Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free...

Invent. math. 115, 347 389 (1994) ~nventio~es mathematicae �9 Springer-Verlag 1994

Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index *

Florin R~dulescu **'***

Institute of Mathematics of the Romanian Academy, Bucharest, Romania

Oblatum V11-1992 & 7-V1-1993

We introduce in this paper a noncommutative probability approach (in the sense considered by D. Voiculescu in [28]) to the algebras that are associated to certain amalgamated free products. In this way we find that the type 11~ factors associated to the free, noncommutative groups F N , N > 2 have a rich lattice of irreducible subfactors of noninteger index. Our main result states that many index values for irreducible subfactors of the hyperfinite I1~ factor are also index values for irreducible subfactors of ~ ( F N ) . This answers a question raised by V.F.R. Jones in [13].

Theorem Let ~ = (A D 13 ~ D; A D C D D) be a commuting square ([20]) ~ n i t e dimensional algebras, which is irreducible (i.e. the centers of A, I3 and respectively C, D have trivial intersection) and A-Markov ([21],[31]), (i.e. there exists a A-Markov trace, in the sense of Jones ([I 1])for C c A, which restricts to a A-Markov trace for D C 13). Let N > 2 be any natural number.

Then there exists a subfactor. /~ C ~ ( F N ) of index A I with relative commutant t3 N C'. In addition

We note that the same statement holds true if T~ is one of the atomic infinite dimensional commuting squares considered by U. Haagerup and J. Schou in [ 10], [26] (see Remark 5.4).

Here % ( F N ) , N E I L N > 2 is the type 111 factor associated to a free group FN, i.e. the weak closure of the group algebra ~C(FN) acting on the Hilbert space

* Research supported by a combined fellowship from University of Calilomia Los Angeles and Institut des Hautes Etudes Scientiliques

** Miller Research Fellow *** Present address: University of California at Berkeley, Department of Mathematics, Berkeley, CA

94720, USA

348 F. R~dulescu

12(FN) by left translation. These factors were first considered by F.J. Muray and J. yon Neumann in [16]. They proved that they are not isomorphic to the hyperfinite IIl factor (since they do not have the property F). Recently, it appeared ([28]) that these factors are, in a certain sense, the natural algebra of observables, if one starts with Wigner's point of view ([32]) in which one models the observables by random matrices of very large size.

If g is any commuting square such that the iterated basic construction of gives an irreducible subfactor of the hyperfinite IIl factor ([11],[12], [31],[17], [20],[10],[26]), for instance if 'd is one of the commuting squares that are associated with the Jones' subfactors R;~ c / ~ for A -1 E {4cos 2 7v/7~]n > 3}, one obtains:

Corollary For any N C (1,~c] and for any index value A -1 of an irreducible sub/bctor of the hypelfinite II1/actor that may be obtained by iteration of Jones' basic' construction from a commuting square (for instance if A i E {4cos 27v/~l't~ >_ 3}) there exists an irreducible sub/actor, /~ C ~ (FN) of index [ ~ ( F N ) :. ~] = A - I .

Moreover. Z is isomorphic to ~(F~N_II~, ,+,) ~ [ ~(FN)]~,/~.

Recall that for a type II~ factor, g with trace 7- one denotes by. Zt the isomorphism class of the algebra e, Ze (with unit e), where e is any projection in . Z of trace 7-(e) = t.

The higher relative commutant for the algebras in the Jones' tower ([1 I]) of the above inclusion. Z C ~(FN) coincide with the ones for the inclusion of hyperfinite type IIi factors P ~ C Q ~ that is associated with the commuting square T~ (we owe this observation to S. Popa).

Due to our theorem, we are able to compute for ~ ( F , ~ ) one of the invariants that was introduced by V. F. R. Jones in connection with his Galois type classification of subfactors. Recall that for an arbitrary type I l l factor M this invariant is ~ ( M ) , the set of all possible values of indices of subfactors of M. Our result gives that

Corollary The set ~(~/J~(F~)) of all possible values of indices of subfactors of ~ ( F ~ ) is {4cosZ(yr/7~)]~, > 3} �9 [4, cx~) (i.e the same set as for R, me hypetfinite

factor of type I I0 .

Note that the only other type I[1 factors M for which anything is known about the invariant ~ ( M ) are the Connes' property T factors, ([5], [1]) for which, by the results of M.Pimsner and S.Popa [19], this set is countable.

The fact that the continuous line [4, vc) is in 7 " ( ~ ( F ~ ) ) is due to the fact [23] that . ~ ( ~ ( F ~ ) ) = ]~.+/{0} (by Jones' remark in [ l iD. Recall that the fundamental group .7"(M) of a type IIi factor M is the multiplicative subgroup of I~+ \ {0} defined by

. 7 (M) = {t > 0lM~ ~ M}.

Note that by a well known theorem of A. Connes in [3], this group is again countable for property T factors.

By a theorem in group theory ([ 15]), an index k subgroup G of FN is isomorphic to F~N-I)k+l. At the group algebra level this corresponds to the fact that there exists a subfactor ~(F~N-I)k+1) ~ ~ ( G ) C %(FN) of index k (since by [11], the index [ ~ ( F N ) : ~ ( G ) ] is the group index [FN : G]). Thus our main theorem extends in a certain sense the preceding (group theoretic) result to the case of noninteger index.

To give a meaning for the isomorphism class of the subfactors that we find by the construction in the first theorem we introduce a real continuation ~ (F~), r > l

Random matrices, amalgamated flee products and subfactors 349

with type I l l factors for the sequence of algebras ~ (FN), N C ~ ~, N > 1, although there is no meaning for F,. as a group (at least for nonrational values of r).

Such a continuous series appears in a natural way if one tries to find the analogue of Voiculescu's formula

~-~'~ (FN) ~ A//,:('2) @ ~(Ig(N l)k2+l)

or equivalently

(0.1) ~(FN)I/I~. ~ %(F(N lilv2+l) for all k E I L N C i L N >_ 2

for numbers other than 1/k. A first sign that such a continuation may exist, is the fact ([24]) that for 1 / , , ~

instead of 1/k in (0.1) (in which case both terms in (0.1) still make sense as group algebras) the isomorphism:

(0.1)' ~(FN)v~/F ~ ~(Fr l~.<) for all k c ! ! .N ~ i L N > 2

is valid. The series ~(F,.),.ca,,.>l is thus uniquely defined as a real continuation of the

sequence of factors associated to free groups and it is subject to the following conditions:

(0.2) ( 'Z(F, . )) t ~ ~ ( F ( r I)t 2+l),f~ all t > 0, r > 1,r,t E L;,

(0.3) ~(F,.) * ~ ( F , , ) ~ ~(F,+, , ) for all r,r ' > 1.

A similar series with the properties (0.2), (0.3) was considered by K. Dykema in [8]. Formula (0.2) shows in particular (see [23], [28]) that the type 1[i factors

% (FN), N ~ 1 !, N > 2 are stably isomorphic (by tensoring with I4(H), the algebra of all bounded linear operators on a separable infinite dimensional Hilbert space H). Note that Voiculescu's formula (0.1) implies in particular that ~ (F2) and ~ (/~s) are stably isomorphic but, it does not imply, for example, that ~/~ (Fe) and % ( F 0 are stably isomorphic (as fonuula (0.1)' shows).

Note that (by (0.2)) the algebras ~ (FN), N > 1 are also stably isomorphic in the sense that

~(Fo~) 0 ~ ( F N ) ~ ~ ( F ~ ) ~o '/,(Fat) for all N, M > 1.

(we owe this observation to G. Skandalis). This is due to the (general) fact that for all type IIl factors A,/3, we have

At ~ B1/~ ~ A ~ B

and thus by formula (0.2) we get

Corollary The isomorphism class of ~ iF,) @ 'Z ( F~) depends only on

( r - 1 ) ( s - 1) for all r ,s E ( 1 , ~ ] .

Moreover, a simple algebraic manipulation with both formulae (0.2), (0.3) will show that the fundamental group of ~ (Fr) is either ]~.+/{0} or {1}.

350 F. Rfidulescu

In fact we will prove a more precise statement

Corollary One (and only one) of the fi)llowing two statements holds true: (i) For all finite r > 1 the type I l l factor ~ ( F r ) is isomorphic to ~ ( F ~ ) . (ii) The type I[i .[actors ~ (F,.) are mutually nonisomorphic j~br r C (1, oo]. Note that (i) is equivalent to any of the following statements ( i / The isomorphism ~(F, . ) ~'~ B ( H ) ~ %(Fo~)~B(H) holds true for some

(equivalently for all)finite r > I. (i)" The fundamental group . 7 ( Z (F,.)) is nontrivial for some (equivalently for

all) finite r > 1.

At this moment it is still unknown if the type II1 factors

may be isomorphic (this is an old question of R. V. Kadison in the early 50's ([ 14])), but we hope that the results above could serve as evidence for a positive answer to this question.

Amalgamated free products of operator algebras have been intensively studied in [30], from the point of view of noncommutative probability theory. They were introduced in the setting of type II~ factors in [22], where the basic results on the existence of traces, factoriality and computation of relative commutants are obtained.

S. Popa realized that the problem of constructing irreducible subfactors of index > 4 in arbitrary factors naturally leads to the consideration of certain canonical subfactors coming from amalgamated free products involving an "initial algebra" Q and the Jones ' sequence of projections. He uses such subfactors to construct a series of irreducible inclusions of non-F factors N~(Q) c M'~(Q), of any index s ~ {4cos27r/,r~ln _> 3} U [4, oc).

Let R~, C R be the canonical Jones' inclusion of hyperfinite II j factors of index A -~ -- .s C {4cos 27r/nlr~, >_ 3} U [4, oc). The factors N'~(Q) C_ M"(Q) are subfactors in (Q c~ Rx) *re:, R and they depend in a functorial way on the type IIi factor Q.

in fact, when 'g = ~), is a commuting square of relative commutants in the tower of the Jones ' basic construction for R,x C R, A -1 = s c {4cos27r/~Tln > 3} the inclusion described in the statement of our main theorem, coincides with a term in Jones' basic construction of Popa's inclusion N"(Q) c ]~I'~(Q), for Q = ~(F~,.~) (although our approach will be more direct, by making use of the finite

depth properties of the inclusion R;~ C R). In its most general form our construction depends also on an initial algebra Q

and also depends (rather then on a scalar A -~) on a nicely behaved finite dimensional commuting square ~ , called A-Markov (with Ind ~ = ~ 1). We prove:

Theorem Let Q be a type II i factor and let

= ( A D 13 D D; A~_ C ~_ D)

be a commuting square [20] of finite dimensional algebras which is irreducible (i.e the centers of the algebras A, B (respectively C, DI have trivial intersection) and )~- Markov ( i.e. A is endowed with a A-Markov trace for the inclusion C C A (in the sense of Jones [11]L which restricts to a A-Markov trace on D C B). Then

N ~ (o ) = (O | D)*I) C C M ~(Q) = (O | B)*;3 A

Random matrices, amalgamated free products and subfactors 351

is an inclusion o f type I l l fac tors o f index [3d': (Q) : iV ~ (Q)] = A -I attd relative commutant t3 A C' .

In this paper (except when otherwise mentioned) by amalgamated free product of finite von Neumann algebras, we will always understand the reduced, von Neumann algebra, amalgamated free product that is obtained, via the G. N. S. construction, from the trace on the algebraic amalgamated free product of the algebras (see [22]).

The higher relative commutants invariants of the series N ~ (Q) c M Y (Q) may be computed (by Theorem 1.3, in this paper, which is due to S. Popa) and they coincide with the corresponding invariants of the inclusion of hyperfinite Ii~ factors P~ C Q ~ that is canonically associated to the commuting square ~ . This means that the standard invariants ([21]), or the paragroups ([17])

:~ ( N " (Q) c_ ~I ' : (Q))

and .~ ( P ~ c Q ~ ) coincide.

The main body of this paper is concerned with the case Q = %(/:I~). Let A D /3 be an irreducible inclusion of finite dimensional algebras and fix a faithful trace ~- on A. We prove first that there exists a trace preserving embedding of % (F:~)@13 *t~ A into ~ (F~ , ) .A . This will allow to find a random matrix model (in the sense of D. Voiculescu) for the von Neumann algebra ( % ( F ~ ) @ B *R A). The precise form of this model is as follows:

Theorem Let 1 6 /3 C_ A be finite dimensional algebras. Let "r be a faitt~ful, normalized trace on A. Assume that 13 is abelian. Then there exist a (natural) trace preserving embedding o f ( ~ ( F , ~ ) (• 13) *[3 A into ~ ( F ~ ) * A. This embedding is realized explicitly as.follows:

Y ~ . . . . Let ( ).~Gs' be an infinite semicircular family, let (p, )l~_ I he the minimal projections in 13 and let . / 2 be the yon Neumann algebra fi'ee product

{(Y),~es} * A ~ % ( b ~ , ) * A .

D~:fine

X " = ~ T ( p , ) - 1 / 2 p , Y ' p , for all .~ ~ S.

X ~ r,X.,) it, ~ ~ ( F ~ ) ) a n d Then ( " ).~cs is a ./)ee semicircular fami ly (henee ~ ~.s'~ {(X ) ,~s U A} is isomorphic to the algebra

[{(X'~),e5 } ' ' @ 13] *t3 A ~ ( % (F~.) | 13) *t3 A.

Finally we will use the above statement to prove that even i f /3 is not necessarily abelian, a reduced algebra of ( '/, ( F ~ ) ~, B) *B A is isomorphic to ~ (F%). Since by 123] ~ (E~,)~ e ~ (F,~)~, for any idempotent e in ~ (F-~), it follows that ( 'Z (F~.) ~-~ 13) "12 A is isomorphic to ~ (F ,~ ) .

This gives the lines for the proof of the main theorem in the case of infinitely many generators (which is done in the first three paragraphs). To prove the general case for ~ ( F N ) we follow the steps of the preceding proof and perform some supplementary

computations. By the definition of the series ~ (F,-)~.cR,,->I, we only have to count the elements in certain sets of generators (since we already proved the similar result for the case of infinitely many generators). In this way we obtain:

352 F. R~dulescu

T h e o r e m Let 1A ~ 13 C A be an irreducible inclusion of.finite dimensional algebras. Let T be a normalized faithful trace on A, let F be the inclusion matrix of A ~ B and let (s#)~.6~ be the vector of the values of the trace ~- on a set of representatives for the minintal projections in A.

For any real N > 1 or for N = vc, we have:

( % ( F N ) ~ / 3 ) *B A ~ %(Fl+ , :< t l - r~ . ,~> <.~,.~>).

Clearly, the above theorem and the theorem on subfactors in amalgamated free products will complete the proof of the main result.

Finally we point out that an intriguing question that remains open is to describe the structure of the set of all possible values of irreducible subfactors of ~ ( F ~ ) (this is an other invariant for type 111 factors introduced by V.F.R. Jones in [l I 1). On one side this set should coincide with the similar set for the hyperfinite IIi factor R and on the other hand, the factors in Popa 's series N ~ C M ~ for s c [4, ~ ) are irreducible, nonhyperfinite and have (by [21) Haagerup's approximation property (and thus are close to Z ( F , ~ ) ) .

Another related open question is if one can prove unicity, modulo conjugacy, for the one parameter action of J~' on ~ ( F ~ ) that was constructed in [25]. For the hyperfinite I I ~ factor such an action is unique, modulo conjugacy, by the work of A. Connes ([4]) and U. Haagerup ([9]).

The paper is organized as follows: the next paragraph recalls the definitions we use. Paragraph 0 contains an outline of the proof and the first paragraph contains the proof of the facts concerning subfactors in (Q @ B) */3 A. In the second paragraph we construct a random matrix model for a reduced algebra of ( ~ ( F ~ ) ~ B) *~ A. We use this model in the third paragraph to give a direct proof of the main result in the case of a free group with infinitely many generators.

The fourth paragraph contains the definition of the real continuation for the sequence of algebras associated to free groups (and the consequences of the existence of such a continuation). In the last paragraph we use the results above to prove our main result.

Acknowledgement. We would like to thank Georges Skandalis, Uffe Haagerup and Sorin Popa for useful discussions. Also, we greatfully acknowledge the hospitality of 1. H. E. S. where the final version of this paper was completed.

This paper has been circulated since the winter of 1991 as an 1.H.E.S. Preprint, no. 89/ 1991. The results in this paper have been announced in a note in C. R. Acad. Sci. Paris, t.315, p.57-62, 1992.

Definitions

Let H be a Hilbert space and let B ( H ) be the space of all bounded linear operators acting on H . A weakly closed unital subalgebra M of B ( H ) is called a yon Neumann algebra. When no mention is made of the underlying Hilbert space on which M acts, then 3,1 is called a I,V* -algebra. In this case the weak topology on AI comes from its predual M , ([27]).

Random matrices, amalgamated tree products and subfactors 353

An abelian yon Neumann subalgebra of 3'i is called diffuse if it does not contain minimal projections. As usual i f / 3 is a W* -algebra and ~ is a selfadjoint subset

of /3 , ~ " will be the von Neumann subalgebra o f /3 generated by ~ and I C /3. If /3 �9 /3(H), we denote by ~ ' the commutant of ~ in /3(H). For a subset S i n /3 we denote by Sp(S) the linear span of the set S.

By . J ' ( M ) we denote the set of all projections p = p2 = p. in M and by '-'Z(M) we denote the center of M. For a projection e in 3.1 we denote by eMe or 3'L, the reduced algebra which acts on el l . In fact M,. is a v o n Neumann subalgebra of ~3(ell). If M is a type IIi factor (i.e. ~Z~(M) = U and there exists a normal finite faithful trace 3- on M with 3-(1) = 1) then we denote by M~ the isomorphism class of eMe for any projection e in M of trace 3-(e) t.

If t > 1 then we consider an infinite dimensional separable Hilbert space H, and endow /3(H) with the usual faithful semifinite normal trace that takes value 1 when evaluated on projections of dimension l. Let 3-' be the corresponding tensor product trace on M ~ / 3 ( H ) and let e be any projection in M ~ / 3 ( H ) with 3-'(e) = t. In this case, 3'It is the isomorphism class of (M ~)/3(H))~. It is well known [6], [27] that the isomorphism class of eMe does not depend on the choices made in the selection of the projection e.

Recall some definitions and facts from [28],[29]. Let (A, ~2) be a unital C*-algebra endowed with a normalized trace ~. A family of subalgebras I C A~ C_ A (i c I) is called a free family of subalgebras if ~2((qa2..a,~) = 0 whenever % C A b , ~(%) = 0 for a l l j = l , 2 . . . , n a n d i t r for all 1 < j < ' t ~ I.

A family (~Q,),<I of subsets of A is free if the family of the subalgebras generated by the J'2~ and 1 is free. A family of elements (.]'O,el C A is free if the family {.L},e/ is free.

Moreover a free family (f,)~ is called semicircular if the elements f , are selfadjoint and the distributions of f, with respect to ~ are given by the semicircle law

L' ~?(f~:) = 2/re tA : ( l - t 2 )U2d t , k E ~ ; , k > O .

1

a family ( f0 ,~ , is called circular if the family {.% }~ U {y, }, is semicircular, where .f~ = ;r.t + ~-]-y.~, j E I is the decomposition of .fl into real and imaginary part. By abuse of language, we will say that ( f , ) ,c l is semicircular even if the constant 2/r~, in the formula above, is replaced by a strictly positive constant.

By Theorem 2.1 in [28] the von Neumann algebra generated by a free semicircular family (Y~).~s is isomorphic to the yon Neumann algebra ~ (Fc~rd S) associated to a free group with the same number of generators as the cardinality of S.

We recall some facts from [22]. Let AL, A2 be two yon Neumann algebras endowed with normal, faithful traces 3-1.72 and let /3 be a common von Neumann subalgebra

A~ containing the unit of A, for i = 1,2. Assume that 3-11u = 3-2113 and let E,3 be the corresponding conditional expectation from A~ onto /3 for i = 1,2. Let r be the trace initially defined on the algebraic amalgamated free product Al *u A2 by the condition

3-(alo,2...,~.,,) = 0

if a~ 6 A, ,Et3((z,) = 0 , j = 1 . . . . k il :~ i2 . . . ik-I ~ it,.. The reduced, yon Neumann algebra, amalgamated free product A l* u A2 is the weakly closed subalgebra obtained via G.N.S. construction with respect to the trace r . Moreover r extends to a faithful normal trace on A1 *u A2 (see [22], Lemma 4.1).

354 F. R~.dulescu

We will r e v i e w , for the convenience of the reader, Voiculescu's random matrix picture ([28], Theorem 2.2) for semicircular families. Let (s u) be a probabil i ty space and let Z be the expectation value on ~ , i.e.

( f ) = [ f(cr)du(cr) for all f C N l . f (~ , u). Jo- p > l

Let S be a nonvoid set and for each s E ,5' let

Y ( s , n ) = (o ( i , j ,m "~ C l~ s))~,a= ,, 'n

be a selfadjoint 'n x n matrix, whose entries

a ( i , j , ' m s ) , i , j = 1,...,'rL

are measurable functions with the property that the family

{Rea( i , j , 'n , s ) l l _< i <_ j < ~7, , s ~ S}U

{ I m a ( i , . j , r , , s ) l l < i < j < r, , s E S},

is an independent gaussian family, for each 'rt E l{ and

g (a( i, j , n , s ) ) = O,

~ ( l a ( i , j , rz, s)[ 2) = 1/7~, for all s c S, 1 < i , j <_ n.

Let D(r~,),,~u be a family of diagonal, rt • n matrices. Assume that

sup HDO,)II < c~

and that the family D(rz),~eN has a limit distribution with respect to the normalized traces on ~ x 7/ matrices, when Tz tends to infinity. The elements in the family D(r~),, cN will be identified with constant functions on )3 with values in 3,/,,(C).

Let r• be the trace on ~p>l LP(~') @ a l , , ( ' g ) obtained as the composi t ion of the

normalized trace on M ~ ( C ) with Z . Let (X*). ,~s U {D} be undetermined variables and let ;2 [[(X~).~es, D]] be the noncommutat ive ring over C in the above variables.

Let 4) be the linear functional on C [[(XS).~cs ,, D]] defined by

~h((X.Sl)pl (D)Zl . . , (X,St:)pk(D)~k) =

= lim ~ rz,, ( ( (Y(s l , ~7,)) t'' (DO0) 'q . . . (Y(sk, n)) p* (D(v~)) ~A' ),

fora l l . s~ E S , i / CY{,pj E l t , j = l , . . . , k . Under these assumptions Voiculescu's theorem 2.2 in [29] asserts that the above

limit exists. Moreover, with respect to r ( X " ) . ~ s U{D} is a free family and the distribution of each X 8, ,s E S, is given by a semicircular law.

Following Voiculescu's proof or using [7], one may drop the assumption that the matrices in ((D(7~))~,cN are diagonal. This means that we assume only that the matrices in ( ( D ( n ) ) , , ~ have a limit distribution when n tends to infinity, D(n ) are constant matrix functions on Z' and the min imum of the dimensions of the minimal projections in the algebras generated by D(n), tends to infinity with n. Under these weaker assumptions Voiculescu's theorem still holds true.


Then

The random matrix picture([29]) for semicircular free families was essential (in [28]) in getting a new view of the free group factors. The main device in D. Voiculescu's approach to the structure of the von Neumann algebras of free groups was a new representation for a free semicircular family (X~).~c~, by embedding (X.~).~c.s, into M,~(C) | where/3 is a W*-algebra containing an infinite tree family.

Explicitly~ following [28], Proposition 2.6, let

~ = {g(i , j , , s) l l <_ i , j <_ n,,s ~ S}

be a free circular family in /3, let

aJ2 = { f ( i , s ) l l _< ,~,.~ E S}

be a free semicircular family in /3, so that o.) 1 U 0.2 2 is free and let 1 C D C /3 be C rl a commutative subalgebra of /3 that is free with respect to Wl U w2. Let ( u),,j=l

be the canonical matrix unit in M,~(C), let D,, be the diagonal algebra generated by {ell . . . . ,e,~,,} and let

1<~<.7 <_n l<z<rz

(X.O,,cs c A,I,,(C) @ B

is a free semicircular family that is also free with respect to the algebra D,~ ~.�9 D. Using this, D. Voiculescu proved that there exists an infinite free semicircular

family which generates the reduced algebra

(I ~ ell){(X.O,~cs}"(1 | ~:ll).

This shows that

c/J, (Fk.)(1/N) ~ ~Z(FA, N2 N2+1),

and hence that ~ C - . 7 ( % (F~) ) ([281, Theorem 3.2).

O. Outline of the proof

In the lirst paragraph we prove general results about type l i t factors (Q @ B) . ~ A and their subfactors. Let 7- be a normalized faithful trace on A and let

be an extremal commuting square

'~ = ( A D B D D ; A D _ B D D )

of finite dimensional algebras. Recall that Z( is a commuting square (I201) if the square whose arrows are the conditional expectations (with respect to 7-) between the algebras A, B, C, D, is commutative.

The extremality condition [21] means that ~ is non degenerate (i.e. A = Sp (BC) = Sp (CD)) and irreducible (i.e. the centers of A, /3 and respectively C, D have trivial intersection). Let F be the inclusion matrix for B C_ A and let A be

Ilr, rll -~.

3 5 6 F. R f i d u l e s c u

By [21] there exists a projection f so that A C_ < A, f > (and f ) is the Jones' basic construction fb r /3 C A. Here < A, f > is the (finite dimensional) algebra generated by A and f . The fact that A C < A, f > is the Jones' basic construction implies (by [ 11 ]) that < A, f >= Sp (AfA) and that there exists a trace (also denoted by 7-) extending the trace on A and so that the conditional expectation EA from < A, f > on A (with respect to 7-) has the properties Ea(f) = I f and f x f = E A ( x ) f = f E A ( ; ~ ' )

for all z C< A , f >. Note that in this case A = T(f). In addition, the extremality condition for ~ implies that B C < B,J" > is the

Jones' basic construction for D C_/3 (< B, f >= 13f13 is a subalgebra of < A, f >). Such commuting squares appear, for instance, as relative commutants in the iter-

ated basic construction associated to a finite depth inclusion of hyperfinite type IIl factors.

For example if R:, C_ I~ is the Jones' inclusion corresponding to

A L ff {4cos2(rc/nlln > 3},

(el, e2 , . . . , ) are the corresponding Jones' projections ([ 111) and Aa, Ba, Ca, Da are defined by

I 2 tt A~ = A,,+I = {el, e 2 , . . . , e,,,+l }", B~ = A,,+l = {e2 , . . . , e,~+l }

C~ 1 _ . . . e ~." P~ 2 {ez . . . . , e , } " = A n - { e l , e 2 . . . . . j , = A,, =

then

is an extremal commuting square One may take here f = e,,+2. The main result of the first paragraph is that

N ~ (Q) = (Q @ D)*D C c M " (O) = (Q ~ 13)*B A.

is an inclusion of type II~ factors of index

[ ~ (Q): N ~ (Q)] = A-'= [Ir~rtl

and relative commutant B f3 C ' (in this paper, except when otherwise mentioned, by amalgamated free product for finite yon Neumann algebras, we will always understand the reduced, yon Neumann algebra, amalgamated free product).

The idea to prove this is to show that

M ( ( O ) = (O ~.~ (B, f ) )*(B, f ) (A, .f) and f

is the basic construction for N~(Q) c M ~ (Q). To show the last equality we note that A = Sp (BC) = Sp(CB) , [B, Q] = 0 and thus

M ~ (Q) = weak closure u Sp {AQA. . . AQAIn factors in Q}

= weak closure U Sp {CQC. . . CQA[n factors in Q}.

A similar formula holds true for MI e" (Q) and this in turn shows that

M ~ (Q) = M ~ (Q)IM ~ (Q),

f M ~ (Q)f c_ M ~ (Q)f,


E ~ I > ' ( Q ) ( f ) = .~ = [I /~tF]] 1 = 7 - ( f ) .

By Lemma 1.5.2 in [22], these conditions are sufficient to show that the index of 3~I ~ (Q) in 3Jii g (Q) is k i.

In the next two paragraphs we will prove that ( ~ ( F ~ ) O 13) *B A is isomorphic to ~ (Foo) if t3 C_ A is an irreducible inclusion of finite dimensional algebras. We will do this in two steps. Clearly, since (by [23]) h~Z(F~)h "~ ~Z(Fo~) for all projections h in ~ ( F ~ ) , it is sufficient to show that for a suitable projection g = f e in ( ~ ( F ~ ) ~) 13) *r3 A we have that

We choose a maximal system (f~)zc5 of nonequivalent minimal projections in B and let f = ~'.L. Since /3f is abelian and since

[(Q @ t3) .~ AIj ~ (Q 0 B t) *r~ A.f

it follows that we may assume (to establish the isomorphism) that /3 is abelian and B = {f,}".

We now use the techniques from the papers of D. Voiculescu. We construct a trace preserving isomorphism from ( % ( F ~ ) | *B A onto a yon Neumann subalgebra of the reduced free product ~ (F,~) * A. This isomorphism will allow us to transfer the random matrix representation ([281) of ~ (Foo)* A to the amalgamated free product.

The isomorphism is constructed as follows: Let (Y'~).,c,s' be a free semicircular family that is free with respect to A. Clearly,

Y~ A}". Z ( F ~ ) * A ~ {( ').~es.

Let (XS)seS be the family defined by:

- i / 2 s X ~ = E r ( f z ) f lY .fl for all s C S. /

X s The ( )~es is a free semicircular family (that commutes with B) and the von X s Neumann algebra generated by ( ) .~s and A is isomorphic to ( % ( F ~ ) | *B A.

Thus ( ~ ( F ~ ) | B) *z~ A is identified with a subfactor of ~ ( F ~ ) * A . Let (ek)kcK be a maximal family of mutually orthogonal, nonequivalent, minimal

projections in A, that commute with (.ft)tEl, and let e = ~ e,. We will show that e [ ( Z ( F ~ ) | t3) *B A 1 e is isomorphic to % ( F < ) . We outline here the proof of this statement.

Let r~: be the central support of e~. in A, for all k ~ K and let {e~, }t~i t be a matrix t t t t =

unit for Ark so that epp -~: commutes with (]))t~L and e~l = e~,, for all p = 1 . . . . ,m~., k E K (ink is the dimension of Ar~.). Let F = (a,a)~K,j<L be the inclusion matrix of A D /3 and let N be the reflexive, symmetric relation o n [ ( 2 defined by

l E L

Since the centers of A and /3 have trivial intersection, N contains a single orbit. With this notations, a system of generators for e [ ( % ( F ~ ) Q / 3 ) *~ A] e is Ar

and the set

358 F. R~idulescu

(0.1) " ~ ' = U r I~ y,~e,~ s = lelp -qlJ" C S, p= l , . . . , t k , q 1 , . . . , t ~ } . (m,k)EN

The random matrix picture of Voiculescu allows us to permute the matrix blocks in . ~ ' (we act as if the elements in .Z ' were matrix blocks whose entries are in an independent gaussian family on a probability space). Thus a more convenient way to express the generators for the algebra [ ( ~ ( F ~ ) | B) *u A]~, is given by the following procedure:

We start with an infinite semicircular family (Z~)sc5 , that is free with respect to A~, and a symmetric reflexive relation N on K which contains a single orbit (the last condition is the factoriality condition for e(( ~ ( F ~ ) @ B) *B A)e). The algebra e ( ( ~ ( P ~ ) ~ B) *u A)e is generated by (X*).~cs' and (ek)kcK, where (XS).~es is now defined by

X~= Z ekZ tem, t ES . (k,rn)GN

One could see, by the above picture and by the definition of ~ ( F , . ) , r > 1, that e ( ( Z ( F ~ ) | 13) *u A)e is isomorphic to Z ( F ~ ) . Instead of doing that, we will present (to make the task easier for the reader) in the third paragraph a direct proof of the fact that an algebra with such a system of generators is ~ (F~) .

The proof will consist of considering a reduced algebra of

(( % ( F ~ ) | B) *B A)~

by a projection 9. This time the system of generators for the reduced algebra are obtained from an infinite free semicircular family by deleting blocks from "half" of the elements in the family. The fact that the family is infinite ( by the random matrix picture for semicircular families) makes it possible to fill the corresponding holes by pieces from the other "half" of the elements. This shows that there exists another infinite semicircular family which generates the reduced algebra

[( (/~ ( ~ ) @ /~) "13 Ale 9. In the fourth paragraph we define a real continuation for the sequence

'Z (FN)x>2,xe~ of the type I I j factors associated to free groups. Let (X'~).~cs be a free semicircular family, let e~, f.~ be projections in {X~ which are either mutually orthogonal or equal and let

1 + ~ k~r(e.Or(f~), 'F

where the above factor k~ is 1 if e~ = f,~ and 2 elsewhere. For real r > 1 we define ~(F~,) to be the isomorphism class of

�9 Z = { X ~, (e~X'~f.~)~s/{~}} '',

if the above projections e~, f~ are chosen with the additional property that the finite yon Neumann algebra. Z is a factor.

The correctness of this definition and the fact that indeed, when r E H, we recover in this way the factors associated to free groups relies on elementary properties of free semicircular free families.

The properties (0.2), (0.3) are simple consequences of the definition. As an ap- plication of our definition we will prove that either all ~ ( F r ) are isomorphic for r c (1, oc] or that they are mutually nonisomorphic.


To do that assume that for some s > 2 we have ~ ( F 2 ) ~ ~ ( F O . An immediate consequence would be (by Eq. (0.3)) that ~ (F2) % ~(F~) for any finite s > 2. Let

(F2) be represented as

{ X ~ ' ~ i E i~}".

We use our assumption to replace for each i the generators

Z s by a semicircular family ( �9 ).~cs'~ �9 If card 5'~ = N, is big enough so that

Z 7(P~)2N' = oc

then we would get that % (Fz) is isomorphic to ~/~(F~). In the last paragraph we analyse the isomorphism class of the algebra (% (FN)

B) *z~ A. Here la E B c A is an irreducible inclusion of finite dimensional algebras and we fix a normalized faithful trace 7 on A. Let F = (a,j),CK,.~EL be the inclusion matrix of A D /3 and let (sk)kcK be the vector of the values of the trace "r over a system of representatives for the minimal projections in A.

Clearly the definition of ( ~(F,,))~,~k.,>E

and the procedure that we used to show that

( ~ (Fo~) | *i3 A ~ ~ ( F ~ )

reduces the analysis of the isomorphism class of the algebra ( ~ (FN) 0 / 3 ) *~ A to counting the elements corresponding to ~Z (FN) in the set .Z ' defined by Eq. (0.1).

By this method we prove that (~Z(FN) ~ / 3 ) *B A is isomorphic to ~ ( F M ) if

3,1= I + N < ( F F • > - < s , s > .

Let ~ = (A D C D D; A D /3 D D) be acommuting square as in the first paragraph and let (sk)t~cl<, (14)tel be the vectors of the values of the trace 7- over a system of representatives for the minimal projections in A and C respectively. By the assumptions on ~ (i.e. the extremality condition in Definition 1.1) it follows that < s , s >= ~ < t , t >. Let FI be the inclusion matrix for D C C.

For each P > 1, by the above formula, we obtain that

( ~ (F p ) O /3) *B A ~ (/JJ(FN)

if N - I = P < ( F F I ) s , s > - < s , s > .

The same argument as above shows that

�9 /~ = ( ~ ( F p ) | D ) *D C ~ ~ ( F M ) ,

if M - 1 = P < ( F i F [ ) t , t > - < t , t > .

Since s, ! are Perron -eigenvectors (i.e. ( F t F ) t = )~-l t , ( F ( P l ) s = A-Is) it follows that

( M - I ) / ( N - 1) = A -~

360 F. R~dulescu

or, equivalently, that

M : ( N - 1 ) ~ - ~ + 1 : ( N - I ) [ / F ' F l l + 1.

Thus, by the first paragraph, %(FN) has a subfactor %(FM) of index A -1.

1. Subfactors in amalgamated free products over finite dimensional algebras

In this section we introduce a series of inclusions

N ~ (Q) c_ M ~ (Q),

that are canonically associated to a type I I j factor Q and a commuting square ([20])

= ( A D B D D , A D C D D )

of finite dimensional algebras. The commuting square is assumed to have certain supplementary properties, as for the commuting squares coming from finite depth inclusions ([21], [17], [ l l ] ) .

These series of inclusions may be viewed, by specialization to the case when is the commuting square associated to a Jones' pair R~, C R, for X 1 = ,~ in the discrete series

(4cos2~/. , I~, >- 3},

as an extension of the series of inclusions N'~(Q) C M'~(Q) introduced by S. Popa in [221.

In this case, the relation between the two series is that N g (Q) c M ~ (Q) may be obtained from the inclusion N*(Q) c_ M,~(Q) in [22] by iteration of the basic construction.

The properties of the commuting squares ~ that we are working with are summarized in the next definition.

Definition 1.1 ([20],[21]) Let A D_ 13 D_ D; A D_ C D D be a commuting square of finite yon Neumann algebras which are weakly separable. Let 7- be a normalized, faithful, trace on A. The commuting square is e x t r e m a l (for the value A - l ) i f the following conditions hold.

(i) M a r k o v Irate. There exists a trace ~ on the basic' construction

(A, f ) = weak closure Sp A f A

r the inclusion C C A, that extends the trace 7- on A and so that 7-(fa) = AT-(a)for all a E A. Such a trace is called a A-Markov trace ([31] ,J i l l ) for C C_ A. Moreover

we assume that (f , i"3)" is the basic construction for D C 13 with respect to the trace

"~l<z~,f>- (ii) Non, degeneracy. We assume that

A = weak closure Sp (BC) = weak closure Sp (C13).

(iii) I rreducibi l i ty . We assume that the centers o f the algebras A, 13 and C, D have trivial intersection: ~-~(A) r] ~_(13) = C I and ~2(C) f3 ~2(D) = CI.

Some of the above conditions may be redundant. In particular if A is finite dimensional then the second condition is equivalent to the first ([21], Theorem 1.4). In addition


( (A,f )_D { B , f ) D B , (A, f ) _D A D B)

is again an extremal commuting square. Note that such commuting squares naturally appear in the context of finite depth

inclusion of hyperfinite II l factors ([ 17], [12], [31]). In fact ([17],[20]) any inclusion of finite depth of hyperfinite II1 factors may be obtained from an extremal commuting square of finite dimensional algebras, by iteration of the Jones' basic construction.

Starting with an extremal commuting square g = (A _D B D D; A _D C D D) and a type IIi factor Q we introduce the following inclusion of type II~ factors:

N ~ (Q) = (Q @ D) "19 C _C (Q ~) B) *B A = M ' / (O).

Note that the trace r on A induces a trace 7-1 on Q (<~ B by taking the tensor product of r i b with the normalized trace on the type II1 factor Q. Using the natural identification of 1c2 @ B with B, we clearly have that rt[B = r iB. Hence, by the definition in t221, we may construct the yon Neumann algebra, amalgamated free product (Q @ B) *t~ A. The same is true for the trace rio, and hence we may also construct the yon Neumann algebra, amalgamated free product (Q @ D) *D C.

By the commuting square property, the trace on the algebraic amalgamated free product

(Q @ B ) * ~ A = AI r (Q)

restricts to the trace on the algebraic free product

(Q ~: D) *~) C = N " (Q).

Hence (in the corresponding G.N.S. representation) we obtain indeed an inclusion of type II l factors.

Note that (Q @/3) *L~ A is a factor since (by lemma 1.4.1 in [22])

((O ~ B) *t~ A)' N ((O (~,) B) '13 A) =

O' N ((O =9 B) *I~ A) ~ A' =

B A A' = ~7.(A) n / 3 = s N s = ,21.

With this definitions we have:

Theorem 1.2 Let Q be a weakly separable type IIi factor and let

'f, =(A DCD_D; A D B D D)

be an extremal commuting square (/or the value A) of,finite, countably generated, yon Neumann algebras. Let (A, f ) be the basic construction for A �9 C (so that (I3, .f} is the basic construction for B D D). Consider the following inclusion of type II1 .far'tots:

N ~ ( Q ) = ( Q @ D ) * D C C M ' ~ ( Q ) = ( Q @ B ) , t ~ A .

Then MI / (Q) = (Q ,~) (B, f ) ) *(B,f) (A, f} and f

is the basic construction Jbr the inclusion N ~" (Q) c M 't (Q). In particular N e" (Q) c ]~I ~ (Q) is an inclusion of type I l l factors of index

362 F. R~idulescu

[]~i~ (Q): N~(Q)] =/~- t .

Moreover the relative commutant of N ~ (Q) in M ~ (Q) is B N C'.

Proof. We will verify this by checking the conditions in lemma 1.5.1 from [22]. We have to prove that

Mi ~ (Q) = weak closure [Sp(M ~ (Q)fM ~ (Q))] ,

.IM ~ (Q)f c M 'r" (Q)f,

EA/~ (p)(f) = A = T(f).

This conditions are sufficient to show that the index of M ~ (Q) in A4((Q) is A - l . To prove that the inclusion M ~ (Q) c_ 31((Q) (together with f) is the Jones' basic construction for the inclusion N ~ (Q) c M ~ (Q) we will have to check that also

{ f} ' n M ~ (Q) = N~(Q).

Since A = weak closure [ Sp (/3C)] = weak closure [ Sp (CB)]

and since [B, Q] = 0 it follows that

Ill ~ (Q) = weak closure d~, Sp {AQA.. . AQA]n factors in Q}

= weak closure U,, Sp {CBQA.. . AQA[n factors in Q}

= weak closure U,, Sp {CQBA.. . AQAIr~ factors in Q}

= weak closure U,~ Sp {CQA.. . AQAI'rt factors in Q}

= weak closure U,, Sp {CQCB.. . AQA)~, factors in Q}

= weak closure U,, Sp {CQC... CQAI'n factors in Q}.

Thus

(1.1). l~1~(Q) = weak closure U,~ Sp{CQC...CQAI~t factors in Q}

Similarly one may prove that

All t (Q) = weak closure U,, Sp {AQA.. . AQ < A, f > Irt/'actors in Q}.

Since < A. f >= weak closure [Sp(AfA)] it follows that

3,Ii '~ (Q) = weak closure U,, Sp {AQA.. . AQAfAIn factors in Q}

and henceforth MI" (Q) = weak closure [ Sp (M ~ (Q).fM ~ (Q)]. Moreover by Eq. (1.1) and since f commutes with Q and C, while f B f C Df

and A = weak closure [Sp (CB)] it follows that

f M / (Q)f c weak closure U,, Sp { f (CQC.. . CQCB)f}I'n factors in Q} =

= weak closure U~ Sp {CQC... CQCfBf}tn factors in Q} =

Random matrices, amalgamated free products and sublactors 363

= weak closure U,~ Sp { C Q C . . . C Q C D f } I n factors in Q} =

= [(Q @ D) "1) C] .f = f (Q ~.) D *D C)f .

Thus .r ~ (Q) f c N r (Q) f c M ~ (Q). Finally we have to show that

E ~ (Q)(f) : X.

This is equivalent to show that T(f -- A)z) = 0 for any .r C M " ((2). Using Eq. (1.I) it is thus sufficient to prove that

7( ( f - i )a lq l . . . a~q,a,+l) = 0, for any a, C A, q, c Q.

Recall that the trace ~- on

(QG < B , f >)*<[~,/> < A , f >

is defined by the requirement

(1 .2) T((1,;qtla/l... q;, (gin+l) ---- 0

for all .,', ~< A, f >, (ti C Q with E</3j>(d,) = 0, r(tI~) = 0.

But T(f - A)a lq l . . . a , q , , a ,+ t is a sum of elements as in Eq. (1.2) plus terms of the form T(f A)b. The later terms are vanishing also since E ~ ( f ) = A (because < f , B > is the basic construction for B D D).

Finally it remains to prove that

(1.3) {f} ' N AI': ((2) = NY (Q)

Since f commutes with Q and C it is clear that

7v '~ (Q) c {f} ' n M~((2).

The reverse inclusion is a consequence of the fact:

f M / (Q) f c N ~ ( Q ) f = f N r (Q),

that we already proved. Because of Eq. (1.3), and by using Lemma. 1.5.2 in [22], we obtain that

N~ = {f} ' n M " (Q)

is a type II~ factor and that

f M ~ (Q) f = N~ f = .fN1.

Thus N' / (Q) f = N l f and thus

N~" (Q)= NI = { f } ' N M~(Q)

as f commutes with both N1. N ~" (Q). To end the proof we also have to compute the relative commutant. We have (again

by lemma 1.4.1 in 122])

[(Q ~ D) *D C]' N [(Q % B ) * u A] = Q' N C' N [(Q <~ B ) * u A] = B n C' .

This completes the proof of the theorem.

364 F. Rhdulescu

The following statement is due to S.Popa. It concerns the computation of the higher relative commntants of the above irreducible inclusion.

Theorem 1.3 (S. Popa). The higher relative commutants of the inclusion N ~ (Q) c M '/(Q) are isomorphic with the corresponding higher relative commutants of the inclusion Q~ c P~.. where Q~ c p~ is obtained .from the extremal commuting square '~ by iterating the basic construction. Equivalently the paragroup ([17])

: r ~ (Q) Z M~(Q))

coincides" with the paragmup :r (P~ C Q~). Moreover, the canonical representation of 13 C_ A in N '/(Q) c_ M ~ (Q) can be

extended to a repivsentation of Q~c c P~ such that the higher relative commutants of N / (Q) c M ~" (Q) and Q,~ c p~ coincide.

Proof. Let A,,, B,~ be the iterated steps of the Jones' basic construction for C c A and respectively D C_ 13, and let f,, c B ~ , n c t~ be the corresponding projection ( with fl = f , AI = < A , f >,/31 = < / 3 , f >).

By Lemma 1.4.1 in [22], we have that

Similarly

((Q @ [3) , ~ A)' ( / ( (Q @ B,,) *u,, A~,) =

Q' r] ((Q Q B,,) .;~,, A . ) N A' = B . r~ A'.

((Q @ D) "1) C)' N ((Q @ B , ) *Bn A,,) = B~ N C'.

Note that by the preceding lemma the n - t h step of th basic construction of N ~ (C)) c M" (Q) is

M~ (Q) = (c) ~ B,,) *R,, A..

By [20]. [21 ]. [17], it follows that Por C Q~=, has the same higher relative commutants a s

N 'r (Q) c M Y (Q).

Indeed one may check this as follows : Let P,,, respectively Qn be the n - t h step of the Jones' tower of the inclusion B C A and respectively D C C with the convention that P0 = B, PI = A and Q0 = D, Q1 = C. Clearly we have the inclusion

as Q , is obtained from Q,,-L by adding a projection that commutes with Q,,-2, n > 2 .

Conversely, if x C P,-~ f - /Q~, then x is close to x ~ E Q~,, n P,, and to x" E Q',,+I n P,,+I, for some large r~. But as n tends to infinity the angle between the finite dimensional vector spaces Q',, NP,,, Q~,,+I NP,~+I becomes stationary. Since I j x ' - . e " l L~ is small it follows that there exists

, t p t r"' ~ (Q.+L n ,,+L) n (O., n P,,)

close to both .r', x", thus close to 3'. But Q,+I~ f3 P~, = (r N P0, so that x is arbitrary n t close to an element in Q ~ N P0, i.e.P.,~ Q ~ c_ Q ~ c3 P0.

The rest is trivial by our construction and [221. This ends the proof.


2. The random matrix model for amalgamated free products

In this paragraph we introduce a random matrix model for the amalgamated free products ( % (F~) @ B) .t3 A, where A �9 B are finite dimensional algebras. Let 7- be a faithful, normalized trace on A.

First, we describe this model for commutative 13; the general case is then handled by showing that a similar model is valid for a reduced algebra of ( % (F~,) @/3) "13 A.

For abelian/3 we will construct an isomorphism from ( ~ ( F ~ ) @ 13) *u A onto a von Neumann subalgebra of the reduced free product % ( F ~ ) . A. This isomorphism will allow us to transfer the random matrix representation for % (E~) * A ([28]) to the amalgamated free product.

We prove the above statement by showing that for a suitable system of generators (Y")~cs for % (F~) which is identified with a subalgebra in ~ (F~) * A, the conditional expectations (X'~)~e~ of (Y'~),~cs onto the relative commutant of 13 in ~ ( F ~ ) * A, have the property that (X~).~cs aud A generate a copy of ( ~ ( F ~ ) @ B) *R A in Z ( F ~ ) * A.

Proposition 2.1 Let 1 C 13 C A be f inite dimensional algebras. Let ~- be a faitt~r normalized trace on A. Assume that 13 is abelian. Then there exist a (natural) trace preserving embedding o f ( ~ ( F ~ ) @ 13) *u A into ~ (E~:) * A. This embedding is realized explicitly as fol lows:

Let (Y~).~cs be an infinite semicircular family , let . / ) be the Ji'ee product

{(Y'~),~e,s'}" * A ~ Z ( F ~ ) �9 A

and let (P,)I=I he the minimal projections in 13. Define

X s = ~7(pT , ) -1 /Zp ,~YSp , for all ,s E S.

Then (X"),,ES, is a free semicircular fami ly (hence {(X'),,e~s.}" ~ '/,(F,,~)) and {(X'~)~cs U A} '~ C .;,q is isomorphic to the algebra

[{(X') ,~ , .}" ,~ 13] *u A ~ ( % ( F ~ ) :,~/3) *u A.

Remark . Clearly the above statement is still valid under the weaker assumption that A, B are finite type I v o n Neumann algebras with discrete centers. This may easily be seen if one follows the lines of the proof of Proposition 2.1.

Proq f ( o f Propos i l io 'n 2.1). Recall that by theorem 2.3 in [281 we have that

{T(p~:)--l/2(pt,,Y~pk)}.~ES

is a free semicircular family with respect to the induced trace ~-~,k = ~(P~) i~_ on �9 Y?p~, for each k. In particular

r (p , (X ,~ )h . . . ( X .~. )~,, ) =

366

(2.13

Thus

7-(pOr((y~ )~-, . . . (y~,~)k,,).

F . R ' a d u l e s c u

r((X,~, )k, . . . (X.,,,)k, 0 = ~ r (p , (X.~)k , . . . (X,,0A:~0 z

= Z 7-(P")r((Y'~l)k' . . . (y.~,,)kr~) = r ( (y~ , )k, . . . (y.~0k,,) .

X ~ I n p a r t i c u l a r ( )sex is free and semicircular and by Eq. (2.1) we get

r(/)~(X*' )~" . . . (X '~'~)~'~) =

r(/)Or((Y*' )kf . . . (y.~,, )k,, ) : r (pOr((X* ' )~" . . . (X 8" )~" ).

Thus T(p~x)= r(/) ,)r(x) for all :r E {{X~}.~es) ' ' , i = 1, .., 1. Using the definition of the trace on the amalgamated free products ([22]) it follows

that to complete the proof, we only have to check that

(2.2) r ( f l a l f a a 2 . . . f r , a n + l ) : 0

,s II for all ]'~ E ({X }.~cs) , a, E A with r(f~) = Es(a,.) = 0 for i = 1 , 2 , . . . , n .

We will also have to check the zero trace condition in Eq (2.2) for terms

]'lal.f2,z2... L,a, ,<

with f l = I or with f,,+l = 1; but the computations are the same). To prove Eq. (2.2) we may assume (since [fj ,p,.] = 0 for all i , j ) that there exist

iL, . . . , i ,z C L, so that

Pb aaPO+l = aa for all j = 1 . . . . , n, - 1

and so that r(a a ) = O for all j = l , . . . , n .

Note that the last condition is effective only when i , = ia+ ~. As r( f~) = 0, since

fJPo = P', J) = Pb I ,P , ,

and since / ) , , f p, . is a term of zero trace in {(Y~)~ev P,..}" for each fixed 3, it a .) / ' ~ ' ' a

follows that p, , f,p,~ may be written as a sum of terms of the form

(p , , - r ( p , , ) ) g { (/),, - r ( p , , ) ) g ~ i v , , - r ( / ) , , ) ) . . . .q; ' , , , (/),, - r ( p , , ))

where each g~ is an element of {(Y'~).~es}" of null trace. Moreover Pof f l ) , , may

contain similar terms starting (or ending) directly with g~ (or respectively g}~'b )" Since

r(p 0 - r(p, , ))% (/)0+1 - r(Pb+L ))) :

7 - ( ( P o - - T ( / ) L , ) ) a 3 ) = 7- (aa ( / ) 0 + 1 - - 7 - ( / ) , ~ , + 1 ) ) ) = 7 " ( ~ = 0

it follows that we have proved that all the terms f ln l f2a2 . . . f,~a, in Eq. (2.2) have an expression as a sum o f terms o f the form

bo.q l b i �9 �9 .q,, b,~ + l

Random matrices, amalgamated free products and sabfactors 367

where g, E {(Y'~)~es}', b, c A, and 7-(g~) = T(b~) = 0 for all possible i with the the exception that b0, bn+l may be equal to the unit 1. Since ({Y'~}') .~es , A is a free family o f algebras it fo l lows that (2.2) holds true. This ends the proof.

In general, if /3 is an arbitrary finite dimensional algebra, we choose a maximal system (J))zCL of mutually orthogonal, nonequivalent minimal projections in /3 and let f = L'f , . S i n c e / 3 f is abelian and since

(2.3) [(Q @/3) *B A ]f ~ (Q | ,B s A f

it follows that we may use the random matrix picture from the preceding lemma to describe the reduced algebra [(Q @ B) *t~ A]f. This will be sufficient for our aims since we only want to prove that a reduced algebra of ( ~ (Foo)@B)*BA is isomorphic to %(Fo~). We first prove formula (2.3).

L e m m a 2.2 Let t:3 C_ A be finite dimensional algebra and let Q a ~pe II i factor. Fix a normalized faithful trace "r on A. Let { f k } ~.c l~: be a maximal family of mutually orthogonal, nonequivalent minimal projections in B. Let f = ~ f ~.

Then there exists a trace preserving isomolphism .firm (Q (o /3.1) *13f A f onto ( Q (~ /3) *]3 A ) f , wheJz) the later algebra is endowed with the normalized trace induced f irm the trace on (Q | t3) "l~ A).

Proof. Let g~,i = 1 , . . . , m be a family of projections in B with ~ , g~ = 1 - f and so that there exists partial isometrics v., in 13 with

v,*v~ = L -< f , v,v~" = g~, for i = 1 . . . . . m.

This is always possible because of the choice of f . Any element in f ( (Q @ B) *13 A ) f is a sum of products of elements of the form

x = f a t [ ( l - f ) q l ( l - f ) l a 2 [ ( l - f ) q 2 ( l f ) l . . . l ( 1 - . f ) q , , ( 1 - f ) ] a , , + j f ,

where q, C Q, ~, r A. Since 1 - f = y~.q~ and since [q,,gj] = 0 for all i , j it follows that x itself is a sum of terms of the form:

f o4 (gh qlg, i )a2(g '~2 q e 9 , 2 ) " ' ' (g~ q~g,,, )~1,,+1 f =

.fa~ (u,, v[, q l v,, v:t )ae(v~ z ":2 qzV'z"';2 ) . . . (v~,, v;, q, ,,, ,, v;, )a,+, f .

Since v~ commutes with % (as v, C B, i C I and [Q,/3] = 0) this last term is also equal to

( falv , i )ql(v , lazV,z)q2('t, 2a3~ , ) . . (~1",, * �9 .1an u~,, )q , , ( l ,, a n + l f )

As %*javb+ I belongs to A f (since v* = f v ] , v, = v~f) it follows that any element

in f ( (Q ~2 B) *B A ) f is indeed a sum of elements of the form

aoqlal ...q,~a,~.a~ E A f , q , E Q.f.

To conclude the proof of the lemma it remains to cbc'ck that

T(aoqlal . . . q~a~,) = O. for all a, E A.t, q~ E Q

368 F. R~idulescu

with r (q0 = 0, EB~(a~) = 0 for all i = 0 , . . . ,n,

with the exception of a0, an which may be equal to 1. As EBf(a) = EB(a) for all a E A f , the last condition follows from the similar

properties of the trace on (Q | t3) *R A. This ends the proof.

We use the picture from Proposition 2.l for the algebra I ( ~ ( F ~ ) ~ t 3 ) . ~ AIr

( ~ ( F ~ ) ~ t3f)*Bf A f , when 13 is no longer assumed to be abelian and f is as in the statement of the preceding Proposition. Let e be the sum of a maximal system (ek)keh" of mutually orthogonal, nonequivalent, minimal projections in A f that commute with

(f~)tcI+. We will construct a trace preserving isomorphism from [ ( % ( F ~ ) (�9 13)*B A] fe

onto a yon Neumann subalgebra of ~ (E~) * Af~. This will be essential in the next paragraph, when we show that, g ~ ~(F~ , ) .

Corollary 2.3 Let A D B be.finite dimensional algebras, let r be a faithful normalized trace on A, and let 1" = ( a k l ) l e L , k e K the inclusion matrix of A D t3. Let (ft)l~Lbe a maximal system of mutually orthogonal, nonequivalent, minimal pn)jections in 13 and let (ek )ke l, be a maximal system of mutually orthogonal, nonequivalent, minimal projections in A I that commute with (fDzcL. Denote by e, f the projections defined by f = Yg.fl and e = Sek .

Then there exists a (natural) trace preserving embedding

[ ( % ( F ~ ) | C_ % ( F ~ ) * (Ale),

which is realized as follows: Let (Z t )t<~v be an infinite f iee semicircular Jamily that is fi'ee with respect to the algebra Aye, let

I C L

and let X ~ : ~ e ~ . Z t c , ~ f o r a l l t c T .

( k , T a ) c N

Then {(Xt) tcT, ,Af~} '' is isomorphic to [(~(Foc) Q t3) *~3 A] f , (and the isomorphism preserves the trace).

Proof. Since F is also the inclusion matrix for By C_ A.f and since

[(Q r t3) , ~ A l l ~ (Q | B f ) *Bf A f

for any type II1 factor Q, we may assume that f = 1 and henceforth that/3 is abelian I!

and B = {.5 }leL �9 Let (Y'~),~es be an infinite free semicircular that is free with A, let

.12 = {(Y").<s', ,4}" ~ % ( F ~ ) * A

and let X "~ = E r ( f i ) - l / 2 f i Y * f , for all s E S.

~cL


By proposition 2.1, we conclude that:

= (%(F,~) | B) *z~ A ~ {(X%.~cs,A}".

Let rk be the central support of ~;~ in A, for k E K and let {e~i,q}~q=l be a

matrix unit for A,. k so that 6p,p'k commutes with (fl)l~t. and so that el" I = e~. for all p = 1 , . . . . tk, k E K (t~. is the dimension of A~.~ ).

By Lemma 1 in [25] (see also Lemma 3.1 in [28]) a system of generators for ~,, is

"~'= U te'v r " k ..... ~ql"],scS, P= 1,.. , & , q = l , . . . , t , , , } (m ,,~')C N

and A~. A system of generators for ./~'~ is A, and

r U ~ k Y~e'~'~ = tc]~, .qtl,'6 S , p = l , . , t k ,q 1 . . . . ,tin}. (~rT,,k)G K 7< I'(

In order to compute traces of monomials with variables in the sets ~ and A, (and thus to be able to determine the isomorphism class of the algebra generated by

and A~) we will make use of the random matrix picture of D. Voiculescu for free semicircular families (see also [71).

Assume that (Y'~),~c~' are represented as selfadjoint m • 'm matrices (T~'"D,~,s , with random entries. This means that

T,t..~ ,t,., = (o )1<~ j<,~

where a~,a are measurable functions on a probability space (L', u) so that, for fixed m, the variables in the family:

{Rec~,.~ l1 _ < i < j < m , s � 9 l1 < i < j < m , . ~ c S }

are an independent gaussian family of functions on X' with

(. ,7:7) = o, z (I (,;,i~"1 ~) = ~ / , . .

Recall from the introductory paragraph that E is the expectation value on X given by

o

~(.f) = / . f d ;~ . f c L~(L~,u).

Moreover A,, is represented (asymptotically) by constant matrix functions on L' (we assume that the dimensions of the minimal projections in A,, in the corresponding (asymptotic) representation on 'm, • m matrices, tend to infinity with m). Let also T/,m : ['lp_>l LP(~) ~ 5I.~(~) -+ ,C be the composition of the normalized trace on

m • m matrix functions with the expectation value g . By ([29], see also [7]) the trace of any monomial whose variables belong to

(Y") ,es and A(, is the limit (as the dimension m, tends to infinity) of the value of T/, . , evaluated on the corresponding monomial with variables in (T """) and A~.

This shows that in order to compute traces for monomials with variables in the ,k " "~ (with left support e~, right sets ~" and A~. we may act as if the elements e lpY e,~ I

support el) were matrix blocks of very large size and so that the collection of all the entries in the matrix blocks is an independent gaussian family as above (for fixed m).

370 F. R~dulescu

Hence the elements in the set ~ may be "glued" to another free semicircular system (Zt)t~T in .~c = Y~ that is free with (ek)kcK and so that

{elpY eqltS C S,p = 1 . . . . , tk ,q = 1 . . . . ,t,~}

for all k, m c K (recall that S is infinite). Clearly this shows that the elements in ,Z" (which arc together with A~ a system

of generators for the reduced algebra c_z)~) may be obtained from the elements Z ~ by the procedure described in the statement (after another permutation of the matrix blocks in (Z'~)~cs). This ends the proof of Corollary 2.2.

Remark 2.4 The statement of Corollary 2.3 still holds if one assumes the weaker hypothesis that A, B are finite, type I , yon Neumann algebras with discrete centers.

Proof. Indeed the only complication that occurs in the proof of Corollary 2.3 to this more general statement is the fact that this t i m e / ( may be infinite and therefore we cannot make a simultaneous use of Voiculescu's random matrix picture for all the elements in the set ~ that appeared in the proof of Corollary 2.3.

On the other hand the fact that the elements in the set /)~4g" may be "glued" in a semicircular family is a fact that only concerns traces of monomials of elements in the set ~ and A~. Thus to show that the elements in ~ may be "glued" in a semicircular family it is sufficient to show that the values of the traces of such monomials are exactly those which should be obtained from the frceness relations in the free semicircular family in which the elements will be "glued". As any monomial includes only a finite number of terms, we are clearly reduced to the case of finite K

In the proof of Corollary 2.3 we incidentally proved (and implicitly used) the following statement which is a straightforward consequence of Voiculescu's random matrix picture for semicircular families :

1 ;" U { f l , . . . , f k } ' b e a f i ' e e Lemma 2.5 Let. Z be a type IIi factor and let {Y'~, J,~es family of algebras in. 6 so that (Y~).~es is an infinite semicilrular.family and so that fl . . . . . fk are mutually orthogonal projections with ~ fi = 1. Let {cr(i,j, s)ls C S} be permutations of S, 1 < i < j < k. Then

Z "~ = Z(fiY~'("J'~). f 3 + f jY~ + ~ I,,Y~( ...... )fi, s E S

is a free semicircular family and { Z "~, 1 }~'es O { f l , . . . , fA:}" is again a fi'ee ]amily of algebras.

3. The isomorphism theorem

Let 1 E / 3 C A be finite dimensional algebras whose centers have trivial intersection (by [22], this is the factoriality condition for (Q | B) *B A if Q is a type II i factor). Assume that A is endowed with a normalized faithful trace and let f and e be as in Corollary 2.3. In this paragraph we use the description for the reduced algebra [( ~ ( F ~ ) | B) *B A] f~ that we obtained in Corollary 2.3, to prove that this algebra

is isomorphic to ~ ( F ~ )


Our strategy is to show that a reduced algebra of [ (~(Fo~) | *t~ A] fr is iso-

morphic to ~ (F~). By theorem 6 in [23] (which asserts that g ~(F~)g is isomorphic to ~ ( F ~ ) for any projection g in ~ (F,~)) it will follow that

% ( F ~ ) ~ [ (Z(Foo | B ) * B A]y .

This will be proved by a similar technique to the one used by D. Voiculescu in [28] (see also [25]) for the isomorphism ~Z(Fo~)I/k ~ ~(Fo~).

The first result implies in particular that ~(Foo)*A ~ '/~(Foo) if A is a finite dimensional algebra. This extends the fact ([28]) that ~ ( F ~ ) * ~ ( G ) ~ ~ ( F o c ) if G is a cyclic group.

Lemma 3.1 Let . / ) be a type I I i factor, let .~/~ = { f l , . . . , f~:} be a partition of the unity with pr~)jections in .,/) and let (Y'~).~s be an infinite free semicircular fanlily that is also.free with respect to the algebra generated by .~/).

Fix cr in S and let D be the diffuse, abelian yon Neumann algebra generated by { f , . . . . . .fk } and {.f~Y~ f~li = 1 . . . . . k }. Let (Y%),~es be any fiee semicircular family that is obtained by "gluing" the remaining pieces of (Y'~).~cs', i.e. such that

{LY"~.f3}.~cs = {f~Y~.fj}.~cs for all 1 < i < j _< ,~:

and {.f.,Y"~.f,,}.~cs' = {.f,Y~ f, }~es/{~,}.

Then (Y%).~cs is also,free with D and

{ (Y 'S ) sEs ,D}"= {(YS)sES, f l . . . . , fk} H for a l l / = 1 . . . . . k.

Proof. The last equality is simply a consequence of the fact that both size have the

same generators. The only thing to prove is the fact that {(Y"~).~s, D}" is free. Let

�9 -/) be a larger type I l t factor that contains .)5 ~ and that also contains a semicircular

element Z E . / ] so that ZU (Y~),~sis a free semicircular family which is also free with respect to { f l , . . - , fk}". Let

r Z yO = ~ ,L f, +

Lemma 2.5, then shows that yo O (Y%).~Es is also a free semicircular family, that is also free with respect to { f j , . . . , f~.}l,. The associativity property for free family of yt algebras ([281) implies that {yt) U D}" is free with respect to ( ' ).,es. The result now follows since ,~ k ~ {y0, {(f~Y ./ ,),=t,D} is contained in fJ . . . . . . fk} ' .

The following theorem will be used, together with the results in the first paragraph to prove our main result in the case of Z (F~) . In the proof we will use the description of the reduced algebra [ ( ~ ( F ~ ) | B) *~ A],, that we obtained in Corollary 2.3.

Theorem 3.2 Let 1 r t3 C A be finite dimensional algebras whose centers have trivial intersection. Then (~(Fo~) | 13) * t3 A is isomorphic to ~ ( Fo~).

Proof. Let f = ~ f~. be the sum over a maximal system of mutually orthogonal, nonequivalent, minimal projections in B.

372 F. R~dulescu

By the preceding lemma and by Corollary 2.3 we may use the following description for a reduced algebra of ( ( % ( F ~ ) | *B A)f. Let f = ~Y]fk be the sum over a maximal system of mutually orthogonal, nonequivalent, minimal projections in B.

Let (e0,e~c={t,..,k} be a maximal system of mutually orthogonal, nonequivalent, minimal projections in A f, let e = ~ e~ and let N be the symmetric reflexive relation on K as in the statement of Corollary 2.3. Let Do be the von Neumann algebra generated by (e0~eK=ll,..,k}.

Let ~ a type II1 factor which is generated by a free family of algebras 8 I I {Y }scS U D where D is a diffuse, abelian von Neumann algebra and (Y")~es

is a free semicircular infinite family. We choose (Y'~).~es and D so that Do = { e l , . . . ,e~.}" is a subalgebra of D. Let

X s= ~ e~Y'~ej,sES, 0,3)EN

then we have

�9 - ~ = { ( X ' ~ ) , ~ e s U D } ' ' ~ [ ( Z ( F ~ ) |

Since N contains a single orbit, as a consequence of the fact that the centers of A,B have trivial intersection, we may assume that

( 1 , 2 ) , ( 2 , 3 ) , . . . , ( k - l , k ) c N.

By choosing n big enough we may find a partition of the unity { g l , . . . , 9,7 } i n / / ~ ( D ) so that T ( g 0 = l / n , i = 1 . . . . , n and so that the following inequality holds true:

Z g~Og3 ~ Z erC) es. I~ ~1- <1 } r -s]_<l

Fix ~r in S and let 'v~ be the partial isometry from the polar decomposit ion v~b, of

g~_IY~9,, = 9~_1X~9~ for all i = 2 , . . . , n . .

By Theorem 3.1 in [28] we have

vd~ = fl~-i, v*'v~ = g~, b~ = g~b,,g~ for all i = 2 , . . . , n.

Let t3~ be a semicircular element in g, '~ 9, with

{B , } "= {b,}" c_ g,~.q,

for i = 2 . . . . , n, let a, be a semicircular generator o f giDg~ for i = 1 . . . . . n and let w, = v~v,+l . . . v,~. It is obvious that w,w,* = g~-l , w~w~ = gn, for i = 2 . . . . , n.

By Theorem 3.2 in [28] and its proof (see also lemma 2 in [23]) the elements in the following set are a free semicircular family generating ~q,, :

(.) ~,o = { , , , : Y . ' w , , I~ = 2 . . . . . n, s c s } u

(b) {wto,,_,w,};~=~ u {a,,}U

(c) {Re(w*Y'~wfl , Im(w~Y'~wfll2 < j < i < n,s r S/{cr}}U


(d) {Re ( w * Y ~ w , ) , Im(w~Y'~w.) )12 <_ j < i < n , j r i - l}U

We will describe the generators for the reduced algebra. /-3,~ with a less confusing notation. Let .f2 be the type II~ factor generated by w0. Then the elements on the lines (a), (b), (e) are also elements of, g,j,, while for the elements on the lines (b), (c), to get generators for. Zg,~, we have to delete certain blocks. Indeed, for i , j = 2 , . . . n we have:

* ,s * ~* Z w~ X w j = w~ g~ _ I X "~ gj _ ~ w j = u~ (g~ - i e,,) Y ~ ( g.~ _ 1 e ~ )'u b = (r,s)CN

(r,s)GN

By hypothesis

E g*@gJ <- E e r | I,-Jl<l I~-~l<l

and henceforth g.,fi. ~ 0 for utmost two consecutive values of r (for fixed O. Thus we proved that w ~ X ' ~ w j is obtained from w ~ Y ~ w j as a sum of blocks of the form �9 I * , I I * 7 It g w ; X ' ~ w j h where g ~ {u~a~_l~t,~} , h C { u j a v _ , u j } .

The less confusing notation we promised above is now as follows: we assume that w0 is the union of the infinite free semicircular family (x,,),~Es with the infinite circular family (Y.0,~eT'. We assume that S has the partition S = { e l , . . . , a , } O So, where x " t , . . . , x ~''~ correspond to the elements on the line (b) while (Y~),~CT correspond to the elements on lines (c), (d) and (x.,),es0 correspond to the remaining elements. Moreover we are given a finite family of projections {gj } jeJ , containing 0, 1 and so that each 93 belongs to {S~O }" for some ap3 E {oh . . . . , a,~} for all j E J .

The generators wl for. /~q,, are now obtained from w0 by the following procedure: for each s C T we are given i~,j.~ in J Consider the family

wl = (x~),~cs u {g,,~y~.qj~ }.

We will prove that w~ generates a factor which is isomorphic to ~ ( F ~ ) . To that end we consider a partition with infinite sets

S 0 = U(g,j)E,I2St,j

and show that for fixed (i, j ) E .12 the algebra generated by

(x~).~es~,j U {:r~P3, x~P~ } U {g,Y,~g3 li,~ = i, j,~ = j , s C S } ,

is isomorphic to ~ ( F ~ ) . Moreover we may choose this isomorphism so as to act identically on { S v J , x~'p~ }". Clearly this will complete the proof.

To prove the existence of such an isomorphism we may also assume that SP.7 = x~ ' , . Indeed, let w be any unitary in the type I I i factor {x~P3, x"P~ }" so that w9~ = g~w where

To complete the proof it remains to show that the algebra generated by

374 F. R~dulescu

is isomorphic to ~ ( F ~ ) . As the family {wyS}sET is again circular and since now g~, 9j C {:c~'PJ }" we are left to prove the following:

Lemma 3.3 Let{Zs}seT~uT2 U {x0} be a free family where {z.~}seT, oT2 is a circular family and xo is a semicircular element. Assume that T1,772 are infinite sets. Let 9, h be projections in {:Co}".

Then

= {X0, (Yt)tGTI,, (h~,]tg)tGT2 }" is isommphic to %(For

Proof. We simply use the random matrix picture from [29]. Consequently we may represent (asymptotically) the elements in the set {Re y.~, Im Y8 }~CTlOT? by large matrices, with random entries (of size tending to infinity) and that x0 is (asymptotically) represented by constant matrices. Clearly we may permute the blocks in the family (Yt)teT'l so as to fill the holes in Yt for t E T2 without changing the isomorphism class of the algebra.

As a consequence of Theorem 3.2 and of paragraph 1, we have now proved our main result for ~ (Fo~). We state it separately.

Theorem 3.4 Let ~ = (A D B D_ C ; A D C D_ D) be a commuting square ([20]) of finite dimensional algebras. Assume that ~ is irreducible (i.e. the centers of the algebras A, t3 and respectively C, D have trivial intersection) and that ~ is A-Markov (i.e. there exists a A-Markov trace ([ll]) for C C A which restricts to a A-Markov trace for D C t3).

Then there exists an sub]actor. ~ C ~/~(Foc) of index A - l and the relative corn- mutant. "Z ~ N ~ (Fo~) is isomorphic to [3 N C'. In addition. Z ~ ~/5~(Foc).

In addition, if ~ is any commuting square which (by iteration of the Jones' basic construction) yields a subfactor of the hyperfinite factor, then the higher relative commutants of the inclusion. Z C_ ~ ( F ~ ) coincide with the ones for the inclusion of type II1, hyperfinite factors Po~ C Qo~ that is associated with the commuting square ~ (see Theorem 1.3).

4. A real continuation of the sequence ( ~ (FN))N>_2,NCN

In this section we introduce a continuous series of type 111 factors ~(F, . ) , r E ~,, r > 1 which extends the sequence of the type 111 factors (Z(FN))N>2,NcN that are associated to the noncommutative free groups FN.

This series appears naturally in the analysis of the isomorphism class of the algebras ~ ( F N ) | Alp(C). Indeed by Voiculescu's formula (see Theorem 3.2 in [28]) we have

(4.l) ( % ( F N h / p ~ ~(F(N_I)p2+I) for all p > 1,p E H

or equivalently


~ (F(N_|)p2+I ) @ ]~Lp(C) "~ ~ (FN).

Formula (4.1) suggests to define ~/~(F(N_I)t-2+L ) as the isomorphism class of the reduced algebra ( ~/~(FN))t, for all ~ > 0, N C H, N > 2.

The first indication that such a definition would be possible was the following generalization of Eq. (4.1) (]25], Th. 6 ):

(4.2) ~/~(FN)I/./~ "~ ~(t~(N 1)/~+1), k C I { , N E I { ,N > 2.

This isomorphism shows in particular that the von Neumann algebras

are stably isomorphic (i.e. that the isomorphism class of ~ ( F N ) | B(H) is independent of N for all finite N > 1). Note that formula (4.1) implies in particular that ~(F2) and ~ (Fs) are stably isomorphic but does not imply that ~ (F2) and ~ (FD are.

The construction used in [25], [28] suggest that a natural way to define ( ~(F,.)),.ca,,.>l is to use infinite free semicircular families (X~).~cs. Let ~ be a fixed element in S', let (e~, f.0.~cs\{,~} be a family of projections in {X~'} ' ' which are either mutually orthogonal or equal and let

r = 1 + ~ k~7-(e~)T(f.~)

where k~ = 1 if e~ = f.~ and k~ = 2 if e~.f~ = 0. Then %(F,.) is the isomorphism class of the algebra

�9 ~Z : { x % ( e ~ x L f . , ) . ~ s \ ( ~ } } ' ,

if e~, .f,~ are chosen so that the algebra, /~ is a factor. With this definition we prove that the following formulae hold true:

(4.3) %(F,.)t ~ Z(F(,. 1)t-2+l), for all r > 1, t > 0

(4.4) %(F,.) * %(F~) ~ %(E,.+T.,), for all r , r ' > 1.

A similar series was considered in [8], and the formulae (4.3), (4.4) were also proved there.

The direct consequence of the two formulae (4.3), (4.4) is the fact that the fundamental group . 7 ( Z ( F ~ ) ) is either 1 or F:+ \ {0}, independently on r E (1, oc).

Thus the algebras ( ~ (F~r))~cR,T.>l are either all isomorphic or they are mutually nonisomorphic. In fact the first situation occurs if for some (equivalently for all r > 1) ~(F , - ) ~ ~ ( F ~ ) (using [23]). In this paragraph we will prove that the above definition is independent on the choice of the projections e~, f.~. After proving this result, the formulae (4.3), (4.4) will follow immediately.

Finally in the last part of the paragraph we will present a slight variation on the definition of (~(F~)),.ER,T>I, which will allow us to conclude that for all finite dimensional algebras B C A, whose centers have trivial intersection, the algebras ( Z ( F N ) | B) *~ A belong to the the series ( ~(Fr)),-~R,,->I.

Our first result will be used to show that the isomorphism class of ~ (F , . ) is independent on the choice of the projections e.~, f.~.

376 F. Radulescu

Theorem 4.1 Let A, t?, C be yon Neumann subalgebras (~a type IIi Jactor . /] which are fiee with respect to the trace r on .2~. Assume that A is a factor and that 13 is generated by an infinite fi'ee semicircular family (Y ~ ).~c s'. Let e~ , f ~ be projections in A which are either mutually orthogonal or equal. Let k~ = 2 or ks = 1 correspondingly.

Then the isomorphism class of

�9 / = {A, (e~X~f.~)scs, C}"

depends only on ~.~ l%v(e~)r(fs)

Proof. The proof is done into two steps. First step is to consider another family of projections (e~, f~) .~s (which are either mutually orthogonal or equal) with r (eO = r(e~D, r ( L ) = r(f~) and so that e~ = L iff. e~ = f,~.

Step 1 The algebra �9 ~ 1 t l l! = {A, (e.~Xf~),cs, C}

is isomorphic to. L

Proof. Indeed by hypothesis, there exists unitaries (w,).~es in A with

e'w~ = w,e. , , s = w~f.~, s C S.

Then the algebra. / is generated by A,

w,e~X'~f.~w*~ = e ' (w~X%,*)s s e S,

and C. As (w~X~wD,~es is again a free semicircular family and since

{A}, ( { w , X " w : } " ) ~ s , {C}

is a free family of algebras, it follows that

�9 z = {A, e ' ( ~ , , X * < ) . g , c y '

is isomorphic to . gl.

In the second step we prove that if we change the "shape" of the projection ~.~ e.~ | f., (when e~ | f , are mutually orthogonal), without modifying the "surface" ~.~ l%r(eOr(f .O, then the isomorphism class of the algebra. / remains unchanged.

The content of the second step is summarized as follows:

Step 2 Let S = U,,r be a partition of S with nonvoid sets and fix cr~, in b',, Let S[' = {,s G sr'les = f.~} and let S~' = S n \ S~'. Assume that the projections in the family

{e~ @ f.s}scS[' U {e.~| f~, L @e,,~ I * e s~ ~}

are mutually orthogonal. Then. /. is isomorphic to the algebra

{A, ( Z e ,X"f . , + E (e,X"f,~ + f~X~e~)),,eN, C}". ,<s;' .<s~'


Proo f . It is clear that it is sufficient to prove this isomorphism under the additional assumption that card [ = 1. Moreover we may assume that card S.,, is finite. The case of finite ,5'~ is then simply a consequence of Lemma 2.5.

The reduction to the case of finite Sn is a consequence of the so-convergence of the sum considered above, which in turn is a consequence of the finiteness of

Z ~ X s 2 11 .~ f.4[2+ Y~ 211<~xLL~[i 2 = .<s[, .<s~,

r (e .Or ( f , ) + 2 ~ r(e~)r( f .O < oo.

Clearly a (finite) repeated use of the steps l and 2 completes the proof of the theorem.

We are now able to state (and prove in the same time the correctness) the definition of ~(F~).

X * infinite f i e e semieircular.[amily. Fix a f inite subset Definition 4.2 Let ( " ) ,cs ' be an o f distinct indices {o0, c r , , . . . o , z} in S. For each s in So = S \ {cro,~rl, . . . ,o , ,} , let e~, fs be projections in "tr X ~ ~ " "Tor some ~.~ C { 1 , 2 , . . . , rz }, which are either mutually orthogonal or equal. Let h be a f ixed nonzero projeetion in { Y ~ }". Let

= n, + 2r(tOr(1 - h,) + Z k , r ( e , ) r ( f .O , 7 , s

where k, = 1 if e , = f.~ and/% = 2 if e,~f, = O. Then ~ (F~) is the isomorphism class o f the type II1 fac tor

{ X ~' , X ~ X ~'~ , hX~~ - h), (e~X~ f.O.~c&l }".

Note that {X ~t, hX~o(1 - h)}" is a factor. By the preceding theorem (in fact by the first step) we may assume in the above definition, without any restriction of generality, that rz = 1, Thus we may assume that all of the projections e~,f,~ are in { Y ~ }". Hence an equivalent definition for ~ ( F 7 ) is the following:

Definition 4.2 , Let ( X * ) , c s be an infinite f ree semicircular family , let ao, ol be two distinct elements in S and for each s E 5' \ {ol , at)}, let e.~, f.~ be projections in { X ~l }" , which are either mutually orthogonal or equal. Let h be a nonzero projection in { X ~ }". Let

~/= 1 + 2-c(h)r(1 - h) + Z k~r(e~)r( f~) , ,,~E S\{o-~ ,ao}

where k , = 1 if e~ = f~ and k~ = 2 if e , f~ = 0. Then ~ (F~ ) is the isomoqyhism class o f the type I l l factor:

{ x ~, , h X ~ , , ( l - h,), ( e . ~ X ' ~ L ) . < s \ { ~ l , ~ O } } ' ' .

One of the reasons for considering the more complicated Definition 4.2, (instead of the Definition 4 .U) is the fact that this definition makes trivial the proof of formula

378 F. R~.dulescu

(4.4). We now complete the proof of the correctness of Definition 4.2. Theorem 4.1 proves that ~(F.~) is well defined except for one point: to show that the relation

~(F.~) % {X ~ , hX'~~ - h)}" if "7 = 1 + 2T(h)r(1 -- h),

is consistent with the previous definition. This is done in the following lemma.

Lemma 4.3 Let (Y~ ) s~sbe a f ree semicircular family, let (71, crz be two distinct elements in S and let h, g be two nonzero projections in { Y ~ ' }" . For all s E S \ {oh, or2} let es, .fs, be projections in { y a i }" which are either mutually orthogonal or equal.

Assume that 7-(h)r(I - h) = r(g)~-(1 - g) + Y~-s ksr(e .Or( fs) , with k~ as above. Then the type I I , fac tor { Y ~ , hy'~2(1 - h)}" is isomorphic to

{y~l , gyCr2(1 _ g), ( e . y ~ fo~cs \ {~ l ,~2} }".

P r o o f . Clearly it is sufficient to prove the statement for r(g)r(1 - g) small enough. Let h l , h 2 be nonzero projections in { y ~ l } , , with hi <_ h, h2 < 1 - h and so that hi r h and h2 # 1 - h.

It is also clear that {yo-~, hy,~2(1 _ h) - h l y ~ 2 h 2 } '' is a factor. Hence we may use of Theorem 4.1, to show that an equivalent system of generators for

, g = {y~l ,hy~2(1 _ h)}"

is

{Y'~J, hy~2(1 - h) - h l y~2h2 , gYCr3(1 -- g)}.

Fix o-3 E S \ {at , o-2} and let g be a nonzero projection in {y~L },, with

r ( g ) r ( l - 9) = r (h l ) r (h2) .

By Theorem 4.1 this ends the proof of Lemma 4.3 and this completes the proof of the correctness of Definition 4.2.

The Definition 4.2 directly implies the following statement:

Proposition 4.4 For all r , r ' > 1 the yon Neumann algebra, reduced free product ~(F~.) �9 ~ ( F ' ) is isomorphic to ~(Fr+,-,).

Formula (4.3) is a little bit more difficult to prove than formula (4.4) but if we choose the right generators for the algebra ~ (F,) then the computations for the isomorphism class of ~(F , . ) t are less involved.

Proposition 4.5 The type 111 fac tor ~ (Fr)t is isomorphic to ~ (F~,._ 1)t-2+1 ), f o r all t > 0 , r > l.

Proof .Let (Y'~).~s be an infinite free semicircular family. Clearly it is sufficient to prove the statement for t sufficiently close to 1. Assume that ~ ( F r ) is generated as in Definition 4.2 by

where es, f s are projections in {Yr }" which are either mutually orthogonal or equal and h is a nonzero projection { Y ~ }". We may also assume (by Theorem 4.1) that


7-(h) = t > 1/2 and that es, f.~ <_ h. It will be sufficient to find the isomorphism class of h ~/~ (Fr)h.

Let f be any projection in {Y~ '}" with f _< h, T(f) = T(1 - h). Let w be the partial isometry from the polar decomposit ion

( 1 - h)Y ~' f = wb.

Note that by Theorem 2.5 in [28] we have w = (l - h)wf . Let /3 = f / 3 f be a semicircular element generating the same yon Neumann algebra as { f b f } ' a n d let A = (l - h)A(l - h) be a semicircular generator for {(1 - h)X ~t (I - h ) } ' .

Then h ~ (F~.)h is generated by

{hY ~' h,/3, (h - f )X~2w, w*Aw, e~X* f , } .

and let k, be as in Definition 4.2. By the Definition 4.2, the isomorphism class of the algebra h ~ (F,,)h is thus (Z(F~d) with

M = 1 + ( r (h)) -2 [2(r(1 - h)) 2 + 2T(1 - h) [~-(h) - (1 - T(h)] + 3'] =

1 + t 2(2(1 - t) 2 + 2(1 t)(2t - 1) + 7) = 1 + t 2(2t(1 - t) + 7).

On the other hand r = 7 + 2t(1 t) since t = r (h) . To complete the proof it remains to check the conditions in the statement of

Definition 4.2. We have to show that there exists a semicircular family (Zt)r with

Z t = h Z t h , t c T ,

containing (hX~h)~es and so that there exist distinct elements

Z tO, Z tl , Z t2 E (Z t ) t6T \ (hX'~h)~es

with /3 = f Z t ~ (h - f )X~2w = (h - f )Z t2 f , w*Aw = f Z q f .

This is a direct consequence of Theorem 2.5 in [28] and of the proof of Theorem 3.2 in [281 once we observe that to prove the last statement, we may assume that t = r (h ) = 1 /n for some n c I,L In this form, the last statement to prove is a general fact that concerns only the semicircular family (Y~).~cx. This completes the proof.

The consequence of the two formulae (4.3), (4.4) is the fact that the isomorphism class of ~ ( F ~ ) is either the same for all finite r > 1 or that the algebras ( %(F,-)),-ea,,->j are mutually nonisomorphic.

Corol lary 4.6 The fundamental glvup of ~ (F,,), for.finite r > 1 is either .Yg+/{0} or 1, independently of r.

Proof. Clearly, if x _> Y -> r > 1, then ~(-F,~) ~ %(b~v) is equivalent to the fact

that [ ( y - l ) / ( x - 1)] i/2 is in .;;r(~(F,,)). Thus if . 7 ( ~ ( F , ) ) is non trivial then Z ( E x ) ~ ~ (/wv) for some x > y and thus by Proposition 4.4

~(F~+a) ~ ~(Fy+o),a >_ O.

It follows that . 7 ( ~ (Fr)) contains a line and hence . 7 ( ~ (F~)) = F,+/{0} (since . 7 ( ~ (FT)) is a multiplicative group).

380 F. R~idulescu

The next corollary gives a more precise statement than the previous one.

Corollary 4.7 One (and only one) of the following two statements holds" true: (i) The type [lt ,factor ~ (Fr) is isomorphic" to ~ ( F ~ ) for all (equivalently for

one) finite r > 1. (ii) The type II~ ,factors ~/~(F~), r E (1, vo] are mutually non-isomorphic. Note that statement in (i) is equivalent to the following weaker statement: (i') ~ (F~.) | B ( H ) ~ ~Z (F~o)| H) for some finite r > 1. (if ' The fundamental group . 7 ( ~ (Fr)) is nontrivial for some (equivalently for

all) finite r > 1. Recall that by Corollary 4.5, the algebras ?5(F,-) @/3(H) are all isomorphic for

finite r > 1.

Proof . By the preceding corollary, we only have to prove that the hypothesis that the type I I l factors c/~ (F~) are all isomorphic for finite s would imply that ~ (F , . ) is isomorphic to ~ ( F o c ) for all r.

Let (XS).~cs be an infinite free semicircular family and fix a finite r > 2. Let

{~o,~, . . . . } u {~o,- , , . . .} be an infinite subset of distinct indices in S with infinite complement and let (P~)~=0,1 .... be projections in (X~0)" with

r = 2 + 2 E r(P~)2" "t>l

By definition ~ (F,.) is isomorphic to

�9 /, = {X~~176 XV~p. i E U}".

Let S = U,,eN S~ be a partition of S with infinite sets and let S~, be finite subsets of S~, of cardinality card Si = N~ and so that cry, u, E & for all i. In this case our assumption (that all ~ (F,.) are isomorphic for finite 7" > 1) implies that there exists a semicircular family (Z'~)~e& C {~o~X~ QoiX~,p,)} '', with Z "~ = p,Z'~p,, s E S, and such that

{ ( Z % ~ s , } " = {(p,X~'p~), (p,X%,~)}",

for all i. Since (Z").~ES ~ C_ {(P,X*p~).~E& }", the random matrix picture of Voiculescu (see

the next lemma), shows that there exists a free semicircular family

s o that

and

Thus

(TS),~eu,,s,~ C (X*)~es

Z ~ =p,T*p~ for all s E S~,i E rt

TaO = XaO,T'O = XVO.

%(F~) = { x ~o, x "o, (p,X~p~), (p~X"~p,~), i E H}" =

{X ~ X % (Z~Le&, i E N}" =


{T ~~ T "~ (p,T'~p0.~E&, i E H}".

By the Definition 4.2 we obtain that %(F, , ) is isomorphic to ~ ( F , , , ) where

r ' > ~ N , r ( p , ) 2.

By choosing N, big enough, for each i, we get r ' = oc.

The procedure used in the construction of (T~),Eu,,.&~ is explained in the following lemma

L e m m a 4.8 Let (z'~).~c,s , be an infinite f ree semicircular family. Fix a parti t ion ,5' = S ~ U S " with infinite S " and f ix three distinct indices crl), cr, t, C S ' and a nonzero projection p in {a:~ ' ' . Let ( z t ) t eT be a Jree semicircular fami ly in {pgd" p, pa:" p} ".

Then there exists a free semicircular fami ly (at) t~T in {(x~),Es,\{~,o} }" which is free with respect to {(zs).~ES,,} ' ' U {:c,C~0 } ' ' and so that z ~ = pcffp for t E T.

P r o o f . It is clearly sufficient to prove the statement under the additional assumption that S ' is infinite. Also we may assume that r (p) = 1 / n (eventually by considering a semicircular family (z~)~es and a project ion/5 in { ~ ' ~ } " so that z " =/52'~/5 for all s E S and by proving first the similar result for the later family).

Fix 0 in S'/{cro, or, u} and let (e,)','__ 1 be a partition of the unity in { S 0 } ' ' , with r(ez) = 1 /n for i = 1 , . . . r~ and el = p.

As in the proof of Theorem 3.2 in [28], let uh0 be the matrix unit w, o = w ' w j , where w~ is the partial isometry from the polar decomposit ion of

ezz~ = / / ) ; bz , 2 < i < r~

and let "{1) I = el. Let /3~ be a semicircular element generating the same von Neumann algebra as

b, and let d~ be a semicircular generator of e~{.v~o}"e, for all i = 1 , . . . , rT. Denote

go = {u';a'~ _< i _< rz} u {,%11 < i _< r,,}u

{ w ; ( R e a : ~ w~'( Im:r~ _< i < j < ,z}

and

g = go U {w~(Rez '~)w~, w ? ( I m z " ) ~ t b l l _< i < j _< n , s E S/{O,c~o}}U

{7,/; s.~,,,ll < i < ,,...~ ~ s/{o,~,o} }u

{ < < , < 1 _< ~ <_ ,-~}.

Also denote by ~ ' the set obtained by replacing S by S ' , everywhere in the above definition of the set K. Let U," = ~ \ U, ".

By Theorem 2.6 in [28], ~ is a free semicircular family and thus so 'is

~ ( = ( ~ ' U { z t } < r ) \ {p:c~p,p:c"p}.

Moreover go' U g " is a free semicircular family. To simplify the notation write

{c" I 7- E R} = ~o' \ [ {w;d ,w, II < i < r,} U {z ' } ,er U K/~]

and fix an injective map ?~ : T x { 1 , . . . , n,} 2 ~ R.

382 F. R'~dulescu

We may now paste back the elements in ~o t in a new semicircular family:

l l<z.<j<7~

l ~ z < J < n

+ E w~cX(*'i't)w~ + p z t p ' t E T. 2<z<n

The same argument as above (i.e. Theorem 2.6 in[28]) shows that (a~)teT is free with respect to (x*).,~s,, U { S 0 , x ~ (using also the fact that g0 ~ is free with respect to ~ " ) . This ends the proof of the Lemma 4.8.

The construction of T '~ in Proposition 4.7 runs as follows : By induction, assume that we constructed up to a given n c ~ { the family

with the required properties, i.e. such that (TS)seu~ls~ is semicircular and free with

respect to {x ~, xV} H and so that

We apply the above lemma with x ~o = X ~o,

{.~.'~}.,~s,, = {x,'0} u (T%e~=,s~ u {X~l~ ~ S.; , i = ~ + 2 . . . . } ,

{x~},~s, = {x",,} u { x " } ~ , : , + , ,

P = P77+I, X ~ = X ~r'~+l ~ X u = X u'+l

and Z s ( z t ) teT = ( ),~es,,.~.

T ~ ( a ) t o t given by Lemma 4.8, This completes the We take ( ' ),~es,,+~ be the family t proof of Corollary 4.7.

Finally we state a slightly different form of the definition of ~ (F. 0 , which will occur in the analysis of the algebras ( ~ ( F ~ ) | B) *B A, where B C A are finite dimensional algebras with disjoint centers.

L e m m a 4.9 . Let (X~),~es be an infinite free semicircular family, let Do be a discrete yon Neumann abelian algebra with minimal projections (e~)~Et and let . )9 be the von

{(X )sc,s'} * Do. Neumann algebra free product s tt Fix an element a in S and let p.~, h~ be projections in the diffuse abelian yon

Neumann algebra D = {Do, ( e , X ~ e~)~K }" which are either mutually orthogonal or equal and let ks = 1 if h, = p.~ and k~ = 2 i f p.~h.~ = O.

Assume that N is a reflexive symmetric relation on K • K with a single orbit and let

s ( i ,3)EN

Then


={D0, ~ e X ~ " X ' h " " �9 ~z e T~ ~/)s '.s)sCSl ( z , J ) c N

is isomorphic to % (F~).

Proof. Let k be the cardinality of K (k may be infinite). If k = 1 the statement is obvious. Thus we may assume that k > 2, K = { 1 , 2 . . . , k} and that

( 1 , 2 ) , ( 2 , 3 ) , . . . , ( k l , k ) �9 N.

We use the same kind of arguments as in Proposition 4.3. Replacing .72 with eventually a larger type II l factor, we may assume that there

exists semicircular elements X ~'0 , X ~n , X ~'2 in .,,r so that

X 8 ( ) , ~ s u { x ~ " , x " ' x ~ }

is a free semicircular family. For all s C S let p(~, tzl~ C {X"~ ' ' be projections that are either mutually

orthogonal or equal and have the property that

v-(p:) = r(p,0, r(h~) = 7-(h~) and pl~lz" = 0 iff. p~tz~ = 0 for all s �9 S.

We choose a partition of the unity (e'~),ci in {X ~0 }" with r (e , ) = r(c'~) and let .q~, 92 be two non zero projections in {X(~o} '' with g, < e',, .(;~ # e~ for i = 1,2.

By Lemma 3.1 the algebra

71 ~I l ! .s I It x ' , , % X %,(p.~X h~).~cs/t~} } Ir-pl_> l ,(r',p)C N

is isomorphic t o . /,. Let k0 be any projection in {Xm~}" with 2r(ko)r(1 ko) = 2r(gl ) "/-(.q2).

Since

.q2X m } A = {X ~~ Z e'Xmc"r P __ . 0 ,1Xcr l g 2 __ crl ,,'

]r'-- pl > l ,('r,p)G N

is a factor (due to the choice for gl,92), by Theorem 4.1, it follows tha t . /, is also isomorphic to the algebra

cr 0 1 cr I ! ! .s ! t! { X ' Z e r X e p - g l X ~ r t g 2 - g 2 X c q g l , k o X C r 2 ( 1 - k o ) , ( p s X l z s ) s E S / { c r } } .

rCp

The Definition 4.2 shows that the isomorphism class o f . /~ is thus % (F,.) with

r = l + [ Zv(e~)r (ep) - -27(g j ) r (g2)] +2r (ko )7 ( l - k o ) + Z r ( h j v ( p s ) . r#p s

This completes the proof.

384 F. RS.dulescu

5. Subfactors in ~ (FN)

In this paragraph we prove our main result for ~ ( F N ) , in the case of finite real N > 1. In fact due to Theorem 1.2, we only have to check that any ~ ( F N ) , for finite N > 1, is isomorphic to the amalgamated free product (Q ~ B) *B A for some Q = %(FM) depen~ling on N. This is done in the first theorem.

The expression that we find for M (as a function of N) will be used in the proof of our main result to determine the isomorphism class of the subfactors given by Theorem 1.2.

Theorem 5.1 Let 1A E t3 C A be an inclusion of finite dimensional algebras and assume that the centers of A and t3 have trivial intersection. Let ~- be a normalized faithful trace on A, let F be the inclusion matrix of A ~ B and let (.sk)~:eK be the vector of the values of the trace 7- on a set of representatives fi)r the minimal projections in A. Then

( ~ ( F N ) | ~ ( F M ) ,

and M = I + N < (FF~)s,s > - < s , s > .

Remark 5.2 Note that the above statement holds true under the weaker assumptions that A, B are finite type I yon Neumann algebras with discrete centers.

Proof o f Theorem 5.1. Let {ft }~e L be a maximal set of mutually orthogonal, minimal, nonequivalent projections in B and let {tl}rcL be the vector of the values of the trace ~- on {ft}16L. Let f = ~ f j . Clearly the inclusion matrix of By C_ A f is again F and Bf is abelian.

By the proof of Lemma 2.2 we have that that

[( % (FN) | B) *B A] ~ tt ~ ( ~ (FN) | B f) *u.r A f .

Let (ek)~,cK be a maximal set of mutually orthogonal, minimal, nonequivalent projections in A f, that commute with {.fl}leL and let e = ~ e , Note that the relative traces (Tt(e0),~h - of the projections (e,),c,~ ~ in the reduced algebra Af~. are now given by the formula

~-'(e,) = ( ~ t~)- ~ ( ~ .~k)-~ ~(e~) =

(5.1) = (2_ t~)-' Q_. ,~)-'.~,, i "--" ~--" c K.

We use the procedure described in the Proposition 3.2 to find a convenient system of generators for the yon Neumann algebra

[(~/~(FN)| B I )*Bs Af]~ ' �9

We will first prove the theorem under the additional assumption that N > 2 is an integer.

Let (Y'~),~=I,2,....N be a free semicircular family that is also free with A S and let

Z "~ = E T ( f t ) - ' / 2 f t Y ~ f l , s = 1,2 . . . . ,N.


Recall the notations from Corollary 2.3: rk is the central support of ek in A for k commutes k E K and l e k ttk is a matrix unit for Ar~, with the property that epp t ~pqJpq=l

with (fl)lec and @1 = ek for all p = / , . . . , ink, k C K (ink is the dimension of Ark). Let No be the reflexive symmetric relation on K 2 defined by

tEL

Note that No contains a single orbit, since the centers of A and B have trivial intersection.

Recall from Proposition 2.1 that (ZS).~=I,...,N and A I is a system of generators for ( ~ ( F N ) @ Bf ) *Bf A f . Let

'A/ /= U c k ~.s m~s= = l e l p L eql I, 1 ,2 , . . . ,N ,p l , . . . , t k , q = l , . . . , t , ~ } . (m,kIEN 0

By Corollary 2.3 a system of generators for

((Q | B) *B A)S~ '~ ((Q ~ BS) *Bf A.f)~

(with Q = % (FN)) is . ~ ' and A.f,,. If we fix (k, m) E K 2 then there are exactly

~ amlakl IcL

nonzero blocks of the form elpmk . . . . . . eq I in the set . ~ ' for each fixed s = l, 2 . . . . , N (since Z '~ commutes with f t , l C L).

By the Lemma 4.9 and Eq. (5.1), if we sum the "area" of these elements, we obtain that ( ( % ( F N ) | Bf)*t~f Af)~. is isomorphic to % ( F , ) where r, is given by the formula

. = l + x y ~ ' ~ a ~ , a S - ' ( e Y % ) - y ~ ~-'(~k) ~ IEL z,jcK2 kCK

By Proposition 4.5 in the preceding paragraph (the reduction formula) we deduce that ( ( /~(FN)@ Bf)*B.f Af is isomorphic to %(Fp) where P is determined by

% (F~,).~ . . . . . k ~ % (F~).

Consequently

or

Hence

( P - 1)(~-~ sk) -2 = ~ - 1

f = I + N [< (rr~).~,.~ > - < .~,.~ >] ( ~ t ~ ) -~

[( ~ (FN) | B) *u A] f ~ ( ~ (FN) | Bf) , u S Af ~ ~/~ (Fp),

386 F. R'adulescu

with P as above. The Proposition 4.5 and Lemma 2.2 implies again that (~/~(FN)| B) *B A is isomorphic to ~ ( F M ) where AI is determined by the relation

( M - l ) ( ~ t t ) - 2 = P - 1

whence the formula in the statement. We now complete the proof of the theorem by dropping the assumption that N

is natural. Clearly it is sufficient to prove that (~(F~,) | B) *u A is isomorphic to ~(FM(~,)) with M(tJ) an affine function depending on t/.

Let e, f be as above and let .)9 be a type I1~ factor that contains AS and an infinite free semicircular family (Y~)~s. that is free with respect to Af. Let (&)teL, {at}teL be disjoint, nonvoid subsets of S and let g~, h~ be projections (either mutually orthogonal or equal) in {(ftY~z ft} ", s E St,l c L, so that

r(fD2(u - 1) = E k~r(.q~)r(hts) for all l C L s C S I

and /% 2 " z t = if 9shs = 0 or k.~ = l if 9s = hs. By Theorem 4.1, by Proposition 2.1 and by Lemma 4.9, a possible system of generators for

is

Note that

�9 Z = ( ~ ( F . ) | s Ay

AS, (ftYal.fl)tcL and (gsYShs),~esz.

is a copy of % (F~,) for each I. For each l E L fix a projection ekl E (ek)kelr with ek~ <_ ft. Since {Ak, ftY'~lft} ''

is a factor, by Theorem 4.1, we do not change the isomorphism class o f . /, by replacing for s E St, the projections g~, h i with projections g ' , h'~ in {ek~Y~tekz }" that are either mutually orthogonal or equal with

T(g~) = T(gs), T(hs) = 7 " ( h ~ ) , g~h~ = 0 iff.gsh,~ = 0 for all s c St, 1 E L

and

~ ( f 0 2 ( . - l) = ~ k~r@)TO(O for aU ~ C L. scS~

This time the generators for the reduced algebra..g~ are: (e~:)keK, the elements

' '~ ' L} { g . J h~ls ~ & , l c

and a fixed set of generators O C_. Jgr corresponding to f lX~zft , for all 1 E L. Note that by hypothesis O and (ek)k~K are generating a factor.

For each k E /x" fix a nonzero element of the form e k X % ek and let D be the diffuse abelian von Neumann algebra generated by(ek)k~tr and e k X % e k , k C K. We use again Theorem 4.1 and replace this time the projections g'~, h'~ by similar projections g~', h~ E .J~(D) with

r ( g : ) " = r ( g ~ ) , r(h~) " = r(h,~),


and l! l! 9.~h~ = 0 iff..q.~t~ = 0 for all s E S t , l E L.

Thus. g~ has the following system of generators : D, the set O' (obtained from O by deleting the elements used to construct D) and

. . . . . L}. {g,~ Y" h~ Is c S~, l

As O ~ is independent on u and since

sES~IEL

is an affine function of u (recall that ~-~ is the induced trace on , Ze), the defnition 4.2 shows that the isomorphism class o f . ~ is %(Fly(,,)) where N(u) is an affine function of u. The reduction formula (4.4) shows that the same is true for . Z and thus for ( ~ (F,,) @/3) .,~ A.

This completes the proof of our theorem. The proof of the Remark 5.2 now follows from the Remark 2.4.

By the above results and by Theorem 1.2 we get:

Theorem 5.3 Let ~ = (A D B ~ C; A D C D D) be a commuting square (120]) of .finite dimensional algebras. Assume that ~ is irreducible (i.e. the centers of A, B and respectively C, D have trivial intersection) and A-Markov (i.e. there exists a A-Markov trace ([l l]) for C C_ A which restricts to a A-Markov trace for D C_ B). Let N >_ 2 be any natural number (or more generally let N > 1 be any real number).

Then there exists a subfactor. /~ C ~ ( F N ) ~?f index A -I . In addition the relative eommutant. / / n ~ (FN) is isomorphic to 13 N C r and

' ~ '~ ~ ( F ( N - 1 ) A 1+1) ~ [~(FN)].XI/2"

Before proving the theorem we note the following remark which shows that one may also construct subfactors for the yon Neumann algebra of a free group starting from the infinite dimensional commuting squares considered by U. Haagerup and J. Schou ([10], [26]). We state it separately to make the task easier for the reader. The proof of this remark will be identical to the proof of Theorem 5.3, by making use of the Remark 5.2.

Remark 5.4 The statement of Theorem 5.3 holds true for the infinite dimensional commuting squares considered by U. Haagerup and C. Schou ([26],[ 10]).

The assumptions are now that A,/3, C, D are finite type I yon Neumann algebras with discrete centers and that all the corresponding inclusion matrices have finite nornl .

Moreover one assumes that there exists a normalized faithful trace 7- on A which is a A-Markov trace for C C A and that ~- restricts to a A-Markov trace for the inclusion D c / 3 . In addition there exists u > 0 so that "I- is a u-Markov trace for the other two inclusions.

P r o o f ( o f Theoreru 5.3). Let (s~.)keK, (t t)ter be the vectors of the values of the trace r over a system of representatives for the minimal projections in A and C respectively. Let F be the inclusion matrix of A ~ B and let N > 1 be any real number (possibly infinite). By the assumptions on ~ it follows that < s, s >= A < t, t >.

388 F. R~dulescu

Indeed if X is the inclusion matrix for C C_ A then the A-Markov proper ty of the trace implies ([12], [31]) that s is e igenvector wi th e igenvalue A i for X t X . Hence

< t , t > = < X s , X s > = < X t X s , s > = A - I < s , 8 > .

By the preceding theorem, there exis ts a real P > 1 so that

( ~ ( F p ) • B ) *u A ~' ~ ( F N )

and N = l + P < ( F F t ) s , s > - < s , s > .

On the o ther hand, if the inclusion mat r ix for C _~ D is F t . then

�9 g = ( ~ ( F p ) ~ D ) *o C ~ 5/f(FM),

and, by the same a rguments as above, we have:

M = I + P < (F~F~) t , t > - < t , t > .

By Proposi t ion 1.4 in [2 l ], s, t are Pert'on -e igenvectors for the same eigenvalue u, i.e.

(F1F~)t = v ' - t t, ( F F ~ ) s = u - i s ,

and u - j = IIFII 2 = III'1112. H e n c e it f o l lows that

( M - 1 ) / ( N - l ) = ; ~ - t

or equivalent ly M = ( N - I)A - l + 1.

This completes the proof.

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