1 1 in 2 L (R V LG , {) with the element U n of an orthonormal basis for n &n F 1 )) L 2(R V LG , {), we have that {(H * 1 U E 1 ) (and, similarly, {(E k U tends to 0 as n tends to infinity. At the same time,

[E k U &nF 1 &{(E k U &nF 1 ) I : n # Z],

n n [H * E 1 ) I : n # Z] 1 U E 1 &{(H *U 1

are (norm-) bounded sets. It follows that the preceding sum tends to 0 as n tends to infinity. Hence U n'U &n is weak-operator convergent to 0 as n tends to infinity. Let  ' # (O"F) * ' ' be  kj=1 * 'j ' j +y, where &y& is less than a preassigned positive = (in L 2(R V LG , {)). As noted, for each unitary V in R V LG , the mapping B  VBV* extends to a unitary operator on L 2(R V LG , {). With v a unit vector in L 2(R V LG , {) and ( , ) { the inner product on L 2(R V LG , {), we have, for large n,

}

* ' ', v

:

} {

' # (O"F)

}U \ : * '+ U , v }  U } \ : * ' + U I, v } + |( U yU =

n

&n

'

{

' # (O"F) k

n

&n

'j

j=1

n

&n

j

, v) { |

{

2= from the CauchySchwartz inequality and since U n( kj=1 * 'j ' j ) U &n is weak-operator convergent to 0 in L2(R V LG , {). Thus ( ' # (O"F) * ' ', v) { =0 for each unit vector v, and  ' # (O"F) * ' '=0. Hence A= ! # F * ! ! and A # R. Q.E.D Corollary 2.6. With the notation of Lemma 2.5, if S is a non-atomic von Neumann subalgebra of R, then S$ & (R V LG )R. Proof.

A maximal abelian subalgebra A of S is non-atomic and S$ & (R V LG )A$ & (R V LG )R

from Lemma 2.5.

Q.E.D

(2.3) Ultraproducts and Central Sequences. Next, we review the construction and some basic facts concerning ultraproducts of finite von Neumann algebras (see [Di2], [Mc] and [Co2]). If M is a finite von Neumann algebra with a normal faithful trace { acting on the Hilbert space H=L 2(M, {), then l (N, M) (all norm bounded sequences in M indexed

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by N) is a von Neumann algebra acting on  n # N H. For a given free ultrafilter | (element of ;(N)"N), we define a linear functional { | on l (N, M) as follows: Let [M n ] n # N (or simply [M n ]), with M n # M, be an element in l (N, M). Then [{(M n )] n # N , a bounded sequence on N with values in C, extends to a continuous function on ;(N). (Compare our procedure in (2.1) for defining the Schwartz projections.) Define { |([M n ] n ) as the value of this continuous function at |. It is easy to see that { | is a tracial state on l (N, M). Hence, the left and right kernel of { | coincide and is a two-sided ideal K| in the algebra. Denote by M | the quotient l (N, M)K| . This quotient M | is called an ultraproduct algebra of M. The algebra M | is a finite von Neumann algebra and { | is a normal faithful trace on it [Sa1]. We embed M in M | as the constant sequences. If S is a von Neumann subalgebra of M, the restriction { 0 of { to S is a faithful normal trace on S. With the free ultrafilter |, we can form S | relative to { 0. In this case, l (N, S) is factored by K 0| , the kernel of { 0| . While l (N, S) is a subalgebra of l (N, M), S | is not a subalgebra of M |. Now, X # K| if and only if { |(X*X )=0 for X in l (N, M), and Y # K 0| if and only if { 0|(Y*Y )=0 for Y in l (N, S). Thus K 0| =K| & l (N, S). For A in S |, choose A in l (N, S) such that A =A+K 0| . Let .(A ) be A+K| . Then . is well-defined since K 0| K| . At the same time, . is injective since K 0| =K| & l (N, S). It follows, at once, from the definitions of . and the traces { 0| and { | , that . is a tracepreserving * isomorphism of S | into M |. We prove, generally, that such an isomorphism has as its image a von Neumann algebra. Lemma 2.7. Let S and R be a finite von Neumann algebras with faithful normal traces { and {$, respectively. Let  be a * isomorphism of S into R such that {$ b ={. Then (S) is a von Neumann subalgebra of R and  is ultraweakly bicontinuous. Proof. Applying the GNS construction to { and {$, we may assume that S and R act on Hilbert spaces H and K with (separating and) generating vectors u and v, respectively, such that | u | S={ and | v | R={$, where | x is the vector state associated with vector x. Then Au  (A) v extends to a unitary transformation U of H onto K0 , the closure in K of (S) v. Moreover, for A and B in S, UAU &1((B) v)=UABu=(AB) v=(A)((B) v). Thus UAU &1 =(A) | K0 . Since (S) | K0 is unitarily equivalent to S acting on H, (S) | K0 is a von Neumann algebra. As K0 is stable under (S), it is stable under (S) &, the strongoperator closure of (S). The mapping T  T | K0 of (S) & onto (S) & | K0 is a strong-operator continuous * isomorphism since v is

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separating for R (containing (S) & ). By continuity, (S) & maps into the strong-operator closure of (S) | K0 under this restriction mapping. Since the mapping is one-to-one, (S) & =(S). Thus (S) is a von Neumann subalgebra of R. From [K-R II; Remark 7.4.4], * isomorphisms between von Neumann algebras are ultraweak homeomorphisms. Thus  is ultraweakly bicontinuous. Q.E.D Thus ., defined preceding the above lemma, is an ultraweakly continuous * isomorphism of S | into M |. When we speak of S | as a von Neumann ``subalgebra'' of M |, we are referring to this embedding of S | in M |. These considerations will appear throughout Sections 4 and 5. When R1 and R2 are independent subalgebras of M relative to the trace { on M, then R |1 and R |2 are independent subalgebras of M | relative to { | . (If [A n ] in R |1 and [B n ] in R | 2 have traces { | equal to 0, then {(A n ) and {(B n ) tend to 0 at |. Since R1 and R2 are independent, {(A n B n )= {(A n ) {(B n ) also tends to 0 at |. Thus the trace of [A n ][B n ] is 0.) An element [M n ] in l (N, M) is called a central sequence if &M n A&AM n& 2  n 0 for all A # M. A central sequence [M n ] is nontrivial if lim inf n &M n &{(M n ) I& 2 >0. A non-trivial central sequence represents a non-scalar element in M$ & M |. The central sequences of II 1 factors are closely related to the property 1 of factors defined by Murray and von Neumann [M-N2]. Here we mention some results concerning central sequences and ultraproducts. These results will be used frequently in Sections 4 and 5. In the next paragraph, we assume that M has a separable predual. First, recall that a II 1 factor M has property 1 if and only if M$ & M | {CI and in this case M$ & M | is non-atomic [Co1]. Those II 1 factors that don't have property 1 are called full factors. Also McDuff [Mc] proves that M is * isomorphic to M  R0 if and only if M$ & M | is non-commutative, especially R$0 & R |0 is non-commutative, where R0 is the hyperfinite II 1 factor. If [M n ] represents an element in M$ & M |, then there is a subsequence [N k ]=[M nk ] of [M n ] such that [N k ] is a central sequence in M. (2.4) Slice Maps. Finally we state the main ingredient of the slice map results [To2]. The formulation of slice map results in [K-R IV] is especially suited to our needs. We quote from [K-R IV; Exercise 12.4.36] as a proposition. Proposition 2.8. Let R and S be von Neumann algebras and \ and _ be non-zero elements of R* and S* , respectively. Then (it is ``Show that'' in the original exercise)

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(i) there is a unique element \  _ of (R  S) * such that (\  _) (R  S)=\(R) _(S ) for each R in R and S in S and that &\ _&= &\& &_&; (ii) there are unique operators 8 _(T ) and 9 \(T ) in R and S, respectively, corresponding to each T in R  S, satisfying \$(8 _(T ))=( \$ _)(T ),

_$(9 \(T ))=( \ _$)(T )

for each \$ in R* and each _$ in S* ; (iii) 8 _ and 9 \ (as defined by (ii)) are ultraweakly continuous linear mappings of R  S onto R and S, respectively, satisfying 8 _((A I ) T(B I ))=A8 _(T ) B, 9 \((I  C) T(I D))=C9 \(T ) D for each T in R   S, A, B, in R, and C, D in S, and (that) 8 _(R  S)=_(S ) R,

9 \(R S )=\(R) S

when R # R and S # S; (iv) 8 _(T ) # R0 and 9 \(T ) # S0 if T # R0  S0 , where R0 and S0 are von Neumann subalgebras of R and S, respectively; (v) T # R0  S0 if 8 _$(T ) # R0 and 9 \$(T ) # S0 for each _$ in S* and each \$ in R* . We formulate some results appearing in the solution of this exercise as ``(vi)''. (vi) 8 _(T ) appears, naturally, as a bounded linear functional on R* (thus, it is an element of R). It is defined by: 8 _(T ): \$  ( \$ _)(T )

( \$ # R* )

and |(8 _(T ))( \$)| = |(\$ _)(T )| &\$& &_& &T &. We also know that, for a given T in R   S, 8 _(T ) is linear in T and _. From the above inequalities, we know that 8 _(T ) tends to 8 _$(T ) in norm if _ tends to _$ in norm, where _, _$ # S* . Similar results hold for 9 \(T ). Thus (v) is valid under the assumption that _$ and \$ may be any elements of families of normal states that generate, linearly, norm-dense subspaces of S and R, respectively.

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The preceding statements are true for general von Neumann algebras. In the following section, we restrict our considerations to the case in which R is a factor of type II 1 and S is a finite von Neumann algebra. 3. A Splitting Theorem for Subalgebras of Tensor Products With R a factor of type II 1 and S an arbitrary finite von Neumann algebra, let {~, {, and {$ be the normalized traces on R   S, R, and S, respectively. Passing to the GNS representations of R and S, on H and K engendered by { and {$, respectively, we may assume that R  S acts on H  K, and that u and u$ are generating trace vectors for R and S in H and K, respectively. Thus u  u$ (=u~ ) is a generating trace vector for R  S in H  K. From [K-R II; Theorem 7.2.3], each normal state | of R is a vector state | x | R of R for some (unit) vector x in H (since u is separating for R). As u is generating for R, there is a sequence [A n ] of operators in R such that [&A n u&x&] tends to 0. Multiplying by an appropriate positive scalar, we may assume that &A n u&=&x&=1. If | n is the vector state of R corresponding to A n u, then [&| n &|&] tends to 0. (Note that &| x &| y& &x+y& &x&y&, for vectors x and y in H.) Thus, from (vi) in Proposition 2.8, the vector states of R corresponding to unit vectors Au, with A in R, will suffice for the conclusion of (v) in [K-R IV; Exercise 12.4.36] (as quoted in (v) of Proposition 2.8). Now, with T in R, ( TAu, Au) ={(A*TA)= {(AA*T ). If Au is a unit vector, {(AA*)=1. Letting H be AA*, we denote by ``{ H '' the vector state of R corresponding to Au (with value {(HT ) at T in R). Our main theorem in this section follows. Theorem 3.1. If R is a factor of type II 1 and S is a finite von Neumann algebra, then each von Neumann subalgebra B of R  S that contains R  CI has the form R  T for some von Neumann subalgebra T of S. Proof. Let T be (R  CI )$ & B. It is a remarkable consequence of the slice-map technique (essentially, (vi) of Proposition 2.8 in Section 2) that the relative commutant of a tensor product of von Neumann algebras contained in another tensor product of von Neumann algebras is the tensor product of the relative commutants. (See [K-R IV; Exercise 12.4.37(i)].) Thus (R  CI)$ & (R  S)=CI  S and T  CI  S. It follows that T =CI  T, where T is a von Neumann subalgebra of S. Since R  CI and T are subalgebras of B, the von Neumann algebra R  T they generate is contained in B.

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Suppose B # B. Since 8 _ maps R  S onto R, for each _ in S* , of course 8 _(B) is in R. We show that 9 {H (B) # T, for each positive H in R, whence B # R  T, from the discussion preceding this theorem (in essence, Proposition 2.8 (v) of Section 2). It will follow that B=R   T, as asserted. It remains to show that 9 {H (B) # T. Since 9 {H (B) # S, we have that I 9 {H (B) is in (R  CI$). We must show that I 9 {H (B) # B. With T in R  S, let 9 {H (T ) be I  9 {H (T ). We show that 9 { (=9 {I ) lifts the trace {~ | (CI  S) to the trace {~ on R  S, that is, {~ b 9 { ={~. Since {~ b 9 { and {~ are linear and ultraweakly continuous, it suffices to establish this equality on simple tensors A  B, where A # R and B # S. As ({~ b 9 { )(A  B)={~({(A) I B)={(A) {$(B)={~(A B), the stated equality follows. On the other hand, by a deep result of S. Popa (see [Po1]) there is a hyperfinite subfactor R0 of R with trivial relative commutant in R. R0  CI is hyperfinite and has associated with it Schwartz projections onto (R0  CI )$ (as described in Section 2 (2.1)), each of which is proper. It follows from Theorem 2.3 in Section 2 that the restrictions of each of these Schwartz projections to R   S coincides with 9 { . (We use, again, the equality (R0  CI )$ & (R  S)=(R$0 & R)  ((CI)$ & S)=CI  S.) In particular, each of these restrictions is ultraweakly continuous. At the same time, 9 { is proper and 9 {(T ) # co R 0  CI (T ) & co R  CI (T ) &. Thus, with B in B, 9 {(B) # B since R  CIB. We conclude by noting that 9 {H (T )=9 {((H I ) T ).

(5)

To see this, observe that the terms on each side of this equality are linear and ultraweakly continuous in T. It suffices, therefore, to establish the equality when T is a simple tensor A  B in R  S. In that case, 9 {H (A  B)={(HA) I B=9 {((H  I)(A B)). If we apply (5) with B in place of T (and H in R), then (H  I ) B # B and 9 {H (B)=9 {((H I ) B) # B.

Q.E.D.

Some extended versions of the above theorem will appear in [G-K] by applying more powerful techniques of [Ta2] on conditional expectations and Dixmier approximation property for states. However the following example tells us that the factorial restriction to the tensor component contained in the subalgebra is necessary.

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Example. Let A be a von Neumann algebra with a non-trivial center, B any von Neumann algebra with dimension greater than 5. Choose a central projection P in A and a projection Q in B such that P, Q{0, I and QBQ has dimension at least 2. Let B1 and B2 be the von Neumann subalgebras of B generated by QBQ (together with I in B) and (I&Q)_ B(I&Q) (together with I in B) respectively. We consider the von Neumann subalgebra A of A  B generated by (PA)  B1 and ((I P ) A) B2 . Then A  CI is contained in A. But there is no von Neumann subalgebra C in B such that A=A  C. To see this, let Q$ be a non-zero projection in QBQ different from Q. Then (I&P)  Q$ is not contained in A. The following corollary answers a question raised by Popa [Po2; Problem 4.5 (1)]. Corollary 3.2. Let R0 be the hyperfinite II 1 von Neumann algebra and S a type II 1 von Neumann algebra with a maximal injective von Neumann subalgebra B. Then R0  B is maximal injective in R0  S. Proof. We know that R0  B is injective in R0  S. Suppose that there is an injective von Neumann subalgebra N of R0  S containing R0  B. From our theorem, there is a von Neumann subalgebra B1 of S such that N=R0  B1 and B1 contains B. Using the center-valued trace and the techniques indicated in (2.1) in Section 2, the conditional expectation 8 can be defined for finite von Neumann algebras and subalgebras, in particular, for N and B1 . It follows, since N is injective, that B1 is injective. But B is maximal injective. Thus B=B1 . Hence R0  B is maximal injective. Q.E.D. Theorem 3.1 and Corollary 3.2 together with examples in [Po2] enlarge the classes of maximal injective subalgebras of certain factors of type II 1 . We give more examples on maximal injective subalgebras in II 1 factors in the following two sections.

4. Maximal Injective Subalgebras of Free Products In [Po2], Popa shows that each generator of a (non-abelian) free group corresponds to a unitary operator that generates a maximal injective subalgebra of the free group factor generated by the left regular representation of the free group. That answers a longstanding problem of Kadison. In this section, we extend Popa's result by showing that any non-atomic finite injective von Neumann algebra with separable predual is maximal injective in its free product with each von Neumann algebra associated with a countable discrete group.

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Let R be a non-atomic finite injective von Neumann algebra with separable predual and { 0 a faithful, normal trace on R. From [Kap], [Co2] and [K-R; Sec. 9.4], there is a unique decomposition of R:

\

+

R=(Z0  R0 )   Zj  M nj (C) , j1

(6)

where each Zj is an abelian von Neumann algebra for j0, R0 is the hyperfinite II 1 factor and M nj (C) is the n j _n j matrix algebra over C. Since R is non-atomic, Zj is non-atomic for j1. We write R1 for the hyperfinite II 1 factor represented as the ultraweak closure of an irrational rotation algebra generated by two unitary generators U 1 and V 1 , such that U 1 V 1 =e 2?i%V 1 U 1 for some irrational number %. (The C*-algebra generated by two such unitary operators is called an irrational rotation C*-algebra. For details about embeddings and representations of the irrational rotation C*-algebra in the hyperfinite II 1 factor, we refer to [Br] and [P-V].) From the properties of the trace on R1 , we know that { 1(U 1m V 1n )=0 if |m| + |n| {0, where { 1 is the unique trace on R1 . Moreover [U 1m V 1n : m, n # Z] forms an orthonormal basis of L 2(R1 , { 1 ). Recall that a Cartan subalgebra A of a finite von Neumann algebra S is a maximal abelian von Neumann subalgebra of S whose normalizer generates S, where the normalizer of A is the set of unitary operators U in S such that UAU*/A. Each of U 1 and V 1 generates a Cartan subalgebra of R1 (these algebras are maximal abelian and each of the unitaries normalizes the algebra generated by the other). Lemma 4.1. With R and R1 as described before, there are a Cartan subalgebra A0 of R and a (trace-preserving) * isomorphism . of R into R1 such that .(A0 ) is the Cartan subalgebra of R1 generated by U 1 . Proof. If Dn is a Cartan subalgebra of M n(C) and R=Z  M n(C) with Z non-atomic abelian, then Z  Dn is a Cartan subalgebra of R. Since Z is * isomorphic to any Cartan subalgebra of the hyperfinite II 1 factor R2 and R2  M n(C) (=R1 ) is again hyperfinite, there is a * isomorphism from Z  M n(C) into R1 that maps Z  Dn onto a Cartan subalgebra of R1 . When M n(C) is replaced by the hyperfinite II 1 factor R0 , the same result holds. From the decomposition (6) of R, let P j be the central supports of Zj in R for j0 (in other words, P j are the identities in Zj ), and E 0 be a projection in Z0 such that E 0 Z0 is non-atomic and (P 0 &E 0 ) Z0 is completely atomic. Atoms in (P 0 &E 0 ) Z0 are denoted by E k for k1. Then  k0 E k =P 0 ,  j0 P j =I and

\

+ \

+

R=((E 0 Z0 )  R0 )   CE k  R0   Zj  M nj (C) . k1

j1

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Now [E k , P j : k0, j1] is a partition of the identity I in R and is a countable set of central projections. We know that {(E k ), {(P j ) for k0, j1 are positive numbers, denote them by * l for l1, then  l * l =1. Let a k be  kl=1 * l and a 0 =0. Then lim k a k =1. Let [M 2n(C)] n1 be an ascending sequence of matrix subalgebras of R1 with union ultraweakly dense in R1 . Suppose that [E (n) jk ] are compatible matrix unit systems for M 2n(C) for j, k=1, 2, ..., 2 n. (Compatible in the (n) (n&1) (n) (n) =E (n) sense that E (n&1) 11 11 +E 22 , ..., E 2n&12n&1 =E 2n &1, 2n &1 +E 2n2n .) Let A (n) be the ultraweak closure of the algebra generated by all E kk for n1 and k=1, ..., 2 n. Then A is a Cartan subalgebra of R1 . Let Aj be the ultraweak n closure of the algebra generated by those E (n) kk such that a j&1 k2 a j for all n1. Then A= j1 Aj . If Q j are the identities in Aj (they are projections in R1 ), then { 1(Q j )=a j &a j&1 =* j . From the construction, we see that for a given j, [Q j R1 Q j & M 2n(C)] n forms an ascending sequence of matrix algebras and the ultraweak closure of their union gives us Q j R1 Q j which is again hyperfinite. Moreover Aj , the ultraweak closure of the union of all diagonal matrices in [Q j R1 Q j & M 2n(C)] n , is a Cartan subalgebra of Q j R1 Q j . Since [{ 1(Q j ) : j1] is the same as the set [{(E k ), {(P j ): k0, j1], there is a (trace-preserving) one-one map from the family of orthogonal projections [E k , P j : k0, j1] (central projections in R) onto [Q j : j1] (in R1 ). Each E k R( =CE k  R0 ) is the hyperfinite II 1 factor for k1, E 0 R=E 0 Z0  R0 and P j R=Zj  M nj (C). Each of these algebras is * isomorphic to a von Neumann subalgebra of the hyperfinite II 1 factor and the image of a Cartan subalgebra of the algebra is Cartan in the hyperfinite factor. Thus for each P # [E k , P j : k0, j1], there is a corresponding Q j0 # [Q j : j1] such that {(P)={ 1(Q j0 ) and there is a * isomorphism from PR into Q j0 R1 Q j0 and this isomorphism maps a Cartan subalgebra AP of PR onto a Cartan subalgebra of Q j0 R1 Q j0 . From [C-F-W], we may assume that this Cartan subalgebra of Q j0 R1 Q j0 is Aj0 . Thus there is a * isomorphism . of R into  j Q j R1 Q j R1 . It is clear that the sum of AP for P # [E k , P j : k0, j1] is a Cartan subalgebra A0 of R. And from [C-F-W] again, we may assume that .(A0 ) is the Cartan Q.E.D. subalgebra of R1 generated by U 1 . From this lemma, we may consider R as a subalgebra of the hyperfinite II 1 factor R1 with a common Cartan subalgebra generated by U 1 . The restriction to R of { 1 on R1 yields a faithful normal trace on R. Let LG be the von Neumann algebra generated by the left regular representation of a countable discrete group G on the Hilbert space l 2(G). The canonical trace on LG is denoted by { e , where e is the identity in G. We may identify L 2(LG , { e ) with l 2(G) and the action of LG on l 2(G) with left multiplication on L 2(LG , { e ). Let G be [L g : g # G], the group of unitary operators in LG induced by the left multiplication of elements in G. Then G is an orthonormal

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basis for L 2(LG , { e ). Let M1 be the free product R1 V LG with trace { (={ 1 V { e ). Then M=R V LG is a von Neumann subalgebra of M1 . Throughout this section, we denote by | a given free ultrafilter | on N. Let 8 R and 8 R | be the conditional expectations of M and M | onto R and R |, respectively; 8 R 1 and 8 R |1 be the conditional expectations of M1 and M |1 onto R1 and R | 1 respectively. See Section 2 (2.1) and (2.3) for details. Lemma 4.2. Suppose that R is a non-atomic finite injective von Neumann algebra with a separable predual and G is a countable discrete group. Let M be as above. If A is an element in M | that commutes with U1 , then for each W in M such that 8 R (W )=0, the vectors W(A&8 R |(A)), (A&8 R |(A)) W, and W8 R | (A) &8 R | (A) W are mutually orthogonal in L2(M |, { | ). In particular &WA&AW& 22 &W(A&8 R |(A))& 22 +&(A&8 R |(A)) W & 22 . We state a special case of this lemma as our next lemma, the case where R is the hyperfinite II 1 factor. Lemma 4.3. Suppose that R1 is the hyperfinite II 1 factor and G is a countable discrete group. Suppose A is an element in M |1 (=(R1 V LG ) |, as defined above) that commutes with U 1 in R1 . Then for any element W in M1 such that 8 R 1(W )=0, the vectors W(A&8 R |1 (A)), (A&8 R |1 (A)) W, and W8 R |1 (A)&8 R |1 (A) W are mutually orthogonal in L 2(M | 1 , { | ). In particular &WA&AW& 22 &W(A&8 R |1 (A))& 22 +&(A&8 R |1 (A)) W& 22 . Note that M is a von Neumann subalgebra of M1 and when we consider M | and R | as subalgebras of M |1 , for B in M |, we show that 8 R |1 (B)= 8 R |(B) and then Lemma 4.2 follows from Lemma 4.3. Let [B n ] be a representing sequence for B, where B n # M. Then [8 R 1(B n )] and [8 R (B n )] are representing sequences for 8 R |1 (B) and 8 R |(B), respectively. We show that 8 R 1(B n )=8 R (B n ) for all n. Since RR1 and R1 has a separable predual, using an argument similar to that of the proof of Lemma 2.5, we can choose an orthonormal basis B 1 of L 2(R1 , { 1 ) that is an extension of an orthonormal basis B of L 2(R, { 1 | R) such that B 1 R1 and I # B. Let B 2 be the set free product of B 1 "[I ] and [L g : g # G, g{e] and B 3 be the set free product of B"[I ] and [L g : g # G, g{e]. Then B _ [I ] is an orthonormal basis of L 2(M1 , {) and B 3 _ [I ] is an orthonormal basis of L 2(M, { | M) (a subspace of L 2(M1 , {)). With T in M, 8 R (T ) # R and T&8 R (T ) is orthogonal to L 2(R1 , { 1 | R) (L 2(M1 , {)). Hence T& 8 R (T ) is orthogonal to B and lies in the (closed) span of B 3"Bwhich is orthogonal to R1 . Thus 8 R 1(T&8 R (T ))=0 and 8 R 1(T )=8 R (T ) for each T in M. In particular, 8 R 1(B n )=8 R (B n ) for all n, as we wished to show.

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The estimate in Lemmas 4.2 and 4.3 is the key to the proof of the main theorem of this section. Before we prove Lemma 4.3, we need some notation and preparation. Let F be an orthonormal basis of L 2(M1 , {) consisting of the identity I n and all unitary elements in the free product of [U m 1 V 1 : m, n # Z]"[I ] and G"[I ]. It is easy to show that all finite linear combinations of unitary elements in F, denoted by CF, form an ultraweakly dense * subalgebra of M1 . (Although F is not * closed, CF is.) The support of an element A in M1 is a subset of F that consists of all elements U in F such that {(U*A){0; the length of an element in F is defined as follows: the length n m1 n1 mk nk mk+1 V n1k+1 is of U m 1 V 1 is |m|; the length of U 1 V 1 W 1 } } } U 1 V 1 W k U 1 mj |m 1 | + |m 2 | + } } } + |m k+1 | where W j ({I) # G for 1 jk and U 1 V n1j {I for 2 jk. (Any element in F can be expressed in one of these two forms.) We extend the concept of length to all unitary operators in [ ;U: ; # C, | ;| =1, U # F] (this set of unitary operators forms a group) by defining the length of ;U to be the length of U. We denote the length of ;U by l( ;U). Then it is easy to see that |l(W )&l(V )| l(WV )l(W )+(V ), for all W, V # [ ;U: ; # C, | ;| =1, U # F]. If F0 is a subset of CF (or M1 ) and A is an element in M1 , then A F0 is the image of A under the orthogonal projection from L 2(M1 , {) onto m the Hilbert subspace spanned by F0 . Let F1 be the subset F"[U m 1 V1 : m, n # Z] of F. Note that when we have an expression of the form A S for some subset S of CF, we can treat A S as a vector in the Hilbert space L 2(M1 , {) and regard TA S as the image of A S under T. At the same time, A S may, for appropriate choices of S, correspond to an operator in M1 , and T may be an operator in M1 . In this case, the product of T with A S as operators corresponds to the same vector as the image of A S acted on by T. However, we shall also use expressions of the form TA S R. In this case, the vector interpretation holds when we replace R in M1 by its canonical image in M$1 under reflection about the (canonical) trace vector (I in L 2(M1 , {)). On the other hand, we can regard T, A S , and R as elements of the algebra of (unbounded) operators affiliated with M1 (see [M-N1] and [K-R IV; 8.7.60]), where multiplication is the standard product of unbounded operators followed by forming the closure. With T and R in M1 (=L (M1 , {)) this product TA S R # L 2(M1 , {) since A S # L 2(M1 , {). Proof of Lemma 4.3. Given any =>0 and with W given as in the lemma, since CF is an ultraweakly dense * subalgebra of M1 , by the Kaplansky density theorem there is an element W $0 # CF such that &W $0&&W& and &W$0 &W & 2 <=. Let W 0 be W $0 &8 R 1(W$0 ). Then 8 R 1(W 0 )=0 and &W 0 &&W$0 &+&8 R 1(W $0 )&2&W $0 &2&W &. Because

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8 R 1(W )=8 R 1(W 0 )=0, W 0 &W is orthogonal to 8 R 1(W$0 ) as vectors in L 2(M1 , {). Thus &W 0 &W & 2 =&W $0 &8 R 1(W $0 )&W & 2 &W $0 &W & 2 <=. Each T in M1 has an expansion in terms of the basis of F for L 2(M1 , {) and . R 1(T ) is expressed in terms of those unitary operators U n1 V 1m in F. Since W $0 is a finite linear combination of elements in F, 8 R 1(W $0 ) is again a finite sum. Thus W 0 (=W$0 &8 R 1(W$0 )) is in CF. Let n 0 &1 be the maximal length of the elements in the support of W 0 . Denote by S 1 the subset of F1 consisting of all unitary elements U that have the form: n1 mk nk mk+1 1 V n1k+1 , where W j ( {I) # G for 1 jk and Um 1 V 1 W1 } } } U 1 V 1 Wk U 1 mj nj n1 mk+1 1 V n1k+1 U 1 V 1 {I for 2 jk, such that |m 1 | 2n 0 &1 (U m 1 V 1 and U 1 may be equal to I ). Denote by S 2 (the ``dual'' of S 1 ) the subset of F1 n1 1 consisting of all elements U that can be expressed as: U m 1 V 1 W1 } } } mk nk mk+1 nk+1 m1 U 1 V 1 W k U 1 V 1 , where W j ( {I ) # G for 1 jk and U 1 V n1j {I for 2 jk, such that |m k+1 | 2n 0 &1. Let S 0 be S 1 _ S 2 and S be F1"S 0 . Now we choose a representing sequence [A n ] for A (in M |1 ) with A n # M1 m and assume that lim n  | &A n U m 1 &U 1 A n& 2 =0 for all m # Z. We may also assume that &A&1 and &A n&1, otherwise replace A by A&A& and A n by A n &A n&, respectively. In this case, &A n& 2 &A n&1. (i) We show that there is a neighborhood U of | in ;(N) such that &(A n ) S 0 & 2 = when n # U & N. Let n$0 be an integer multiple of 4n 0 such that n$0 2 5= &2n 0 . Then there is a neighborhood U (depending on = and n$0 ) of | in ;(N) such that &U k1 A n U &k 1 &A n& 2 <

= 4

when |k| n$0 and n # U & N. Then, when k # Z, 4n 0 |k| n$0 , and n # U & N, we have that 0k 0k &U 4n (A n ) S 1 U &4n &(A n ) U 4n 0 k S U &4n0 k & 2 1 1 1 1 1 0k 0k =&(U 4n A n U &4n &A n ) U 4n 0 k S U &4n0 k & 2 1 1 1 1 1

= 0k 0k &U 4n A n U &4n &A n& 2 < . 1 1 4 Using the parallelogram identity in the Hilbert space L 2(M1 , {) and the above inequalities, we get 0k 2 &(A n ) S 1& 22 =&U 14n0 k(A n ) S 1 U &4n &2 1 0k 0k 2 &U 4n (A n ) S 1 U &4n &2 1 1

2 +&U 14n0 k(A n ) S 1 U 1&4n0 k &2(A n ) U 4n 0 k S U &4n0 k & 2 1 1 1

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2 0k 0k =2 &U 4n (A n ) S 1 U &4n &(A n ) U 4n1 0 k S1 U &4n 0k &2 1 1 1 2 +2 &(A n ) U 4n 0 k S U &4n0 k & 2 1 1 1

=2 2 < +2 &(A n ) U 4n1 0 k S1 U &4n 0k &2. 1 8 n By the definition of S 1 (any element in it begins with U m 1 V 1 such that |m| 2n 0 &1) and comparing the beginnings of elements in 0k 0k S 1 U &4n ] for different k, we know that the linear subspaces of CF [U 4n 1 1 4n0 k 0k for k # Z are orthogonal to each other. Taking spanned by U 1 S 1 U &4n 1 the sum of above inequalities over those k with 0<4n 0 |k| n$0 , we have

2n$0 = 2 2n$0 +2 &A n& 22 . &(A n ) S 1& 22 < 4n 0 4n 0 8 Thus, since n$0 2 5= &2n 0 , we have &(A n ) S 1& 22 <= 28+4n 0 n$0 = 24. Similarly &(A n ) S 2& 2 <=2 and hence we have &(A n ) S 0& 22 &(A n ) S 1& 22 + &(A n ) S 2& 22 <= 2. So &(A n ) S 0& 2 =. This proves (i). (ii) We prove that for any n1, W 0(A n ) S , (A n ) S W 0 and W 0 8 R1(A n )& 8 R 1(A n ) W 0 are mutually orthogonal vectors in L 2(M, {). Suppose that U, U$ # F are in the support of W 0 . Then l(U ) (the length of U ) and l(U$) are less than n 0 . From 8 R 1(W 0 )=0, we know that U and U$ lie in F1 . Since any element in S begins and ends with elements of the n forms U m 1 V 1 with |m| 2n 0 , it follows that U S and SU$ span two orthogonal subspaces of CF by comparing the beginnings of their elements; that the linear spans of US and (F & R1 ) U$ are orthogonal by comparing the ends of their elements; that the linear spans of U S and U$(F & R1 ) are orthogonal since an element of U$(F & R1 ) has at most one U 1 term with power of absolute value great than n 0 (at or just before the last term), while each element of U S has at least two such U 1 terms; and that SU is orthogonal to (F & R1 ) U$ and U$(F & R1 ) by similar comparison. This proves that the supports of W 0(A n ) S , (A n ) S W 0 and W 0 8 R 1(A n )&8 R 1(A n ) W 0 are mutually orthogonal. Hence (ii) follows. By the definition of { | and L 2(M |1 , { | ), we see that [W 0(A n ) S ], [(A n ) S W 0 ] and [W 0 8 R 1(A n )&8 R 1(A n ) W 0 ] are mutually orthogonal in L 2(M |1 , { | ). For the computation that follows, recall the discussion preceding the proof of this lemma regarding the interpretation of (A n ) S , a vector in L 2(M1 , {), as an operator affiliated with M1 . &W(A&8 R |1 (A))&[W 0(A n ) S ]& 2  sup &W(A n &8 R 1(A n ))&W 0(A n ) S& 2 n # U&N

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 sup

[&(W&W 0 )(A n &8 R 1(A n ))& 2

n # U&N

+&W 0(A n &8 R 1(A n )&(A n ) S )& 2 ] = sup

[&(W&W 0 )(A n &8 R 1(A n ))& 2 +&W 0(A n ) S 0& 2 ]

n # U&N

 sup &(W&W 0 )(A n &8 R 1(A n ))& 2 + sup &W 0(A n ) S 0& 2 n # U&N

n#U&N

=(sup (&A n &8 R 1(A n )&+&W 0&))=(2 sup &A n&+&W&). n

n

Similarly, we have &(A&8 R |1 ) W&[(A n ) S W 0 ]& 2 =(2 sup &A n&+&W &). n

Also we have that &W8 R |1 (A)&8 R |1 (A) W&[W 0 8 R 1(A n )&8 R 1(A n ) W 0 ]& 2  sup &W8 R 1(A n )&8 R 1(A n ) W&(W 0 8 R 1(A n )&8 R 1(A n ) W 0 )& 2 n # U&N

 sup &(W&W 0 ) 8 R 1(A n )& 2 + sup &8 R 1(A n )(W&W 0 )& 2 n # U&N

n # U&N

2= sup &A n&. n

As W(A&8 R |1 (A)), (A&8 R |1 (A)) W, and W8 R |1 (A)&8 R |1 (A) W are arbitrary close to mutually orthogonal vectors in L 2(M | 1 , { | ), they are , { ). The inequality in this lemma follows mutually orthogonal in L 2(M | | 1 from the equality &WA&AW& 22 =&W(A&8 R |1 (A))& 22 +&(A&8 R |1 (A)) W& 22 +&W8 R |1 (A)&8 R |1 (A) W & 22 . The proof is complete.

Q.E.D.

The next lemma generalizes Lemma 3.1 in [Po2] and is used in proving Theorem 4.5. Lemma 4.4. Suppose that R is a finite non-atomic injective von Neumann algebra with a separable predual and G is a countable discrete group. Let M (=R V LG ) be as in Lemma 4.2. If B is a von Neumann subalgebra of M that contains R, then there is an orthogonal family [E j ] j0 of central projections in B with sum I such that BE 0 =RE 0 and BE j is a factor containing

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RE j as a proper subalgebra when j1 and E j {0. Moreover, (B$ & R | ) E j has a non-zero atomic part when j1. Proof. Note that the center Z of B is contained in the commutant of R. Thus, from Corollary 2.6, ZR. Let E 0 be the union of all projections P # Z such that BP=RP. First, we show that Z(I&E 0 ) is an atomic algebra. Assume, to the contrary, that there is a projection F in Z(I&E 0 ) such that ZF is non-atomic. Let A be R(I&F )+ZF. Then A is nonatomic. From Corollary 2.6, A$ & (R V LG )R. Since ZF is in the center of B, BF commutes with A. Thus BF is contained in RF and BF=RF. This contradicts the maximality of E 0 . Let [E j ] j1 be the atoms of Z(I&E 0 ) (so that  j0 E j =I ). Then BE j is a factor for each j1. As RE j BE j and RE j is non-atomic, BE j is a factor of type II 1 for each j1. Suppose that (B$ & R | ) E j is non-atomic for some j1 and let B1 be the algebra (B$ & R | ) E j +R |(I&E j ) (R | ). Since R is non-atomic and finite, the same is true of R |. Thus B1 is also non-atomic. Let U be the set of all unitary operators U in R such that {(U )=0. Since { is faithful and R is non-atomic, each self-adjoint operator in R is a linear combination of I and two unitary operators in R of trace 0. (Argue measure theoretically in a maximal abelian von Neumann subalgebra of R containing the selfadjoint operator.) Thus R is the set of all finite linear combinations of elements in U _ [I ]. Let W be the set free product of U with the set [L g : g # G, g{e]. Then W together with I generates an ultraweakly dense linear subspace of R V LG . Elements of W are unitary operators in R V LG (=M). For any W in W"U, we show that R | and WR |W* are independent subalgebras of M |. From (2.3), we only need to show that {(WA*B)=0 when {(A)={(B)=0 for A, B # R. We assume that W has the form U 1 W 1 } } } U k W k U k+1 with U j in U and W j in [L g : g # G, g{e] (when W has other forms, the proof is similar). Then by the freeness of R and LG , * U 1* B) {(WAW*B)={(U 1 W 1 } } } U k W k U k+1 AU* k+1 W* k U* k } } } W1 ={(U 1 W 1 } } } U k W k U k+1 AU* k+1 W* k U* k } } } W *(U * B&{(U *B) I )+{(U * 1 1 1 1 B) I ) *)=0. ={(U 1* B) {(U 1 W 1 } } } U k W k U k+1 AU* k+1 W* k U* k } } } W1 Thus B1 (R | ) and WB1 W* are independent subalgebras for any W in W"U. As B1 is non-atomic, it contains a finite-dimensional, von Neumann subalgebra A = such that { |(E )<= for a preassigned positive = and each minimal projection E in A = . Of course, A = and W A = W* are independent. Lemma 2.4 applies, and W is trace-orthogonal to B$1 & (R V LG ) |, in particular, to BE j ( B$1 ). We know that W"U generates a dense linear

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subspace of L 2(R V LG , {)  L 2(R, {). Thus BE j is contained in R, and Q.E.D. BE j =RE j . Again, this contradicts the maximality of E 0 . Theorem 4.5. Each non-atomic injective finite von Neumann algebra with a separable predual is maximal injective in its free product with any group von Neumann algebra associated with a countable discrete group. Proof. We adopt the notation of the preceding lemma. Suppose that R is a non-atomic injective finite von Neumann algebra with a separable predual and G is a countable discrete group. We want to show that R is maximal injective in M (=R V LG ). Suppose, to the contrary, that R is not maximal injective. Then there is an injective von Neumann subalgebra B of M containing R properly. From Lemma 4.4, we know that there is a central projection E in B (also in R) such that BE is a factor and (B$ & R | ) E has a non-zero atomic part. Since B is injective and BE contains the non-atomic RE, the factor BE is isomorphic to the hyperfinite II 1 factor. Hence (BE )$ & (BE) | has no atoms (see (2.3) in Section 2), and (B$ & R | ) E is a proper subalgebra of (BE )$ & (BE) | ((BE)$ & (EME) | ). Choose B in (BE )$ & (EME ) | and not in R |E. Since E is central in B (and in R), this B (=BE ) commutes with R and BE. In particular, for any element W in BE that is traceorthogonal to R, we have, from Lemma 4.2, that 0=&BW&WB& 2  &(B&8 R |(B)) W& 2 . Thus (B&8 R |(B)) W=0. Since RE is a proper subalgebra of (the II 1 factor) BE, with W 0 in BE "RE, we have that W 0 &8 RE (W 0 ) (=W ) in BE is trace-orthogonal to RE. Thus (B&8 R |(B)) P W =0, where P W is the range projection of W in BE. For any unitary element V in RE (considered as a von Neumann algebra with identity E ), VWV* is in BE and trace-orthogonal to R. Thus (B&8 R |(B)) P VWV* =0, where P VWV* is the range projection of VWV* in BE. Let E$ be the union of the range projections P W of all W in BE that are trace-orthogonal to R. Then E$W=W, E$E, E$ commutes with RE (since E$=VE$V* for all unitary operators V in RE), hence, with R, and (B&8 R |(B)) E$=0. From Corollary 2.6, E$ # R, hence E$ # RE. Since BE is a factor, if E${E, there is a non-zero partial isometry W in BE such that WW*(E&E$) and W*WE$. Then W=(E&E$) WE$. If T # R, then TWTW=TWE$T(E&E$) W=0. Thus { |(TW )=0, and W is traceorthogonal to R. Hence W=E$W=0, contradicting the choice of W. So E$=E, and (B&8 R |(B)) E=0. It follows that B=BE=8 R |(B) E=8 R |(BE )=8 R |(B), since E # R |. Thus B lies in R |, contradicting our choice of B. Hence R is maximal injective in M. Q.E.D.

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The last paragraph of the preceding argument uses the assumption that B (containing R properly) is injective only to conclude that (BE)$ & (BE ) | has no atoms. If we know that BE is a factor that is not full (i.e. that has property 1 ), then we have the same conclusion. The remainder of that argument produces the same contradiction. Thus BE must be full. The following corollary summarizes this discussion. Corollary 4.6. With the notation in Lemma 4.4, when B is a von Neumann subalgebra of M (=R V LG ) containing R and [E j ] is the orthogonal family of projections given in Lemma 4.4, then BE j must be a full factor when E j {0 for j1. In particular, if B is a subfactor of M containing R properly, then B is a full factor. Corollary 4.7 Every non-atomic finite injective von Neumann algebra is * isomorphic to a maximal injective subalgebra of each free group factor. Proof. If F n is the free group on n generators and R is a non-atomic finite injective von Neumann algebra with a faithful normal trace {, then R V LFn is * isomorphic to the free group factor on n+1 generators (see [Dy]). Q.E.D.

5. Maximal Injective Subfactors of Certain II 1 Factors In this section, we study maximal injective (or hyperfinite) subfactors of some II 1 factors. From [Kap] and [Co], we know that each injective type II 1 von Neumann algebra is isomorphic to the tensor product of its center with the hyperfinite II 1 factor. So, the isomorphism classes of maximal injective type II 1 von Neumann subalgebras of a type II 1 factor are classified by their centers. Theorem 4.5 tells us that any non-atomic finite injective algebra is isomorphic to a maximal injective subalgebra of some II 1 factors. Corollary 4.7 tells us that each non-atomic finite injective von Neumann algebra appears as a maximal injective subalgebra of certain non-injective II 1 factors; finiteness and non-atomicity are the sole algebraic restrictions. How these algebras are situated in some non-injective II 1 factor, up to * automorphism of the factor (that is, the description of the conjugacy classes of a given non-atomic injective algebra appearing as a maximal injective subalgebra of the II 1 factor), is a challenging problem akin to the conjugacy-class problem for maximal abelian subalgebras of factors. It was a longstanding question of Kadison whether a maximal hyperfinite subfactors of a II 1 factor can have a non-trivial relative commutant in the factor. Popa [Po2] first answered this question. He showed that a maximal hyperfinite subfactor may have an n-dimensional relative commutant. He

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asked in [Po2] whether a maximal hyperfinite subfactor of a II 1 factor together with its relative commutant generates a maximal injective subalgebra of the factor. We shall see that this question has a negative answer even when the relative commutant is two-dimensional and that the relative commutant may fail to be injective. The following lemma describes another special property of free products. (Compare it to Lemma 2.5.) We believe that these properties hold for free products of more general finite von Neumann algebras. Lemma 5.1. Let R1 be the hyperfinite II 1 factor, LG be a group von Neumann algebra associated with a countable discrete group G, and M1 be R1 V LG (with respect to the trace { 1 on R1 and the canonical trace { e on LG ). If A is a non-atomic abelian von Neumann subalgebra of LG and A is an element of M1 such that AAA*R1 , then A=0. Proof. As described earlier, R1 can be viewed as the strong-operator closure of an irrational rotation algebra generated by unitary operators U 1 , V 1 such that U 1 V 1 =e 2?i%V 1 U 1 with % an irrational number. Let F0 be the set of unitary operators in M1 consisting of I and the free product of n [U m 1 V 1 : m, n # Z]"[I ] and [L g : g # G, g{e]. Let { be { 1 V { 2 , the trace on n M1 . As R1 and LG are free, for any U in F0 , UU m 1 V 1 U* and L g are 2 orthogonal in L (M1 , {) when g{e. Thus UR1 U* and LG are independent subalgebras of M1 . From our assumption AAA*R1 , whence UAAA*U* UR1 U*. So {(B 1 UAB 2 A*U*)={(B 1 ){(UAB 2 A*U*) for B 1 # LG and B 2 # A. Since A is non-atomic, for any positive =, there are projections E 1 , ..., E n in A with sum I, such that {(E j )<= for each j. Then

} \\: E + UA+} = } : {(E UA) } : |{(E UAE )|

|{(UA)| = {

j

j

j

j

j

j

j

: {(E j UAE j A*U*E j ) 12 =: {(E j ) 12 {(UAE j A*U*) 12 j

j

=: &AE j & 2 {(E j ) 12 &A& : {(E j ) 32 j

j

= 12 &A& : {(E j )== 12 &A&. j

As this inequality holds for each positive =, {(UA)=0 for each U in F0 . But F0 is an orthonormal basis for L 2(M1 , {) and A is (among other things) an Q.E.D. element of L 2(M1 , {). Thus &A& 2 =0, and A=0. Now suppose that the group G is the direct product of an i.c.c. amenable countable discrete group H with a countable discrete group G 1 . Then

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LG =LH  LG1 and LH is the hyperfinite II 1 factor. We denote LH by R2 and choose unitary generators U 2 and V 2 in R2 such that U 2 V 2 = n e 2?i% $V 2 U 2 , where %$ is an irrational number. Then G=[U m 2 V 2 L g : m, n # Z, g # G 1 ], a set of unitary operators in LG , forms an orthonormal basis for L 2(LG , { e ). Let F be the set of unitary operators in R1 V LG (=M1 ) n consisting of I and the free product of [U m 1 V 1 : m, n # Z]"[I ] with G"[I ]. 2 Then F is an orthonormal basis of L (M1 , {) where {={ 1 V { e is the trace on M1 . In this section, R2 will play an important role in our construction. We define the length (different from that of the last section) of an element in F as follows: the length of I is 0; the length of an element of the form n1 k1 l1 m2 n2 k2 l2 mp np 1 Um 1 V 1 U 2 V 2 L g1U 1 V 1 U 2 V 2 L g2 } } } U 1 V 1

(or

n1 k1 l1 mp np kp lp 1 Um 1 V 1 U 2 V 2 L g1 } } } U 1 V 1 U 2 V 2 L gp ,

or

n2 k2 l2 mp np 2 U k21 V l21 L g1U m 1 V 1 U 2 V 2 L g2 } } } U 1 V 1 ,

or

n2 k2 l2 mp np kp lp 2 U k21 V l21 L g1U m 1 V 1 U 2 V 2 L g2 } } } U 1 V 1 U 2 V 2 L gp )

is |m 1 | + |k 1 | + |m 2 | + |k 2 | + } } } + |m p | (or |m 1 | + |k 1 | + |m 2 | + |k 2 | + } } } + |m p | + |k p |, or |k 1 | + |m 2 | + |k 2 | + } } } + |m p |, or |k 1 | + |m 2 | + |k 2 | + } } } + nj kj lj j |m p | + |k p |, respectively), where none of U m 1 V 1 , or U 2 V 2 L gj , for all j1, is equal to I. We extend the notion of length to the group of all unitary operators [*U : * # C, |*| =1, U # F] (in the same way as we did in Section 4): the length of *U, denoted by l(*U ), is equal to the length of U. If we choose % $ carefully, the mapping :: U 1  U 2 extends to a * isomorphism of R1 into R2 . For example, when 2%$=% (recall that U 1 V 1 =e 2?i%V 1 U 1 ), the mapping :: U 1  U 2 , V 1  V 22 determines an isomorphism. Let : be one such isomorphism. (Note that : is defined only on R1 and that :(R1 ) is contained in LH .) Since :(R1 ), a subfactor of R2 , contains a Cartan subalgebra of R2 , :(R1 )$ & R2 consists of scalars. Then (:(R1 )  CI )$ & LG =(:(R1 )   LG1 )=CI  LG1 (see the proof of Theorem 3.1). We denote CI )$ & (R2  CI  LG1 by C. Let N (=M 2(M1 )=M 2(C)  M1 ) be the 2_2 full matrix algebra with entries in M1 . Let E be ( 0I 00 ) in N and R the subalgebra of 0 ) for A # R1 . Then R is a hyperfinite N consisting of all elements ( A0 :(A) subfactor of N and ER=RE$R1 . Theorem 5.2. With the above definition and notation, R is a maximal hyperfinite subfactor of N, that is, there is no hyperfinite subfactor of N containing R properly, and R$ & N=CE+C(I&E) where I, here, is the identity of N.

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Proof. First, we show that R$ & N=CE+C(I&E). Suppose X= 11 X12 (X X21 X22 ) is in R$ & N. Then for A # R1 , X 11 X 12 X 22 21

\X

A

+\ 0

0 X 11 A X 12 :(A) = :(A) X 21 A X 22 :(A)

+ \ + A 0 X X = \ 0 :(A)+\ X X + AX AX = \:(A) X :(A) X + . 11

11

12

21

22

12

21

22

So X 11 A=AX 11 , X 12 :(A)=AX 12 , X 21 A=:(A) X 21 and X 22 :(A)=:(A) X 22 for all A # R1 . Since R1 has trivial relative commutant in M1 (this follows from Corollary 2.6) and :(R1 )  CI has relative commutant C in M1 , it follows that X 11 is a scalar multiple of the identity and X 22 is in C. From for all n in the equality X 12 :(U n1 )=U n1 X 12 , we have that X 12 =U n1 X 12 U &n 2 n &n n UU ) for all n in Z. As [U Z. With U in F, {(X 12 U )={(X 12 U &n 2 1 2 UU 1 : 2 n # Z] is an infinite subset F0 of F, the expansion of X* 12 (in L (M1 , {)) relative to the orthonormal basis F has some of the coefficients of elements of F0 arbitrarily near 0. Thus {(X 12 U )=0 for all U in F, and X 12 =0. Similarly, X 21 =0. Thus R$ & NCE+C(I&E ). The reverse inclusion is immediate. Now we want to show that R is a maximal injective subfactor of N. Assume the contrary. Then there is an injective subfactor N0 of N containing R properly. Note that N$0 & N is contained in CE+C(I&E ). We divide the remainder of the proof into three steps. (i) We show that EN0 E{( R01 00 ) and N0 has a central sequence [X n ] (n) (n) X X such that X n =( X 11(n) X 12(n) ) # N=M 2(M1 ), X n is selfadjoint, &X n&1 and 21 22 &X (n) 12 & 2 $>0 for all n1. 11 X12 First, we show that there are selfadjoint elements X=( X * X22 ) in N0 , X 12 with X jk in M1 , j, k=1, 2, such that X 12 {0. Otherwise, E commutes with N0 , and EN0 is injective. From Theorem 4.5, EN0 =ER1 (we regard EN0 and ER1 as subalgebras of M1 ). But N0 {R and ER=ER1 , thus, there is 0 ). Since an element ( A0 B0 ) in N0 such that B{:(A). Let Y be ( 00 B&:(A) RN0 , Y is in N0 . The element Y generates a non-trivial two-sided ideal in N0 (this ideal is contained in (I&E ) N0 ). But N0 , a II 1 factor, is simple. Thus there are selfadjoint elements X in N0 as described. Now ( R01 00 )EN0 E since RN0 (although E is not necessarily in N0 ). For A in R1 ,

X 11

\X*

12

X 12 X 22

A

+\ 0

0 :(A)

X 11

+\X*

12

X 12 X AX 11 +X 12 :(A) X* 12 = 11 X 22 V

+ \

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V , V

+

MAXIMAL INJECTIVE SUBALGEBRAS

63

which lies in N0 . If X 11 # R1 and X 11 AX 11 +X 12 :(A) X* 12 # R1 , then  CI is a non-atomic subalgebra X 12 :(A) X* 12 # R1 for all A in R1 . As :(R1 )  of LG and free with R1 , Lemma 5.1 would tell us that X 12 =0. Thus EN0 E{( R01 00 ) (otherwise X 11 and X 11 AX 11 +X 12 :(A) X*12 would lie in R1 ). Next, we show that E does not commute with N$0 & N |0 (where | is, for the remainder of the proof, a given free ultrafilter in ;(N)). Assume that E commutes with N$0 & N |0 . Since N$0 & N |0 is a type II 1 von Neumann algebra (from the fact that N0 is the hyperfinite II 1 factorsee Section 2, (2.3)), (N$0 & N | 0 ) E is also a type II 1 von Neumann algebra (with identity E). If M2 is the von Neumann subalgebra of ENE generated by EN0 E (here, we identity ENE with M1 ), then R1 /M1 and, since R1 has trivial relative commutant in M1 , M2 is a factor. From Corollary 4.6, we know that M2 is a full factor. So that M$2 & M | 2 is trivial (see, again, Section 2, (2.3) or [Co1]). On the other hand, M$2 & M |2 contains (N$0 & N |0 ) E which is non-atomic (because N0 is the hyperfinite II 1 factor). This contradicts our assumption. Let [Y n ] represent a selfadjoint element Y in N$0 & N |0 not commuting with E, and let f ( p) be &Y p E&EY p& 2 for all p # N. Then f extends to a continuous function on ;(N) and f (|)>0. Thus there is a positive $ 0 and a neighborhood U of | such that &Y p E&EY p & 2 $ 0 for all p in U & N. From this, we can choose another representative for Y (denote it, again, by [Y n ]) such that each Y n is selfadjoint and &Y n E&EY n& 2 $ 0 for all n in N. Since [Y n ] represents a non-scalar element in N$0 & N | 0 , as noted in Section 2 (2.3), there is a subsequence [Y nk ] (=[X k ]) that is a non-trivial central sequence in N0 . We know that [X n ] is a uniformly bounded sequence, so [X n r], r=sup[&X n& : n # N], gives the central sequence described at the beginning of (i). (ii) We prove, now, that there are a selfadjoint element Y )) in N0 , such that &Y &1, a selfadjoint central sequence [X n ] (=( YY1121 YY12 22 in N0 and a constant c$ such that &Y 21 X (n) 12 & 2 c$>0, for all n1, and (n) (n) X 11 X 12 (n) &X n&1, where X n =( X (n) X (n) ), X jk , Y jk # M1 , j, k=1, 2. 21

22

From (i), there is a selfadjoint central sequence [X n ] in N0 such that &X n&1 and &X (n) 12 & 2 $>0 for n1. We prove that there are a positive (nk ) ] k # N of [X (n) that c$, and n 0 in N and a subsequence [X 12 12 ] such (n) (n0 ) (nk ) 0 Y1 &(X 12 )* X 12 & 2 c$>0 for all k1. Let Y 1 be the sequence [( 0 0 )] in (n) N, where Y (n) 1 =X 12 for each n in N. Suppose there are no such c, n 0 and (nk ) subsequence [X 12 ]. Then, for each k in N, there is an n k (>n k&1 ) such (m) &k , when mn k . We may assume that n k k for that &(Y (k) 1 )* Y 1 & 2 <2 (k) (nk ) for all k. Let Y 2 be the subsequence [( 00 Y02 )] k # N of Y 1 , where Y (k) 2 =Y 1 each k in N. When mk, (m) (k) (nm ) & 2 <2 &k &(Y (k) 1 )* Y 2 & 2 =&(Y 1 )* Y 1

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(V)

64

LIMING GE

(k) since n m n k . Thus lim k  | &(Y (k)  1 and Y 2 are the 1 )* Y 2 & 2 =0. If Y |  2 =0. elements in N represented by Y 1 and Y 2 , respectively, then Y * 1 Y From (V), for each k in N, (m) (nk ) (m) &nk 2 &k, &(Y (k) 2 )* Y 2 & 2 =&(Y 1 )*Y 2 & 2 <2

(since n k k), when mn k . Hence we are in precisely the same situation (even notationally) with the subsequence Y 2 of Y 1 that we were in with the (k) sequence Y 1 . Let Y 3 be the subsequence [( 00 Y03 )] k # N of Y 2 , where (nk ) (m) &k Y (k) for each k in N. Again, &(Y (k) when mk. 3 =Y 2 2 )* Y 3 & 2 <2 Continuing in this way, we find Y 1 , Y 2 , Y 3 , ... such that Y j is the sub(k) Y (nk ) sequence [( 00 0j )] k # N of Y j&1 , where Y (k) j =Y j&1 for each k in N and (k) (m) &k each j in [2, 3, ...], and &(Y j )* Y j+1& 2 <2 when mk. When j< j$, (m$) =Y , where m$m. Thus if mk, then Y (m) j$ j+1 (m) (k) (m$) &k &(Y (k) j )* Y j $ & 2 =&(Y j )* Y j+1& 2 <2 (k) &k since m$mk. In particular, &(Y (k) for all k in N, when j )* Y j $ & 2 <2  j is the element of j< j$. It follows that Y *Y j  j $ =0, when j< j $, where Y N | represented by the sequence Y j in N. In addition, &Y j &1 (since &X n&1 for all n), &Y j & 2 (- 22) $. Thus the range projection Q j (a projection in N | ) of Y j (and of Y j Y *) has trace not less than 12 $ 2, and Q j , j j=1, 2, ..., n, are mutually orthogonal projections. Since N | is a type II 1 von Neumann algebra, 1={ |(I ){ |( nj=1 Q j ) 12 n$ 2 for all n # N. From this contradiction, there are a c$ and a selfadjoint central sequence as described. To establish (ii), let Y be X n0 and [X n ] (in the statement of (ii)) be the subsequence [X nk ] just constructed.

(iii) We, now, derive a contradiction from the assumption that R is a proper subset of the hyperfinite II 1 factor N0 . Let [X n ] be the selfadjoint central sequence and Y be the selfadjoint element described in (ii). Since [X n ] is a central sequence in N0 and N0 contains R, lim n

"\

U k1 0

0 :(U k1 )

+\

X (n) X (n) X (n) 11 12 11 & (n) (n) X 21 X 22 X (n) 21

+ \

X (n) 12 X (n) 22

+\

0 U k1 0 :(U k1 )

+" =0, 2

for all k in Z. This gives us that &k (n) &U k1 X (n) 11 U 1 &X 11 & 2 n 0, &k &X (n) &:(U 1 ) k X (n) 22 :(U 1 ) 22 & 2 n 0, &k (n) &:(U 1 ) k X (n) 21 U 1 &X 21 & 2 n 0, &k &X (n) &U k1 X (n) 12 :(U 1 ) 12 & 2 n 0,

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(7)

MAXIMAL INJECTIVE SUBALGEBRAS

65

for all k in Z. Similarly, since lim n &YX n &X n Y& 2 =0, we have that (n) (n) (n) &Y 22 X (n) 22 +Y 21 X 12 &X 22 Y 22 &X 21 Y 12& 2 n 0.

(8)

We shall use computations similar to those in the proof of Lemma 4.3 to derive a contradiction to (8). We work in M1 . Our information about &Y 21 X (n) 12 & 2 is the key to this argument. With F and the ``length'' of elements of F as defined following the proof of Lemma 5.1, the set CF of all complex, finite, linear combinations of elements of F is an ultraweakly dense * subalgebra of M1 . By the Kaplansky density theorem, for each positive =$, there is a selfadjoint Y $ in N with each Y $jk in CF such that &Y jk &Y$jk & 2 <=$ and &Y $jk &&Y jk &, where Y $ is the 2_2 matrix with entries Y $jk , for 1 j, k2. Let n 0 &1 be the maximal length of all elements (in F) appearing in the supports of Y $jk , 1 j, k2. Let n$0 be an integer multiple of 4n 0 such that n$0 >2=$ &2n 0 . Let c be c$17, n 1 (=n 1(=$, n$0 )) be such that when nn 1 and |k| n$0 , we have &k (n) &U k1 X (n) 11 U 1 &X 11 & 2 <=$, &k &:(U 1 ) k X (n) &X (n) 22 :(U 1 ) 22 & 2 <=$, &k (n) &:(U 1 ) k X (n) 21 U 1 &X 21 & 2 <=$,

(7$)

&k &U k1 X (n) &X (n) 12 :(U 1 ) 12 & 2 <=$.

Since &X n&1 for all n, we have that &X (n) jk &1, 1 j, k2. Thus with =$(
(8$)

Note, too, that &Y $21 X (n) 12 & 2 c$&c. Let S 1 be the subset of F that consists all elements that begin with n Um 1 V 1 such that 0 |m| 2n 0 &1; S 2 be the subset of F that consists of n all elements that end with U m 2 V 2 L g such that 0 |m| 2n 0 &1. Let S 0 be 4n0 k 0k of F span orthogonal subspaces of S 1 _ S 2 . The subsets U 1 S 1 U &4n 2 2 L (M1 , {) for distinct k in Z. (Compare the beginnings of elements in different subsets.) Recall that :(U 1 )=U 2 # R2 . For k in Z, when 4n 0 |k| n$0 , we have &4n0 k 0k 4n k &4n k (X (n) &(X (n) &U 4n 1 12 ) S 1 U 2 12 ) U 1 0 S 1U 2 0 & 2 &4n0 k 0k 4n k &4n k =&(U 4n X (n) &X (n) 1 12 U 2 12 ) U 1 0 S 1U 2 0 & 2 &4n0 k 0k &U 4n X (n) &X (n) 1 12 U 2 12 & 2 &4n0 k 0k &U 4n X (n) &X (n) 1 12 :(U 1 ) 12 & 2
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