The relative dixmier property for inclusions of von Neuman algebras of finite index

Ann. scient. &. Norm. Sup., 4e s&k, t. 32, 1999, p. 743 a 767

THE RELATIVE DIXMIER PROPERTY FOR INCLUSIONS 1OF VON NEUMAN ALGEBRAS OF FINITE INDEX

BY SORIN POPA

“II rze faut pas toujours intigwc I1 faut aussi de’sint&er. ” Eugen Ionesco: La lecon

Dedicated to Professor Jacques Dixmier

ABSTRACT. .- We prove that if an inclusion of van Neumann algebras N c Al has a conditional expectation E:M+ N satisfying the finite index condition E(z) 2 cz,Yz E M+, for some c > 0, then ti c M satisfies the relative version of Dixmier’s property on averaging elements by unitaries in N, i.e., for any z E M, the norm closure of the convex hull of {UZU* 1 ~unitaty element in N} contains elements of Af’ n M. Moreover, in the case N, M are factors of type 111 and N has separable predual, the finiteness of the index of the inclusion is proved equivalent to the relative Dixmier property and to the property that a normal state on N has only normal state extensions to M. We give applications of these results. 0 Elsevier, Paris

RBsur&. - !Uous demontrons que si une inclusion d’algebres de von Neumann N c M est d’indice fini, i.e., si elle aldmet une esperance conditionnelle E : M + JV satisfaisant la condition E(z) 2 cz.k+‘z E M+, pour un certain c :. 0, alors N C M satisfait la version relative de la propriete de Dixmier, i.e., pour tout z E M, la fermeture en norme de l’ensemble convexe CO{UZX”; / u Clement unitaire dans N} contient des elements de N’ n M. Si en plus N, M sont des facteurs de type 111 et ni a un predual separable, alors on demontre que, reciproquement, si JV c M a la propriCk? de Dixmier relative alors N c M est d’indice fini. De m&me, on demontre que la finitude de l’indice d’une inclusion de facteurs de type 111, JV c M, est Cquivalente au fait que les Ctats normaux du sow-facteur N n’ont que des extensions normales a M. D’autres resultats et quelques applications sent aussi donnes. 0 Elsevier, Paris

We prove in this paper a version for inclusions of von Neumann algebras of finite index of Dixmier’s classical result on the norm closure of “averaging” elements by unitaries, as follows:

THEOREM. - Let hf c M be an inclusion of von Neumann algebras with a conditional expectation & : M -+ n/ of$nite index, i.e., 3c > 0 such that E(x) 2 cx, ‘dx E M+. Then n/ c M h#as the relative Dixmier property, i.e., for any x E M, we have CO~{UXU* 1 u unitary element in N} n N’ n M # 0.

In tlhe laist part of the paper we present several applications of this theorem, notably a result showing that for type 111 subfactors with separable preduals, the finiteness of the Jones’ index is in fact equivalent to the relative Dixmier property.

(*I Supported by NSF-Grant 9500882

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To prove the above theorem, we treat separately the finite and properly infinite cases. The proof consists then in a series of applications of the Hahn-Banach theorem and of the classical Dixmier theorem for single von Neumann algebras [Dl] (including its refined form in [SZl]), in some analysis specific to finite index situations, in convexity arguments and in reduction-disintegration type techniques. Also, to prove the properly infinite, semifinite case we make use of the general result ([PoR]) on derivations of von Neumann algebras into the ideal generated by the finite projections of a semifinite von Neumann algebra.

For the elements of reduction theory needed in the properly infinite case we refer the reader to [SZl], [II] and [ApZ]. For basics of finite index analysis, see [PiPo], [Po2,4].

II. Some Preliminaries

We begin by making several considerations that will reduce the proof of the theorem to proving it for selfadjoint elements “orthogonal” to N V (.ni’ fl M), with JV, .M being both finite or both properly infinite von Neumann algebras.

But first of all, let us note that an expectation E satisfying the condition l(z) > CCC, V’n: 2 0 is automatically normal and faithful (cf. remark 1.1.2, (i) in [POT]):

1.1 Proposition. If MO C Ml is an inclusion of von Neumann algebras with a conditional expectation I of Ml onto MO satisfying L?(x) >_ cx,Vx E Ml+, for some c > 0, then G is automatically normal and faith&d.

Proof. The faithfulness is clear from the condition. Let n/r;* C MT* be the inclusion between the biduals of Mel Ml ([S]) induced by the

initial inclusion and E** : MT* -+ M;* the bidual of E. Thus E** is a normal conditional expectation onto M G*, it satisfies the same inequality as I does and coincides with & when restricted to Ml.

For each i = 0,l let pi E Z(Mz*) d enote the “normal” projection for Mi, i.e. piM%* = piMi and II; E Mi, xpi = 0 implies z = 0 (this latter condition simply means that Mi 3 z H zpi E M;p; is an isomorphism; note that p; can also be characterized as the largest projection p in Z(Mff*) such that Mr*p = Mip). Then clearly pa > pl and z E MO, xpr = 0 implies z = 0 (both conditions due to MO being a von Neumann subalgebra of Ml). In particular it follows that E**(pi) lies in Z(Ma) and has support pa.

Let then E’ : MT*pl -+ MG*pr be defined by

E’(Xpl) = I**(Xpl)(E**(pl))-$1,X E MT*.

Since pl < pr(E**(pi))-’ 5 c-lpi and E** is normal, it follows that E’ is normal. Also, &**(Xpl) > cXpi,VX E MT>, so that (IE’(Xpi)lI > /~cE**(X~,)-~/~X~~E’(X~~)-~/~/I 2 I/cXpill. But MT*pl N Mp, and M;E*pl crl MO. Thus, by [BaDH] (see also [POT]), E’ implements a normal conditional expectation of Ml onto MO, still denoted E’, satisfying r’(x) > CX,‘V’X E Ml+.

Let us show that this implies pi = po. To this end let cp E (MT*),, q~ 2 0 be such that s(p) 5 ~0. Thus ‘pl,uO is normal (since s(cplM;*) 5 pa). But then 47 o I’ is normal on Ml, and since E’(z) 2 ~2, we also have ~(3:) < c-‘p o 1’(x): ‘v’x E Mi+. Thus cp is normal on Ml, so s(q) 5 pr. This shows that p. = pr, so by the formula defining E’ we have & = 8’. Since E’ is normal, it follows that E is normal.

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THE RELATIVE DIXMIER PROPERTY 745

Next we ,will show that we only need to prove the relative Dixmier property for I(: E M satisfying the condition II: E Z(N’)’ n M. To see that we may do so we need a technical lemma. Its proof is elementary but, “fame de mieux”, it is somewhat long and tedious.

1.2 .Lemma. Let Z be a commutative von Neumann algebra and Zi C Z, i = 0, 1, be von Neumann subalgebras such that Zo V 251 = Z and such that there exist conditional expectations Ei : Z + Zi with IndEi < 00, i = 0,l. Then there exists ajnite dimensional *-subalgebra A0 c ,250 such that A0 V Z1 = C;piZl = Z, where {pi}i are the minimal projections of Ao.

ProoJ: Before any other consideration, let us note that if A0 i= A is an inclusion of abelian von Neumann algebras with a normal conditional expectation F : A -+ A0 such that IndF g(max{ c 1 3’(z) 2 CI(;, Vx E A+})-l, then IndF can alternatively be described by (IndF)-’ = inf{F(p)(f(s)) 1 s E R,p E P(A),p(s) # 0}, where R is the spectrum of A, Ru is the spectrum of A0 and f : R --+ flu is the continuous surjection implementing the inclusion C(Rc) = Aa c A = C(R). Also, if we denote a! = (IndF)-I, then for any p E P(A) the spectral projection e of F(p) corresponding to the open interval (0,l) satisfies ae 5 F(p)e 2 (1 - a)e, in other words if SO E Ro is so that 0 < F(p)(sa) < 1 then a! 5 F(p)( so) 5 1 - Q. M oreover, note that by ([J], [BDH]) we always have IndF 2 1 and if IndjC < 2 then IndF = 1, i.e., A0 = A.

Let then V denote the set of all isomorphism classes of 5-tuples

in whilch Z is a commutative von Neumann, Z; c Z, i = 0,l are von Neumann subalgebras with Za V Zi = Z and Ei : Z + Zi are conditional expectations with IndEi < co, i = 0,l.

By the preceding observation, for any V = (Z, Zo; Zi , fo, El) E V, any constant c’ > 0 with IndE; 2 d-l, i = 1,2, and any p E P(Za), the element Eo( El (p)) has gaps in its spectrum, between 0 and c’~ and between 1 - c’~ and 1, i.e., if so E R. dsf spectrum(Za) is so that O < Eo(El(p))(so) < I then c’~ 5 Eo(El(p))(so) 5 1 - c’~.

To prove the lemma we need to show: (*) ‘W q = (Z, Za, 21, la, El) E V, 3Ao C ZO finite dimensional *-subalgebra such that

A0 v Z1 == Z. Let V. := {V E V 1 (*) d oesn’t hold true for V}. Assume, on the contrary, that

Va # 0 and let ,8 = inf{IndV 1 V E Vo}, where for V = (Z,ZO,Z1,Eo,E1) E V we set IrrdV == IndEa + IndEl.

Let V = (Z, ZO, Zi, Eo, Ei) E V. be such that IndV < p + p-“/2. Let PO be the set of projections q in P(Zo) such that Eo(E1(q)) 5 (1 - c2)1, where c = (IndV)-‘. By the wieak continuity of Eo o El it follows that if {pi}i c P(Za) is an increasing net of projections in PO then its limit is still in PO. Thus Pa is inductively ordered, with respect to the usual order.

Let p E ‘Pa be a maximal element. Let po E P(Zo) be the support of Eo (El (p)). Then p. belongs to Z. n Zi and Zo( 1 - po) = Z1 (1 - po) = Z( 1 - po).

For assume on the contrary that there exists a projection q E Zo( 1 - po) such that Eo(E1(q)) # q and denote qa = q - 41, where q1 is the spectral projection of Eo (El (4)) corresponding to the set { 1). Thus q. is a non-zero projection in Z. and Eo(El(qO)) doesn’t have 1 as an eigenvector, i.e., EC, ( El (40)) I (1 - c”) 1. But then the projection p + 40 E Z.

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lies in ‘PO, i.e., EO(El(p ,- au)) 2 (1 - c”)l. To see that this is indeed the case, note tirsl that since qoEO(E1(p)) = 0 we have yoE1(p) = 0 as well and thus E1(qO)El(p) = 0, showing that El(p) and El(qo) h ave mutually disjoint supports. Thus, if we assume that p + q. is not in PO, then there would exist a non-zero projection C& E X0 satisfying 4; I EO(El(P f 40))3 and thus 46 5 &(p + 40) as well. But for any p’ E % we have (1-p’)El(p’)(l-p’) = (I-p’)-(l-$)&(I-$)(1-p’) 5 (I-c)(l-p’). By applying this to p’ = p+qo it follows that if sb satisfies the above inequalities then CJ~ 5 y + qo. So by replacing if necessary c$, by either q&p or by y&o we may assume that either 4; 5 p or that 4; L 407 while still having all the previous properties for c&. If 41 5 p then the inequality crl, 5 &(p)+G(q~) ilnplies cih = yb(l-q0) 5 (l-q0)El(p)(l-q0>+(l-q0)E1((20)(1-q0)

with (1 - YO)-&(P)(~ - YO) = (1 - HO)& h aving support orthogonal to the support of (1-qo)El(qo)(l-qo) = Cl--YO)&(YO) andwith (1-qo)El(qo)(l-y0) I (1 -c~)(~-Yo)~ thus forcing 4; < El(p) and consequently yh 5 E. (El (p)), contradicting p E PO. Similarily, if qb 5 q. then qh = 4;40 L YOEO(El(P + 4o))Yo = 40-G0(E1(Y0))40~ SQ that 46 < Eo(El(qo)) 5 (1 - c2)1, a contradiction.

These contradictions show that p + q. must in fact lie in PO. But since p $ q. is strictly larger than p this contradicts the maximality of y. We conclude that indeed the support p. of the maximal element p in 7’0 satisfies po E Z. fY Z1 and &jl - po) = Zl(1 - po) = Z(1 -p(J).

Since (*> doesn’t hold true for V = (Z;&;,; Z1;EO;E1), this shows that it doesn’t hold true for (Zp,, Zopoj &PO, Loo: EL) either, with this latter having the same index as V. kVe may thus assume, without loss of generality, that in fact V is so that the maximal projection p E PO satisfies supp&(E1(p)) = 1.

Let then pl E Z1 be the spectral projection of El(p) corresponding to the set (I}, p’ E & the spectral projection of El(p) corresponding to the set [c, 1 - c], p2 = p’p, p3 = pi( 1 - pj E Z and p4 E Z1 the spectral projection of El(p) corresponding to the set {O}. Thus 111 + p2 + p3 + p-‘4 = 1.

For each 1 <_ j < 4 let Zj = Zpj, Z{ = Zipj, % = 0,l. Also, for each I 5 j 5 4 for which pj # 0 we define the expectations @ : ZJ -+ Z,/ by E/(x) = Ei(~pj)(Ei(pj))-lpj E Zzpj = Z;;i = 0; 1,~: E ii?.

It is then immediate to see that IndEi < IndE, and that for j = 1; 4 we have IndEi 5 (1 - c2)IndE0 while for j = 2,3 we have Ind@ 5 (1 - c2)IndEl. Thus, if we iet T/l = (Zj, Z{ j Z{, Ei: Zj) then by using that IndEj 2 1 we get

Indlj = IndEi + IndE{ 2 IndEo + IndEl - c2 = IndV - c2 < /j.

Thus (*) does hold true for each Vjj 1 5 j 5 4, so there exist finite dimensional subalgebras Aj c Zi = Zopj such that Zj = Aj V Zl, ? 5 j 5 4. Let then Ai c .Zo be finite dimensional subalgebras such that A&I, = Aj and define A0 c Z0 to be the algebra generated by p; 1 and the algebras AR, 1 < j 5 4. Then A0 is clearly a finite dimensional von Neumann subalgebra of Z. and it satisfies A0 V Z1 = Z, thus contradicting the fact that (*) doesn’t hold true for V, i.e.: V E PO. This final contradiction completes the proof of the lemma.

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1.3 corollary. iJ”JU C M is m inelusion L$ van Neumann dgebms as in The statement of the theorem then there exists a finite dimensional *-subalgebra A0 of Z(ni) such that


A0 V Z(M) > Z(N). In particular, there exists a jinite number of unitary elements ul,...,~Jm in z(N) such that Cjvjxvj* E Z(N)’ n M, Vx E Jbt.

Pro05 By the previous lemma, it is sufficient to show that if we denote Za gf Z(N), Z1 dzf Z(ibf) and Zdzf Z(N) V Z(M) = Za V Zr then there exist normal conditional expectations Ei of finite index from Z onto Zi; i = 0,l.

Noting that Z is included in N’ n M and that E(N’ n M) c N n N’ = Z(N) = ZO,

it follows that E(Z) = Za. Since IndE,, 5 Indl < 00, we may simply take EO dzf E,Z.

To define El, note that, since E(z) > cx,Vx E M+, it follows that if M? M1 is

the basic construction for N: M, then E,(x) > c’x,‘v’x E Ml+ (see for instance 1.2.1 in [POT]). Note also that if x E M’ n Ml then E,(z) E Zr = Z(M). By canonical conjugation it follows that there exists a conditional expectation El of N’ n M (=the canonical conjugate of M’ n Ml) onto Zr (=the canonical conjugate of itself,) such that El(z) 2 c2x,Yx E (N’ n M)+. In particular, if we still denote by El the restriction of El to Z = Za V Zr, then El(x) 2 c2x,‘dx E Z+, so IndEr < 00.

1.4 Notation. For x E M denote CM(X) = COn{~z~* 1 u E U(N)}. Remark that, with this notation, the relative Dixmier property for the inclusion N c M can be reformulated as follows: “Cn/(x) n N’ n M # 0, Vz E M”. If z E M is a given element for which Cn/(x) n N’ n M # 0, then we will say (occasionally!) that we have the relative Dixmier property for 5.

Note that if y E Cn/(x) then C&J) c CN(X), so that if C,v(y) n N’ n M # 0 then CN(X) n N’ n M # 0. By taking y E Cn/(zc) of the form y = AC,“=, vjxvT, with vi: . . . . vum the unitary elements in Z(M) given by Corolla.ry 1.3, it thus follows that in order to show that CN(X) n N’ n M # 0, V’z E M, it is sufficient to show that C,,.C(Y) n iv’ n M # 0,Vy E Z(N)’ n M.

But this means that we may simply replace the algebra M by the smaller algebra Z(N)’ n ,U, i.e., we only need to check the relative Dixmier property for the inclusion N c .Z(N)’ n M. Since the center of the algebra Z(N)’ n M contains Z(N), this shows that in order to prove the relative Dixmier property for all inclusions of von Neumann algebras with finite index N c M it is sufficient to prove it for those that satisfy Z(N) c Z(M).

Next we will show that we do have the relative Dixmier property for elements 2 E N v A/’ n M. This will enable us to concentrate on x’s that are “perpendicular” to N v A” n M. But first note:

1.5 Lemlma. If JU c M is an inclusion of von Neumann algebras with a normal faithful corzditional expectation & then there exists a unique normal faithful conditional expectation of M onto N V N’ n M, denoted ~N~JJI~M. Moreover this expectation SUtiS$eS f = f 0 fNV~n,k,j.

Proof. Let $a be a normal semifinite faithful weight on N. Then clearly 4 = $a o E is a normall semifinite faithful weight on M and its restriction to N V N’ n M is also semifinite. Moreover, since the modular automorphism group a$ associated with d, satisfies a+(N) = N, it follows that CT~(N’ n M) = N’ n M, so that

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g$(j\/VM’nM) = NVJJ”nM. This shows that the conditions in Tkesaki’s criterion ([T]) for the existence of a &preserving normal conditional expectation of M onto ni VJ@if’ n M are met.

Since (NVJV’nM)‘n Jb’i is included intiVN’flM, by a result of Cannes ([Cl]), the expectation is unique (not only among those expectations that preserve the weight $1. If we denote it by &~v~~,nn/l then 4 = d, o E aud 4 = 4 o EJJ~N,~M, thus 4 = $ o E o EN”N:,-,,L,~. Thus, since both I and L? o &~“n/,~,+t are normal +-preserving conditional expectations onto J\i, it fOllOW§ that E 0 &,VvN~n,&,t = E.

q

For the proof of the next result it is useful to note that if x is an element in M for which there exists a sequence y. = 2; y1 j ys, . . . in M such that yn+l f cN(yun); ‘in > 0 and d(yn, N’ f! M) + 0 then en/(Z) n n/’ f’ M # 0. Indeed, because if d(y,n, i4f’ n M) < E then d(y,N’ n M) < E for all y E C&y,) ( since all unitaries in N used for averaging yn commute with N’ n M) so that diamcN(yn) < ZE, and thus the condition d(yn,ti’ n M) + 0 implies nnCN(&) is a single element set {y’} for some y’ which lies both in CN(X) and in N’ C’ M.

.d Proposition. Zf JV; JU, & are as in the hypothesis of the Theorem then:

a,j. ,V v N’ n M has a finite orthonormal basis over N which is contained in Jff’ n M.

bj.C~(x)nJ\/‘r?M#0~~~ENvN’nJ~andifzEMissothatI~~~~~~~(x)=O then either en/(X) nAfViv’nM = (0) = Cn/(x) nAf’nM or CM(X) fuVVi\/‘nM = (il.

c).CN(x)n~‘nM~0,‘Jx~M~andonZyifCN(x)n~’nM#0,~x=x*~M satisfying &N~N~“M (x) = 0.

Proof. a). Note first that E(ti’ n M) = 2(N) and that any orthonormal basis of W n M over 2(M) (with respect to E) is an orthonormal basis of N V ni’ n M over H (with respect to the same 8).

Thus 2(n/) C n/’ n Jtl is an inclusion of finite index with 2i = 2(n/) abelian. It follows that N’ n M is of type Ifin with the homogeneous type I,% von Neumann algebras entering in its direct sum decomposition satisfying supini < 00 (see 1.1.2 (iii) in [POT]). Thus N’ n M is a finitely generated module over 2 = 2(N’ n M).

Since 21 C 2 has finite index as well, if we define 2, to be equal to 2 and apply Lemma 1.2 to 20, 2i,2, it follows that 2 is finitely generated over 2i. Altogether this shows that ni’ n M is a finitely generated module over Z1 = Z(N). By orthonormalizing a given finite set of generators (see 1.1.7 in [Po2] or 1.13 in [POT]), it follows that Af’ n M has a finite orthonormal basis over 2(N).

b). By a) N C Af V ti’ n M has an orthonormal basis {mj}llj~n c JV’ n M. Thus,

for any x E N V N’ n M, we have x = 5 rnj yi, for some yi E N. By applying the j=l

classical Dixmier theorem to yi , ykj ~ . . , yk E Af, given any E > 0 we can find unitary elements {~k}ilh<~ c U(ti) such that

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1 5 j 5 n, for some ,z~ E Z(N). Thus

1 m

II-

11

m c u,xu; - k=l = II

mjxj < E. j=l

Since $ Cr=, ukxu; is still in N V N’ n M, it follows that we may use the above recursively for E = I/2” to get a sequence of elements yn = x, yi j ys, . . . in N v N’ n M such that Y,,+~ E CN(yn),Yn 1 0 and d(y,, N’ fl M) + 0. Thus, by the observation preceding the statement of the Proposition, CN(Z) n N’ n M # 0.

Now if in turn x E M is so that EN~N,,,M(x) = 0 then clearly EJJ,N/“M(Y) = 0 for any y E CN(X). Thus if CM(n;)nNvN’nM # 0 and we take y E CN(x)nNvN’nM then we have both that y is in N V N’ n M and EN~NJ~M(Y) = 0. But this implies y = 0 and so CN(IC) n N v N’ n M = (0).

c). Assume CM(X) n N’ n M # 0 (so = (0) by part b)) for all x = x* E M satisfying EN~N,“~ (x) = 0. Let us first prove that this implies CN(X) n N’ nM # 0 for all arbitrary selfadjoint elements 2 in M. To this end fix x = x* E M and denote x0 = E~v,v,,,~(x), x1 = x - x0. By assumption it follows that VJE > 0,3vi, 212, . . . . v, E U(N) such that

But then y := A Cl=, V,XOV~ still belongs to N V N’ n M and so by part b) there exist unitary elements uj E U(N), 1 5 j 5 m and some yi E N’ n M such that

il

1 m - c m.

ujyu; - y; < E/2. 3=1 II

Altogether ,this shows that we have

Thus, if we let yi = & Cj,r, ujvkxv;u~ then y1 belongs to CM(X) and llyl - yi]] < E. Thus, by applying all this for E = l/2”, n 2 1, we recursively obtain a sequence

of elements y. = x, y1,y2, . . . in M with yn+l E C,,,(y,), n > 0, d(yn,N’ n M) < l/an, n > 1. But by the observation preceding the statement of the Proposition this implies that CN(X) n N’ n M # 0.

Take now an arbitrary element x in M. By writing x = Re(x) + iIm(z) and using the same argument as in the proof of the classical single algebra case (see [Dl]) together

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with the above relative Dixmier property for selfadjoint elements, it readily follows that in fact CH(Z) n N’ n M # 0 as well.

0 At this point let us recall (see e.g., 1.12 (iii) in [POT]) that Ind(iQ? c &) < 30 implies

there exists a projection y in Z(N) fl Z(M) such that pti c p&I is an inclusion of finite von Neumann algebras and (1 - p)N c (I- p)M is an inclusion of properly infinite von Neumann algebras. Thus, since obviously Nl c Ml, h/z c M2 have the relative Dixmier property if and only if ,Vr @ ,z/, c Ml $ M2 has the relative Dixmier property, we may deal separately with the case ,V c M is an inclusion of finite von Neumann algebras and respectively the case iv C ,ti is an inclusion of properly infinite von Neumann algebras.

Altogether we have shown in this Section that in order to prove the Theorem it is sufficient to prove the following statement:

LT. Let Jif c M be an inclusion of von Neumann algebras with a conditional expectation E : M --+ N of$ nl e ‘t index and sati.s_jjling Z(N) c Z(M). Assume that either Jf? J.4 are both Jinite or both properly infinite. Then given any z = x* E M with E,krv~*,n~(z) = 0, we have C,v(z) n N’ fi J\/f # ID.

Thus, we will assume throughout Sections 2 and 3 below, which contain the proofs of the finite and respectively properly infinite cases of 1.7, that N, JU and x are as in 1.7.

2. Proof of the Finite Case

consider first the case when N, ,ti are finite von Neumann algebras. We will proceed by contradiction, using the Hahn-Banach theorem to separate C*,(z) from N V N’ n M with a functional equal to some linear combination of states that can be taken tracial on ,U. The key observation which will then lead this to a contradiction is that the hypothesis “I(x) 1 cx,‘dx E M,” implies “p < c-iv o E; Vcp E J!'f;", so that, if a state cp on ,RvL is normal on N, it is automatically normal on all M. The argument is carried to an end by using the “weak” version of the realtive Dixmier property in [Poll, showing that Gn/ow n A(’ n iL4 # &‘dx E M.

With this plan in mind, assume, on the contrary, that there exists an element x = x* in M, with EN~,v/~M(x) = 0, such that Chr(x) n N’ n M = 0. By 1.6 this implies CN(X) n N V JV’ n M = B. Moreover, it implies that inf{ IIyIl j y E CN(X)} > 0 and, in fact, inf{ lly - b/l I y E CJJ(Z), b E tiVN’n JU} > 0 as well. Indeed, the first inequality is cleLar (or else 0 E CN(X) so Cn/(z) n N’ n ,U # 0) and for the second one we use the fact that if y E CM(Z) and b E L? = N VW n M then E~“,v,r~~(y - b) = -b,‘dy E C,U(X) so that llbll = jjE~(y - b)ll I jly - b/l, thus l/yII I jly - bll + llbll I 211~ - bll, implying that

inf{ lly - bll I Y E c;\r(x), b E B) 2 2-l inf{llyll I Y E CA,(X)} > 0

This shows that in the Banach quotient space JM/B (which has an isometric “adjoint” operation * inherited from M) the set Chc(x)/B is away from 0. By using the Hahn-Banach theorem in the quotient space M/B it follows that there exists a selfadjoint functional a’ on M/B which separates the self-adjoint set CN(X)/B from 0. By composing @’ with the quotient map from M onto M/B, it follows that there exists a functional Q, = @* on M and e. > 0 such that Q vanishes on B = N V ti’ n M and G(y) 2 Ed, ‘v’y E C.,v(x).

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From these properties of Cp and from the definition of C;v(z), it follows that \Il(rc) >_ E(), V6 E co{@(u . u*) 1 u E U(M)} so that Q(X) 2 Ed; VIJ E CM(@) ~ffyw’,M){dj(U . U*) 1 u E U(N)} and in fact 9(y) > ~0, ‘v’y E C,,,(z) as well. Also, each Q E Cn/(@) vanishes on N V n/’ n M. In order to get a contradiction we will first show that there exists a 9 in Cn/(@) with the property that it can be written as Q = 91 - $2 , with Q1,s E S(M) commuting with Af and being factorial traces when restricted to ti. By using Sakai’s Radon-Nykodim type theorem for Ki?‘1,2 and II, = 9i,s o E, we will then obtain positive elements al,2 E N’ II M+ such that 4 0 E(~I+‘~ . ~2:‘~) - 7j 0 E(u~‘~ . a!j2) is close to 9, and thus, still separates z from 0. But this will eas,ily be seen to lead to a contradiction by using [Poll.

To prove the existence of 9 = 9i - Q2 E Cn/(@) with Pi;2 states commuting with N and being factorial on N, we need the following:

2.1. Lemlma. Let $ = ($1,~2,. . . ,&) E S(M)n, and set

C,,/(G) = ~d(M*Y:Mn) {($i(u. u*))l<i<n I u E u(N)} c (S(M))n.

Then there exists (al;. . , an) E Cn/($) such that cxilz(~) = $ilz(Nl and N is in the centralizer of cq, 1 < i 5 n.

Proof: We first show that Cd($) n Siz/,+(M) # 0, where Sri/,$(M) ef{(crl,. . . , a,) in S(M)n 1 Nile is the unique trace on N such that oi12(~) = $‘;jz(Nj, 1 5 i 5 n}.

Note that (h/L”)* = (M*)“, with the duality being given by

((Xl,. . . Jn), (cpl,. . . > CPTJ) = ~;~l’p&4~

then remark that, in the G((M*)~, Mn) topology on S(M)“, both C,&!J) and sn/,lL v4 are convex and compact. Assume first that N = M, in which case S~,~((rl/) q = {(a:, . . . i ok)}, where ai E S(N) is the unique trace state with

“P,2(N) = $ilz(~), 1 5 i < n. Assume (a$‘, . . . ,ui) $ CN($). By the Hahn-

Banach theorem there exist (xi; . . .z,) E Nn and ~0 > 0 such that 2 cpi(zi) > i=l

i: &Xi) -t- Eo, qfl,. . .,cpn) E Cd$J>. i=l

By the classical Dixmier theorem there exist unitary elements ul, . , u, E U(N) such m

that II& C ,ujziuj* - ctrN zill < ~~/2 n, 1 5 i 5 n, where ctr,v denotes the central trace j=l

on N. ‘Then (cpy,. . . , cpi) %f( $ Jgi $Q(u~ . ~j*))i~i~~ belongs to Cn/($) and we have

Thus we get

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which together with 5 of(~) + i=l

EO 5 $i cpf(~i) (that we have because ((~9)~ E CN($J)

and because all elements in CN($) satisfy this inequality) gives a contradiction.

Now for the general case consider the linear application E : (S(M*))n + (S(N’*))” defined by E((h)lci+) = (b 0 E)lci<,.

Clearly E(Sn/,+(M)) = {(~$)i~;~~) and E(CJJ($J)) = cN(?,!j o E). Note also that E-i({(~~)i<i<~>) = S,v,+(M). By the first part we have E(C,v($)) 3 (c~p)~. But then -- any 4 = (41,. . . , &) E C,V($J) with E(4) = (c$); will belong to C.&$) n &J,+(M), ending the proof of the fact that CN($J) 17 SN,,(M) # 0.

Fix now a ($i)i in c,V($)nSN,+(M). Let K = ~“{(~i(u~u*))l<i<, 1 u E M(M)), the closure being taken in the usual uniform norm on (M*)” = (Mn)*. We want to prove that K is compact in the LT((M~)*, M”}) topology. By Akemann’s compactness criteria [A], in order to show this, we need to prove that if {e,}, is a sequence of mutually orthogonal projections in M, then #(e,) -+ 0 uniformly in 4’ E U; K; where Ki = CO”{ &(u u*) / u E U(N)). It is clearly sufficient to check this uniform convergence for 4’ of the form #i(u . u*), u E U(iV). But &( ue,u*) < c-i&(uE(e,)u*) = c-i&(&(e,)) and the sequence {&(E(e,))}, t en s d t o zero (because Cn&i(E(e,)) 5 1 > independently on U.

Altogether these show that K, with the affine action ($i)i -+ ($I(u . u*))~ of the group zA(ili) on it, satisfies the hypothesis of the Ryll-Nardjewski fixed point theorem (see e.g. [SZ2]). Thus there exists (or, . . . . a,) E K with CY~ = ai(u . u*), \Ju E U(N). Since by its definition K is included in CJJ($), we are done.

For the separating functional (a(= @*) considered before the lemma, let now Q, = @+ - @- be its decomposition into positive and negative parts. Since Q, vanishes on N V N’ fl M, it follows that @+ and G-coincide when restricted to this algebra, in particular @+( 1) = c9- (1). By replacing Q, with @*(1)-l@, we may assume that @* are states. Apply the previous Lemma for n = 2, $1 = a+, & = @- E S(M). It follows that there exists Xi? E cN(@) such that Il[I = !Pi - Xi?12 for some ‘IJ~,~ E S(M) which both have N in their centralizers. Also 9(z) > co > 0 and Q vanishes on N V N’ n M, meaning that @ 1;2 coincide on this algebra.

Let us prove that in addition to these properties P 1,2 can be taken to be factorial traces when restricted to N. To this end, let C = {($i,&) E (S(M))’ 1 (& - &)(x) > E~~$~IN~JJ/,,M = $J&4,fv~@&+t and $Q commute with N}. Thus C # Qi and clearly C is compact in the D((M*)~,M~) topology. Since the function ($~i,$~) -+ ($Q - $J~)(x) is continuous in this a*-topology, it follows that there exist (&, $4) E C such that

($4 - VW4 = SUP-w1 - dG-4 I ($1, $9) E c> ef&l.

Let CO = ((&, $$) E C 1 (&-$&)(x) = pi}. Th us C 0 is convex, &-compact and non- empty. Let ($1, $2) E Cc be an extremal point in CO. We claim that 741,~ are (equal) factor states when restricted to N. To see this, let p E P(Z(N)) be such that p, 1 -p are non-zero and note that both (t-l~~(.p),t-‘lDg(.p)), ((1 - t)-l&(.(1 - p)), (1 - t)-i&(+(1 - p)), where t = 44,2(p), belong to Ca. (Recall all the way that Z(N) c 2(M) !) By extremality, it follows that either $~i,~(p) = 0 or $i,~(l -p) = 0. This shows that $1;212(,v) are (equal) characters on Z(N) and so, since &,4N = $i,z 0 CtrN, ?/Jl,+f fOllOW (eqUd> factor states on N (as in the proof of 2.1; CtrN denotes the central trace on N).

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Denote by w = $1,21~(n/) the corresponding element in the spectrum of Z(N), 0~. Also, set $ = woc:trNol and remark that + = $101 = $zoE. Since X 2 c-l&(X),VX E M+, we have $11:x) 5 c-‘&(E(X)) = c-l+(X), so that 1+5~ 5 c-‘$. Similarly, $2 5 ~$5. We next want to prove the following:

2.2. Lemlma. V’s1 > 0, 3a1,2 E M+, such that ai 5 ~‘1, ll$+ - $(a:/2 . $“)iI < S1 and Iluu&* - a;ll+ < Sl,VU E Lt(N),i = 1,2.

Proof. Let (lip+,, [+, ‘FI+) be the GNS representation for (M, $). Let p E T+(M) be the support projection of the state implemented by (.&,, &,) on r+(M). Since ni is in the centralizer of $ we have p E 7r+ (N)’ n 7r~ (M).

Denote b4” = p$(M)p, n/” = 7r+i(~V), and still denote by $ the state (&,, &,) on M”. Note that [+ is cyclic and separating for M”, so that the norm IlXll+ = 11X(+ II implements the so-topology on the unit ball of M”.

Since NW is in the centralizer of $, there is a unique $-preserving conditional expectation I” of M” onto NW, which is implemented by the orthogonal projection of p’l+ onto q(N)& == NW&,.

Moreover, if X E M is such that I(X) = 0, then we have &(Y*X) = 0,‘v’Y E N, so that (p’~~(~7)p~~,~~(Y)t~) = (~~(x)t+,~+(y)I+) = $(Y*X) = O,‘v’Y EN, implying that Ew@ry5(X)p) = 0. Thus F&r+(X’)p) = r$((E(X’))p, ‘d’x’ E M. In particular, since E(X ‘11 > cX’,‘dX’ E M+, we g et E”(p7r+(X’)p) 2 cp7r+(X’)p,VX’ E M+. By density, E‘+(s’) 1 cz’,V~’ E M”,.

Note that since $ is a normal faithful trace on N”, we have that N” and M” are finite von Neumann algebras (cf. 1.1.3 in [POT]). In fact, note that NW is isomorphic to the type II1 factor N*/[ w 1, h w ere [w] denotes the (maximal) ideal in N generated by w (cf. [Sal).

Moreover, note that the inequality $i 5 c-l4 on M implies that llii implements a state on M”‘, still denoted by $;, satisfying $i (pr+ (X)p) = $;(X) , VX E M, and $; 5 ~‘4, i = 1,2 on M”. Also, as states on M”, $i have N’” in their centralizers.

By the IRadon-Nykodim theorem, it follows that there exist Ai E N”’ n M”, 0 5 A; 2 c:-‘1, such that $; = $(Ai’2 . At’“) on M”, i = 1; 2.

By the density of KG(M) in T@(M) and Kaplansky’s theorem, it follows that ‘d’s > 0 there exists ai E M such that 0 5 ai- < c-l 1 and I17r+(~~‘2)[~ - At’2c+ II < 6. Since [u,A~‘~] = 0, [u, C] = 0,V’u E U(N), this implies that II~~(ua~‘“u* - A,““)[+11 < 6, so that Ilr~a:‘~~* - ~$~ll+ < 26 on M.

Moreover, we have the estimate Illi,(~~‘2&‘2)-$~\l = Ij~(a,“2.aZ1’2)-~(Al’2.A,1’2)II < 2~~6. By taking, from the beginning, 6 to be less than &l/4, this completes the proof.

0 We next want to prove that the positive elements a; E M in the previous lemma are

small perturbations of elements in N’ n M.

2.3. Lemma. rf b E M is such that IIubu* - bll+ 5 6,Vu E U(N), then there exists b’ E N’ n .M such that [lb - b’ll+ 2 6.

Pro@ Let EN,“M be the unique normal conditional expectation onto N’ n M that preserves ctrH o E, i.e., such that ctr,v o E o ~~~~~ = ctr/ o E. By replacing if necessary b by b - &hrlnM(b) (which clearly still satisfies the inequality in the hypothesis) we may

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assume ~?~rt”,q(b) = 0. We will prove tha.t this and the commutation relations in the hypothesis of the lemma imply llbjlQ L: 6.

To this end, let us show that 3~ E M(U) such that Re (ctr,v*(E(b”ubu*))) < (6”/Z)l as operators in Z(N). Indeed, let { (qi, ua))it-l be a maximal family of pairs of elements with 4i mutually orthogonal projections in P(Z(N)) and 74 E U(JV'~~)

such that Re (ctr;‘(8(b*uibur))) 5 (6”/2)q;,‘v’i E I. If 4 = 1 - CiG1qi # 0 then apply (2.3 in [Poll) to the element bq E Mq, with Ec,vq.),nMq(bq) = 0, to obtain a projection 40 E Z(N): 0 # 90 I 4, and a unitary element u. E N-q0 such that Re (ctrN(E(b*uobuG))) 5 (S2/2)q 0, contradicting the maximality of {(qi; ui)liE.[. Thus C;q; = 1 and so u = Ciu; is a unitary element in JV. We then have:

= 2jlbll$, - 2Re w(ctrAr(E(b*ubu*))) 2 2ilbll$ - 2S2/2.

This shows that //bllzi, < 6.

q With these technical results in hand, let us now proceed to end the proof of the finite

case of the theorem.

Thus, let z = z* E M satisfy EN~,v*~~M(z) = 0, Ed = ($1 - $2)(z) > 0, with $I,* states on M that have J\l in their centralizers and are factor states on JV, as before.

By applying Lemmas 2.2, 2.3 above to $1j G2 and $(= $i o E = gz o E), it follows that vs > 0,s < Q/4: 3 al, a2 E ~if' n M, 0 5 ai 5 c-l1 such that /I& - $(a,fj2 a~~/2)ll < 6. Then we get:

$(a;‘2xa:/2) - $(a;‘2xa;‘2) > l/2

h(x) - 742(4 - IlTh - @(a, . “1 l/2)11 - I[$9 - $(a;‘” . cpll > El - 26.

But I = f o EN~,~,~M, so that $ = $ 0 I h’v&*‘nM, a,“2&~vn;~nM(x)a~‘2 = 0, we get

and since l~~,b9~,~ (ai’22ai’2) =

0 2 El - 2s > El - q/2 = El/2 > 0

a contradiction which completes the proof of the finite case. 2.4. Remarks. lo. The proof of the finite factorial case of the Theorem can be made

considerably shorter. Thus, 1.1-l .7, 2.2 and 2.3 are not needed and the Radon-Nykodim argument at the end of the proof can be applied immediately after lemma 2.1, by using that “E(z) > cz;‘v’x E M+ ” i.mplies “cp 5 C-IV o E, ‘dq E MT” (see A.1 in [PoSf for an elementary and short proof of this factorial type 111 case). The more general case when E preserves a normal faithful trace on M can also be given a shorter proof. Most of the difficulties encountered for the general case stem from the case when there exist no trace preserving conditional expectations of finite index, yet there do exist expectations of finite index. For an example of such an inclusion consider the locally trivial Jones subfactors NI, c Mk of index (l/k)-l + (1 - l/k)-’ = k2/(k - 1) and El, : i!Jh + Nk be the expectation of minimal index, IndE,+ = 4. Then Jdf = @kNh c $k Mk = il.4, and the expectation I = $kEk has finite index but the unique trace preserving expectation E of JW onto N has infinite index.

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2”. Note that once we have the Dixmier property for all inclusions of finite von Neumann algebras with conditional expectations having index majorized by a fixed constant c- ‘, it follows that for any E > 0, there exists n = n(c, E) > 1, such that given any inclusion N c M of finite von Neumann algebras, with a normal faithful conditional expectation E : M ---f N satisfying IndE 5 c-l, we have that ‘do E M, IIxII 5 ~,~u~;u~,...,u, E U(N) such that d(kCz=, UkSU;,N’nM) < E. For assume this is not the case. Then %a > 0,&a < 00 such that Yn > 1,3N, c M,, with &, : M, + Nn satisfying &n(X) 2 caX,VX E M+, and 3s, E M,, llxnll 5 1, such that d( i xi=, uLx,uz, NA n M,) > ~a,‘&~; u2; . . ..u. E U(N,). But then applying the relative Dixmier property for N = $,N;, c $,M, = M, E = BYE, and X = &z, we get some V, = $n~l,n; . . . . Vk = $nwk,n E U(N) such that

vjxvj*; N’ n M

This gives ai contradiction if n > t%.

3. Proof of the Properly Infinite Case

Unlike the: finite case, where we used a contradiction argument and the Hahn-Banach theorem, the proof of the properly infinite case will be direct and more or less constructive.

Thus, in this section we will assume that N c M is an inclusion of properly infinite von Neumann algebras, with Z(N) c Z(M) as always.

We first prove the theorem under the additional assumption that JV~ (and thus M) is countably decomposable, or equivalently a-finite, i.e., N has normal faithful states. This assumption will be needed for the reduction theory considerations below.

For X E M and u = (ui,u~, . . . . u,) c U(N), it will be convenient to denote TJX) = ; c”= 3 I 2~jXuT. Note that M 3 2 +-+ T,(x) E M is completely positive and unital, thus Tu(X)*Tu(X) 5 37,(X*X), by Kadison’s inequality.

Let further 3 c M denote the norm closed ideal generated by the finite projections of M. Note that I(J) = 3 n N is the ideal generated by finite projections in N and that I implements a conditional expectation, still denoted by I, of M/J onto N/J. Note also that if N, A4 are purely infinite then J = 0.

Then for )Gi E 3, K > 0 we have E(K) 2 cK and by ([HI or [SZl]) it follows that for any E > 0 there exists u = (ui: . . ..u.) c U(N) such that IIT,(I(K))II < E, thus IIT,KII < EC ‘. Thus 0 E C&Y).

If X E .A4 then we denote by (IX/I, th e norm of X in the “Calkin” algebra M/J. Thus llXlle = /IX/l if M is purely infinite. Note that if X E N then Ilw- n JVll = IIWJ-I(= IIXlle).

For t E RJ~ denote by I, the maximal ideal in N generated by t. Similarly, if s E Rm then denote by 1; the maximal ideal in M generated by s. Recall ([SZl]) that if y E N (resp. z E n4) then the function t -+ Ily/ltII (resp. s 3 Ilz/1~]]) is continuous on Qfl (resp. on 0;2hZ). Also, with the above notations, we have sup{]]y/l,II I t E 0,~) = IIylle, suP{JIx/Q:.I/ I 3 E %4> = 11~11e (cf. w11, since N, M are countably decomposable).

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3.1. Lemma. For to E RN let Jt, = {x E M 1 I(x*x) E It,]. Then Jt, is a closed bilateral ideal in M and JtO+ = {x E M+ 1 E(z) E It,}. Moreover, if{~k}l<k<~ C 0~ is the preimage of to under the surjection f : RM -+ fiN, implemented by the inclusion C(fi~) = Z(N) c Z(M) = C(R,M), then &, = n,&. AZso, ifX E M then thefunction t -+ IIX/Jtl/ is continuous on Sl,v and its supremum equals IIXl/e~

Proof By the way it is defined, JtO is clearly a closed left ideal in M. To prove that it is a right ideal as well, note that if x E Jt, then the function t + 115(x*x) /It /I has a zero at to. Thus YE > 0 there exists an open-closed neighborhood Va of ta in 0~’ such that IIE(x*x)/Itll < c,Qt f VO. Equivalently, if pv, denotes the projection in Z(N) corresponding to the set VO, we have ilE(x*x)pvO Ile < E. Thus, if y E M, then, by using tbat Z(N) c Z(M) ( so in particular pvO E Z(M)) we get:

llf(Y*~*~Yh% l/e L II?/* x*~YPvo Ile I llYl1211~*vwG lie

It thus follows that IIE(y*x*xy)/ItOll < &c-l llyl12. Since E was arbitrary, this shows that IIE(y*x*xy)/l,,ll = 0, thus I(y*x*zy) E I,,, proving that ~9 E Jt,.

If we now take x E &+, then E(x2) E It, and since &(x2) 2 Ed it follows that Ed E I,,, so that E(x) E I,, as well. Conversely, if x 2 0 and l(x) E I,, then x1j2 E Jt, (by the definition of jTt,) so that x E Jt,.

To prove the last part of the statement, let x E Jt,. Since the continuous function t ++ lKx*4ll h as a zero at to, for each E > 0, there exists p E P(Z(N)) such that P(~o) = 1 and IiE(x*x)~ll~ < E. By regarding p as an element in G’(G,) = Z(M) > z(N) = C(flN), we have p(sk) = l,M. But E(x*x)p > cx*xp, so that IIx*zpjl, < c-l&, implying that ~~x*~/I~~~I < c -‘E. Since E was arbitrary, this implies llx/I~, II = 0 so that x E nkl;k.

Conversely, if x E fIJtk;, then, there exists pk E P(Z(M)) such that pk(sk) = 1 and lIxpk[j, < &,Vk. If we denote p = Vkpk, it follows mat p(sk) = l,V’k, and Ilxplj, < E. But by the proof of Lemma 1.2 there exists 4 E P(Z(N)) such that 4 5 p and q(to) = 1. Thus 11~~11~ < E, so that ~~E(x*x)~~~~ < e2. Thus llE(x*x)/ItOll < E and since E > 0 was arbitrary, this implies E(x*x) E It,, i.e., x E Jt,.

The very last assertion in the statement of the lemma follows now by first noting that for X E M and t E R,u we have

And since by lemma 1.2 there exist continous functions gl,g2, . . . i gn : RN + fin such that f o gk = ido,! V’lc (i.e., gk are right inverses for f) and such that

it follows that the function flN 3 t H llX/Jt 11 is the maximum of the continuous functions fl~- 31--t ijX/I~sc,lIl,i = 1, .'., n and thus it is itself continuous. Also, since

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= suP{JJ~/qJ I 3 E S-‘(t), t E av)

= sup{llX/I:,II 1 s f %4} = pqe.

q The next t’echnical lemma, based on a convexity argument, can be regarded as the crucial

point in proving the properly infinite case of the theorem.

3.2. Lemma. Let x = n;* E M and t E 0~. Denote a = inf{sp(E(y2)/&) 1 y E CN(X)}. Then we have: a) There exists y E Cn/(x) such that Il&(y”)/It - all] < E; moreover, given any element y E CA/(x) satisfying Ilf(y2)/It - al/l < E, we have

Ilq(zY>“>/~t - allI I E,VW = (‘ul, **a> 4 c wq

b) If y E C,,/(x) satisfies IIE(y2)/It - all/ < E then

IICY - %Y~;)lJtll I @EC -y2, VW0 E u(N).

c) If y E C,,/(x) satisjies IlE(y2)/It - allI < E and Y’ E CN(Y) then II(Y - v’)/Jtll 2 (8&c-y

Proof a) L,et y E CN(Z) be such that inf{sp(E(y2)/1,) 5 a + E}. By [SZl] there exists u = (Ul, . ..) u,) c U(ti) such that

IIccd~(Y2)) - WLII < F*

Since TU(E(y2)) = &(TU(y2)) we get, by Kadison’s inequality applied as above,

&((T,Y)~) 5 E(Tu(y2)) = T.(f(y2)) 5 a + E, moddo &.

Since TUy E CM(X), by the definition of a we also get E((~!‘,Y)~) 2 al modulo It (i.e., in the quotient (Y-algebra AI’/&). Thus Il(E((TUy)“) -. ul)/Itll 5 E. Since T,y E CN(X), taking lif,y fior y we get the first part of a). Next, if y satisfies lj(&(y”) - ul)/Itll 5 E and w = (WI, “., v,) c IA(N) then again modulo It we have

a I q(zy)2) I K&qY2)) 5 rJ((a + &>l> = a + es

b) Since by a) we have ll(E((TVy)2) - ul)/Itl/ 5 E for any ‘u = (~1, . . . . w,) c U(N), it follows that modulo I, we have

0 5 qG(Y2) - (zY)2) = W(Y2)) - wGY)2) I E

Since E has finite index we get:

Il(?:(Y”) - (zY)2)IJtlI I c-l Il(qG(Y2)) - q(zY)2)),‘411 L 2Ec-1

Take then w. E U(N) and note that (y - vDy$)” = 4(T,(y2) - (5%~)~), where ‘u = (1, Q). From the ablove we then get ll(y - u~~TJ~)/J~II~ 5 ~EC-~.

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c) is now trivial from b).

Let us now show that from the estimates in lemma 3.2 at a point t we can get similar eiiimates in a neighborhood of t:

3.3. Lemma. Let to E 62,~, beBxed and assume y E JIM satisfies jj(uyu* - y)/J,, /( < q,, vu E U(N). 27 zen there exists an open-closed neighborhood V of to such that /I(uYu* - Yhlle < 2EO> ‘du E u(N).

Proof. Assume, on the contrary, that for all neighborhood V of to, 3u E &f’(M) satisfying /I(~yu* - y)pbpjle 2 2~.

Let {(P;,u~>>GI b e a maximal family of pairs of elements with pi mutually orthogonal projections in Z(N) and ui E U(Np,) satisfying ll(uiyu% - y)p;/Jtll 2 &,/2,Vt E I&, where Vpi is the open closed set in QJJ corresponding to pi E Z(N). Let p = Zipi and VP C RN its associated open-closed set,

There are two possibilities: either to E VP or to E RN \ VP.

If to E VP and we put 2~ = Ci’ui + (1 - p) then u is a unitary element in N and thus lI(~yu* - y)/.Jt, 11 < ~0, by hypothesis. Thus, there exists an open-closed subset U c VP such that to E U and / / ( ,y w U* - y)~~ll~ c Ed. Since pu # 0 and pu 5 p = Zipi, it follows that there exists i such that pupi # 0. But then we get

Eo > Ij(uyu* - y)pupiIIe = sup(ll(uyu* - Y)/Jtll I t E 4% n u> 2 3&o/2,

a contradiction. If to E RJJ, \ VP, then by the initial contradiction assumption applied to V = RJJ \ VP it

follows that there exists ‘LL E U(J\/) satisfying /I(uyu* - y)pvIle 2 2~~. It then follows that there exists a non-empty open-closed set V. c V such that if

p0 = pub then

But then ((P~,G,))GI u ((PO, TO)) contradicts the maximality of ((pi7 ui))iEI, This final contradiction completes the proof of the lemma.

0 We will now use 3.2, 3.3 and a compactness argument to obtain an element in CN(ZG)

that globally satisfies the commutation estimates with unitaries in N:

3.4. Lemma. Let :I: = :f E A4. Given any E > 0 there exists y E C&i(z) such that IJwyu* - ylle 5 &,VU E M(N). M oreover, if y’ is an arbitrary element in C,v(y) then lb - Y’lle 1. E.

Proof. By 3.2, Vt E a,~, 3~~ E CM(Z) such that /I(uytu* - y,)/&,il 5 ~/2,‘du E M(N). By 3.3 it follows that there exists an open-closed neighborhood V, of t in $2,~ such that /I (uYtU* - ~,th, I/e < E,~U E u(N).

Since Rj,.f is compact, there exists tl , . ..> i& E fi*r such that I.&V,~ = RN.

Tale pl = pv, and pk = pv, - pv, (C~:;,‘P~) if k 2 2, which will thus be mutually orthogonal projections in Z(N), with sum equal to 1. Let also y = C;=,yt,pk and note that y E CJJ(X), Indeed, this is because if yl; . . . . ym E en/(z) and pl, . . .? p, is a finite

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partition of the identity with projections in Z(N) (which is included in Z(M)) then CjYjPj E cu/(4, a fact that can be easily seen by the same argument as in the single algebra case (see e.g. [H] or [SZl]). Since

it follows that y, defined this way, satisfies all the required conditions.

The last part is trivial, since any y’ E CN(Y) is a norm limit of elements of the form T,y, aad for all such elements we have II y - T,yll, < E.

q Let us now proceed with the proof of the properly infinite case (Jd; M are still assumed

countably decomposable for the moment). As in the finite case we take z = z* ‘E M such thiat EN”N/“M(~) = 0. By Lemma 3.4, there exists yl E CM(Z) such that llyl - wyi21*l], < 1/2,Yu E U(N) and llyr - y’lle < 1/2,‘v’y’ E Clf(yr). More generally, by applying recursively Lemma 3.4 we can construct recursively a sequence of elements {yntn (C C*,(z) such that:(i)y, E c~(y/,-1); (ii) the diameter of Cti(y,) in the “Calkin” algebra M/J is < l/an; (iii) 11~~~ - y,~ll, < 1/2”,Vv E 24(N). But then yn. is Cauchy in M/J so there exists y = y* E M such that I/y - ynlle tends to 0. By (iii) we also have that y commutes with ni modulo J.

Now in the purely infinite case ,7 = 0; ]I /le = II II so we get y E CM(X) n JV’ n M and we are done.

When N, M are properly infinite but semifinite, by (I.1 in [PoR]) it follows that y = y’ + .K for some y’ E N’ n M and K E 3. Thus ‘in > 1,X, E 3 such that l/y’ - (yn + Kn)ll < l/an. By the remark before Lemma 3.1, there exists u = (Ul; . ..., urn) C M(N) such that IjTU(Kn)/I < l/an. Since T,(y,) E CM(X) and TU(y’) = y’ it follows that y’ is at distance < 1/2”-l from C?n/(z),Vn > 1. Thus y’ E C,~(X). But then, as we’ve noticed in 1.6 b), y’ = 0.

This completes the proof of the relative Dixmier property for inclusions of properly infinite, countably decomposable von Neumann algebras with finite index.

To settle the case when Nj M are properly infinite but not necessarily countably decomposable, let us first prove the following:

3.5. Lemlma. Given any normal ekment u E M there exists a family of countably decomposable projections {p;};,~ C N (i.e., each p; is the support of a normal state on M) such that &pi = 1 and {P;}~~~ c {u}’ n N.

ProojF. Since any normal element in M is contained in a von Neumann algebra generated by a unitary element, it is sufficient to prove the statement for u E M unitary.

Begin by noting that if 4 E M is a projection of countable type then Tq def VnEz~nq~-n

and Sq dgf s(E(q)) are also projections of countable type. Indeed, because if 4 is the support of a normal state $ on M then Tq is the support projection of CnEZ2-lnl$(~” . U-“) and Sq is the support projection of $ o E. Also, note that if {qn}7L are of countable type then V,q, is of countable type.

Let tJ?en {pi}ic~ be a maximal family of mutually orthogonal, countably decomposable projections in {u}’ n N and assume p = 1 - &pi # 0. Let 0 # q. < p be the support projection of a normal state on pMp. From the above remark it follows that

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PO fEf vn>O(Sqn(qo) is o countable type as well. Note that q. < l’qo 5 STqa 5 f TSTq0 < . . . / ~0. Also, since (ST)n(qo) E JV,V n, we have pa E ,U. Moreover, for each fixed k E Z we have u~((S’T)~~O)K~ < T(ST)“qa < p. so that ukpo~-k < po. Taking -k for k we get u-“p& < p. as well, so that ulipo& = po,Vk E Z. Thus p. belongs to i~}‘fl~%‘. But then {pi);,~U{po} contradicts the maximality of (pijiEl. Thus, CiE1pi = 1.

By the preceding lemma and by using the fact that if II: = IC* E J\/L is such that E,v~,v~~,M(z) = 0 then E,~~,\f~~,~(piic(pi) = 0; ‘di, where pi are as in 3.5, it follows that in order to have the relative Dixmier property for inclusions of arbitrary properly infinite von Neumann algebras with finite index, it is sufficient to have it in the countable decomposable case, provided we can prove that the (minimal) number of averaging unitaries needed depends only on /Iz/l and E. Thus, all we need in order to conclude the proof of the properly infinite case of the Theorem is the following lemma, whose proof is identical to the proof of Remark 2.4.2”, but which we include here in details, for convenience:

3.6. Lemma. For au.y E > 0, there exists n = n(c, E) > 1, such that given any inclusion j!~’ C J\/I of countably decomposable properly infinite von Neumann algebras, with a normal faithful conditional expectation 8 : M -+ n/ satisfying Ind& 5 c-l, we have that Vz E M, //XI/ < 1,3ul.u2, . . ..uu. E U(ni) such that d(k Xi=1 u~zu~,:V” n M) < E.

Proof. Assume this is not the case. Then 3~ o > 0; 3co < x such that ‘dn 2 1; Eltin C M,, with E,, : Jbf, + tin satisfying E,(X) 2 coX,VX E ,U,, and Zlz, E Mn, I/z,/1 5 1, such that d(h Cz=, u~z,u~,~< n M,) > E~,Yu~,u~~.~.~u, E U(il&). But then applying the relative Dixmier property for the inclusion of countably decomposable, properly infinite von Neumann algebras N = @&i c @,JU,,, = J~Z,

E = @,& (N.B. iv; M are countably decomposable as being direct sums of countably many such von Neumann algebras!) and to the element X = &z, we get some

This gives a contradiction if n > k.

vl = $n~l.n, ...j V, = CB,~IU+~ E U(N) such that

3.7. Remark. 1”. Note that the above proof of the properly infinite case of the Theorem becomes much simpler if we assume the algebras JV, M are properly infinite factors of countable type. Indeed, in this case no reduction theory argument is needed and so, Lemma 3.2 being stated for the factors N. M and the ideal J (instead of Nt, Mt, Jt), the argument following 3.4 can be applied directly (Lemmas 3.3 - 3.6 are not needed).

2”. Let us also present a very simple argument proving the Theorem in the separable factorial type III (purely infinite) case. SO assume N c JU are purely infinite factors with separable preduals and let cp be a normal faithful state on J~Z such that q~ o E = ‘p. If 5 = z* E M is so that I,v,v,br,nn/l(z) = 0 and we denote by Cj’j(:~) the weak closure of CO{UICU* I u E M(N)} (and thus of C,kr(z)), then C;;?(z) n JV’ n M # (4 {for instance by 2.1 in [ILPo]). But E(s) = 0 implies I(y) = 0, V?/ E C,y((x), showing that C,:(z) n N’ n M = (0). By the inferior semicontinuity of the norm ]lyllV with respect to the weak topology, it follows that b’n > 1; 3y, E C,v(z) such that lIynjl+ < l/n’.

4" S6RIE TOME 32 ~ 1999 ~ No 6


Thus, if e, is the spectral projection of lyn 1 corresponding to the interval [l/n, I/X/]] then cp(E(e,)) = cp(e,) < l/n. S ince E(e,) 2 0, this implies that ‘d/~ > 0,3n, > 1 such that

qO,a](f(en)l # 0, where Q = c~/41131\. By the well known properties of the Dixmier sets for elements in purely infinite factors (cf e.g. [SZl]), applied here for the factor ti and the element E(e,) E ti, it follows that there exist u1 j 3~2, . . .> U, E M(H) such that

Since E(e,) > ce,, it follows that

Thus we get

Since yn E Cn/(z) and E was arbitrary, it follows that 0 E C,Q-(x), thus proving the result in this separable, factorial type III case.

4. Applications

We will now mention some consequences of the Theorem. Our first result shows that for inclusions of type II1 factors N c M, the finiteness of the index not only implies the relative Dixmier property, but it is actually equivalent to it. Along the lines, we also prove that the finil:e index condition on N c M is equivalent to the property that states on M that are normal when restricted to N follow normal on all M.

4.1. Corolllary. Let N c M be an inclusion of type III factors.

(i). N c M has finite Jones index, [M : N] < 30, if and only if the only states on M that are normal when restricted to N are the normal states of M.

(ii). If N c M has finite Jones index then N c M has the relative Dixmier property. Zf in addition N has separable predual then, conversely, if N c M has the relative Dixmier property then [M : N] < 30.

(iii). If N is isomorphic to the hype@nite type III factor R, then [M : N] < 00 if and only if any conditional expectation of M onto N is normal (equivalently, if and only if any state on M having N in its centralizer is normal on M).

Pro@ If [M : N] < co, then, by [PiPo], we have EN(Z) > [M : N]-‘z,Vz E M+, where Eni denotes the unique expectation of M onto N that preserves the trace r of M. Thus N c .M has the relative Dixmier property by the theorem, proving the implication * in (ii).

Also, we have already mentioned in Section 2 that the implication + in (i) holds true for all inclusions of von Neumann algebras n/ c M with conditional expectations

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E : ~z/1 -+ :v satisfying the finite index condition “E(z) > cz,‘d.z E J\/1+“. Indeed: for if this condition is satisfied then for any state ;o on A4 we have QG < c-‘p o &, so if cpii\i is normal, then cp is bounded above by the normal state ‘p o & on A4 and thus, by Sakai’s

adon-Nykodim theorem ([Sal), cp is normal itself.

To prove the converse in (i) and (ii), let us first deal with the case when N’ n n/r is infinite dimensional. One can actually prove the following more general:

4.2. Lemma. Let *NC C M be an inclusion of von Neumann algebras and assume there exists a sequence of mutually orthogonal non-zero projections {p,}, in M such that for each m one has the implication “x E iv, zp,, = 0 + x = 0”. Then each normal state on .k’ can be extended to a singular state on M. If in addition N has a separable predual, and an in&ite dimensional irreducible representation then N c M doesn’t have the relative Dixmier property.

Proof. Let $ be a normal state on iv. Since ti is isomorphic to ~4l-p~~ we can view ~4 as a state $,n on Jifp,. Let (Pi be an arbitrary state on pnMp, extending $I~ and still denote by pn the state on ~\/1 defined by compression to pILJ!Ytp,, cp,(z) = (pn(pnxpTL), x E M. Let 4 E J\/L”’ be a weak limit of {v,~}~ E M*. Thus cp is a state on M, cpl,v = $ and cp is singular on M (because cp(l - C,,,P~) = 0: Vm and (1 - C,,,,pn) -+ 1 as m + a>.

If in addition .A has separable predual then let {~~.k}k E Af be a sequence of elements dense in the unit ball of N in the weak operator topology. If we assume, on the contrary, that N c M does have the relative Dixmier property, then in particular the element X = Cn.x,p, in M satisfies CM(X) n JV’ n M # 0. But for each n one has C,v(X)p% = C,~r(x,)p.~~ c hf~,~ and since N’ n M n Npn = N’ n Np, = Z(~b’)p~~, it follows that C,Qr(X) n JL” n M c C,Z(N)p,. Consequently, YE > 0 there exist finitely many Unitary elements Ul : 7L2, . . . , U, E U(N) and some central elements {z,},, in the unit ball of 2(N) such that

ui,xu; - c&&p, Since Xp, = x,p, and uk commute with p,, V’5, n, we have:

ukxu; - Cnznpn u,&x,u; - z, p, k=l ,ii

uk.&u~k - & ,

i

vn.

Thus, ,ul, u2, I... U, E M(N) have the property that

z&x,u; - z, < c,Vn

But then, since {zk}k is dense in the unit ball of N in the weak operator topology, by the inferior semicontinuity of the uniform norm with respect to the wo-topology it follows that Yx in the unit ball of off 3.z in the unit ball of 2(N) such that

4” SI?RIE - TOME 32 - 1999 NO 6


Moreover,, if 7r : n/ + a(‘??) is an irreducible representation of N then, by using again the inferior #semicontinuity of the uniform norm with respect to the wo topology, the wo- density of the unit ball of I in the unit ball of B(K) and the fact that, for z E Z(N), ~(2) is a scalar multiple of identity in 13(7X), it follows that ‘dY E 13(X), 3a E C such that

II

1 m - m c UkY$ - al 5 &.

k=l II

But if x is an infinite dimensional irreducible representation of the von Neumann algebra ti, then the corresponding Hilbert space K is non-separable, and since ul, . . . . U, is a finite set, there exists an (non-zero) invariant separable Hilbert subspace 3-Ia c 7-t which is reducing for all the elements ~1, . . . . u,. Thus, if pa denotes the orthogonal projection of ‘R onto 3-t. then uipouz = pa, V’i. Taking Y = pa it follows that for some scalar Q we have

which is a contradiction if E < l/2, no matter what Q! one tak’es. This completes the proof of 4.2.

Let us now prove the implication + in (i) in the case N’ n M is finite dimensional. Note that if the nlormal states on a von Neumann algebra iii have only normal state extensions to the von Neumann algebra M (3 ti) and p E N’ n M is a projection then the normal states aIn til, have only normal state extensions to $4~. Note also t:hat if dimN’ n M < 00 and [M : Ai] = cc then there exists a non-zero minimal projection p E N’ n M such that [pMp : Np] = cc (cf. [J]).

This shows that in order to prove the implication + in (i) it is sufficient to prove that if an inclusion of type II1 factors N c M satisfies [M : N] = 00 and N’ n M = Cl then there exist singular states on M which restrict to normal states on N.

To see this, recall that by [PiPo] for any E > 0 there exists g E P(M) such that EAT (4) < ~1. Let qE be a maximal projection in M such that EN (qE) 5 E. Then clearly EN(q8) has support 1, because if p would be a non-zero projection in N such that p.G~(q,) = 0 then again by [PiPo] we can find q1 E P(pMp), 0 # q1 < p such that Ep~p (41) 5 ep and so qE + 41 would contradict the maximality of qE. Now let b, = ~~~~~~~~~~~~~~~~~~~~~~~ and note that EN(&) = 1 and r(bE) = 1. Thus b, E L1 (M, 7-)+ and r(. bE) defines a normal state on M. Also, since EN (bE) = 1 we have r(yb,) = r(:y);Vy E N. Furthermore, b, satisfies r(s(b,)) = r(qe) < E.

For each E = 22” take a a, = b2-- this way constructed. Since -r(s(a,)) 5 22”, if for each m we denote p, = V,~~s(a~) then {p,}, is dicreasing to 0. Thus, if $ is a limit point of {T(. b,)}, E M* in the c(M*, M)-topology, then $(l - pn) = 0,Vn and 1 -p, + 1 so that $ is a singular state on M. Thus M has a singular state which restricted to N is equal to the trace, thus finishing the proof of (i).

The implication + in (ii) in the case dimN’ n M < cx follows now immediately from (i) and the result in [P] showing that if M has singular states that restrict to the trace on N and N has separable predual then N c M doesn’t have the relative Dixmier property.

Finally, to prove (iii) assume N is isomorphic to R. If [M : N] < cc and E is a conditiional expectation from M onto N and cp is a normal state on N then ‘p o E is

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normal on &l by (i). Thus, if (zi>i is a weakly convergent net in the unit ball of M then cp(E(si)) + 0, showing that {E(Q)}~ IS weakly convergent to 0 as well, thus E is normal.

Conversely, if [M : Y] = x then by (i> it follows that there exists a singular state y on Ad such that (pp = rllr. Let 24, be a countable discrete amenable subgroup of U(-V) such that M{ = N and put *$ = Jcp(u . u*)dp(u), where p is the invariant mean on the discrete amenable group Ua and the integral is in the usual, Banach limit sense (see e.g., [S]). Since + belongs to the closure in the g(M*, M)-topology of a countable set of singular states on M, by [A] it follows that $ is singular on M. Also, by construction we have $lnr = ~~7 and $(u . u*) = Gi, ‘v’u E Ua. By Connes’ lemma in [C3], it thus follows that $(u , u*) = ,$,Yu E U(N). But then $ is a singular N-hypertrace on M so by the usual construction (see e.g. [C4]) it defines a singular conditional expectation of hf onto N. This completes the proof of the Corollary 4.1.

4.3. Remarks. lo. Note that in the above Corollary the separability condition on :li is essential: indeed, it has been proved in [Po3] that if N c M is an irreducible inclusion of type II1 factors and w is a free ultrafilter then N” c M” does have the relative Dixmier property, even when [M : N] (= [MU : N”]) is infinite.

2”. The hyperfiniteness assumption on N in the last part of the Corollary is also essential. Thus, if the free group on two generators iFz is embedded in the usual way in the free group on three generators 1F 3 and N c M denotes the corresponding inclusion of von Neumann factors L([F,) c L(Fs), then it is easy to see that any conditional expectation of M onto N must coincide with the trace preserving one, while [iW : N] = 30. In fact, by using Day’s trick one can show that if N c &i is an inclusion of type III factors then there exists a singular conditional expectation of M onto N iff YE > o;b&; . ...7& E U(N),3b E Ll(M:T)+ such that: Eiv(b) = 1, I~u&u; - bjj, < E and r(s(b)) < E. In turn, this last condition clearly doesn’t hold true for L([F,) c L(IFs), for instance by ([MvN] or 2.1 in [POT]).

Other examples of inclusions N c M of infinite index for which any expectation of M onto N is normal are obtained by taking N to have the Connes-Kazhdan property T and M an arbitrary type III factor containing N such that N’ n M = Cl and [M : N] = x. For many similar examples, see e.g. [Po3] or [POT]. For instance, if N has the property T, and if P is an arbitrary finite von Neumann algebra with a faithful normal trace and no minimal projections, then there exists a unique conditional expectation of N * P onto N(=N * C), while there always exist singular conditional expectations of R * P onto R !

3”. The proof of 4.1 obviously works beyond the factorial case, for rather genera1 classes of inclusions of finite von Neumann algebras. Let us mention though that if one drops the finiteness condition, even just for M, the problem may become much more difficult to solve. For instance, if one denotes by D the algebra of diagonal operators on the infinite dimensional separable Hilbert space 7-t, then it has been noted by J. Anderson in [An] that the relative Dixmier property for D c B(IFI) IS e q uivalent to the positive solution to an old standing problem of R.V. Kadison and I. Singer ([KSi]), asking whether each pure state on ;r) extends to a unique pure state on B(R). The answer to this is still not known, despite considerable efforts (see e.g., [An], [BHKW], [BoT]). Whiie all the results in these papers are about proving that for certain large classes of operators X E 1?(E) the set CD(X) contains E(X), E being the unique normal conditional expectation of 13(x) onto

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D, thus pointing towards a positive answer to the Kadison-Singer problem, the opinion among experts is that the answer to the problem might as well turn out to be negative.

In connection with this, let us point out that despite IndE = cc in this case, the inclusion 2) c L?(E) does satisfy that a state on B(‘FI) which is normal on 2) is normal on all L?(E) and that 8 is the only possible conditional expectation of B(‘Ft) onto 2). Thus, the extrapolation of the statement 4.1 (ii) to more general inclusions suggests that the Kadison-Siger conjecture might have a negative answer, i.e., that 2, c B(Z) does not have the relative Dixmier property, while part (i) of 4.1 suggests that the conjecture might have a positive answer. On the other hand, it is interesting to note that if there does exist a pure state ‘on 2) which extends to a pure state cp on B(IFI) such that cp # cp o E, then ‘p is still <E))-linear, in fact even ‘D-multiplicative (cf. [An]; one can also prove this by using the same argument as in 2.1 and in the proof following it). For certain fast growing free ultrafilters w on lV it is easy to see that if (01~ = w then this last condition forces cp to factor through E thus leading to a contradiction (see [Re]). For general ultrafilters though, it is not clear that this would still be the case.

The next corollary gives a somewhat surprising consequence of the theorem, which, while related in spirit to Connes’ well known characterization of proper outerness of automorphisms in [C2], may be interesting on its own:

4.4. Corollary. Let Q be an arbitrary von Neumann algebra and 01, . . . . IS~ a jinite set of properly outer automorphisms of & ([Cl]). Then

(0; “‘j 0) E Co”{(w(U*))l<i<n 1 u E LqQ)}.

Proof. This is a trivial consequence of the theorem, applied to the locally trivial inclusion of von Neumann algebras tidGf{x $ nr(z) @ . . . $ a,(z) 1 IC E Q} c

M defM~n+,~x(,+l&G)~ with its obvious expectation E of index (n + l)“, and to the elements {ec,j}i<jjn c M (see [Po2] for the latter notations). -

0 Note that the above result can be applied to the case when Q is the diagonal algebra

D ry Z’(N) mentionned in Remark 4.2.3”, in relation to the Kadison-Singer problem. Thus one obtains that if X E 8(7-L) belongs to the C*-algebra generated in B(7-L) by V and the unitaries in the normalizer of 2) in B(‘Ft) then CD(X) n 27 = {E(X)}. Thus we recover some of the results in [An], [BHKW], [BoT].

4.5. Corollary. Let Q be a type II1 factor and a : G --+ AutQ a properly outer action of a discrete group G on Q. Let M = Q X, G be the associated von Neumann cross- product algebra (thus M is a type ZI1 factor) and denote by MO the reduced C*-cross product algebra, i.e., the C*-algebra generated in M by Q and the unitaries {u~}~~C c M implementing CT. Then Q c MO has the relative Dixmierproperty but if G is infinite and Q has separable predual then Q c M doesn’t have this property.

Proof. The first part is an immediate consequence of Corollary 4.4 while the second part is a trivial c:onsequence of Corollary 4.1 (ii).

0 Finally, let us mention that another significant application of the Theorem is given in

[Po5].

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REFERENCES

IAkl Ch. AKEMAUX, The dual space of an operator algebra. Trans. Amer. IMath. Sac., Vol. 126, 1967, pp. 286-302.

:All] .I. ANDERSON, Extensions, restrictions and representations of states on C--algebras, Tmns. Amer. Math. Sac., Vol. 249, 1979. pp. 303-329.

:ApZ] C. APOSTOL and L. Zsroo, Ideals in W*-algebras and the function rl of A. Brown and C. Pearcy, Revile Roumailze Math. Fur-es Appl., Vol. 18, 1973, pp. 1151-I 170.

.BaDH] M. BAILLET, Y. DENIZEAU and J.-F. HAVET, Indice d’une espCrance conditionnelle, Compositio Mathematics, Vol. 66, 1988, pp. 199-236.

IBHKW] K. BERMAN, H. HALPERN, V. KAFTAL and G. WEISS, Matrix norm inequalities and the relative Dixmier

PoTl

[Cl1

!a

LC31

CC41

IDI1 D23

iHI

[ILPO]

1.U [KSi]

[MW [PiPo]

WI

lPQl1

[PO21

lPo31

IPo41 [PO51

Po61 [PO71

1PoRJ

iReI Isal iSI tsz11

property, ht. Eq. und Op. Th. J., Vol. 11, 1988, pp. 28-48. J. BOURCAIN and L. TZGRIRI, On a problem of Kadison and Singer, J. Reine Angew. Math,, Vol. 420,

1991, pp. l-43. A. CONNES, Une classification des facteurs de type III, Ann. Sci. Ecole Norm. Sup., Vol. 6, 1973,

pp. 133-252. A. CONNES, Outer conjugacy classes of automorphisms of factors, Ann. Ecole iVorm. Sup., Vol. 8, 1975,

pp. 383-419. A. CONNES, Compact metric spaces, Fredholm modules and hyperliniteness, &god. Th, & Dynam. Sys..

1989, pp. 207-220. A. CONNES, Classification of injective factors, Ann. Mafh., Vol. 104, 1976, pp. 73-l 15. .I. DIXMIER, Le3 al&breA d’ope’rateurs duns l’espace hilbertien, Gauthier-Viflars, Paris 1957, 1969. J. DIXIMIER, Quelques propriCtCs des suites centrales dans les facteurs de type 111, Invent. Math., Vol. 7,

1969, pp. 215-225. H. HALPERN, Essential central spectrum and range for elements of a von Neumann algebra. Pac~$c

J. Math., Vol. 43, 1972, pp. 349-380. M. IXMI, R. LONGO and S. POPA, A Galois correspondence for compact groups of automorphisms of von

Neumann algebras with a generalization to Kac algebras, J. Furzct. Analysis, Vol. 155, 1998, pp. 25-63. V.F.R. JONES, Index for subfactors, hvent. Math., Vol. 72, 1983, pp. l-25. R.V. I(/~~x~oN and I.M. SINGER, Extensions of pure states, Amer. J. Math., Vol. Xl, 1959, pp. 383-400. F. MURRAY and .I. van NEUMANN, Rings of operators IV, Ann. of Math., Vol. 44, 1943; pp. 716-808. M. PIMSNER and S. POPA, Entropy and index for subfactors, Ann. Scient. EC. Norm. Sup., Vol. 19, 1986,

pp. 57-106. F. POP. Singular extensions qf the trace and the relative Dixmier property in type III ,factors, preprint

1997. S. POPA, On a problem of R. V. Kadison on maximal abelian subalgebras, Imient. Math., Vol. 65, 1981,

pp. 269-281. S. POPA, Classification of subfactors and their endomorphisms, CAMS Lecture Notes No 86, 199.5. S. POPA, Free independent sequences in type 111 factors and related problems. Ast&sque, Vol. 232. 1995,

pp. 187-202. S. POPA, Classification of amenable subfactors of type II, Acta Math., Vol. 172: 1994, pp. 163-255. S. POP& Some properties of the symmetric enveloping algebras of a subfactor with applications to

amenability and property T, preprint 1997. S. POPA, Correspondences, INCREST preprint 1986. S. POPA, Maximal injective subalgebras in factors associated with free groups, Adv. in Math., Vol. 50,

1983, pp. 27-48. S. POPA and F. RADULESCU, Derivations of von Neumann algebras into the compact ideal space of a

semifinite algebra, Duke Math. J., Vol. 57, 1988, pp. 485-518. G.A. REP, On the Calkin representations, Proc. London Math. Sot., Vol. 23, 1971, pp. 547-564. S. SAKAI, The theory of W*-algebras, Lecture Notes, Yale University, 1962. S. STRATIL& Modular theory, Editura Academiei andAbacus Press, Bucharest and Tunbridge-Wells, 198 1. S. STRATILA and L. ZS~DO, An algebraic reduction theory for W* algebra, Rev. Roum. Math. Pures Appl.,

Vol. IS, 1973, pp. 403-460.

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S. STRATILA and L. ZSIDO, Lectures on von Neumann algebras, Edituru Acudemiei and Abacus Press, Bucharest and Tunbridge-Wells, 1979.

M. TAKESAKI, Conditional expectations in von Neumann algebras, J. Funct. Anal., Vol. 9, 1972, pp. 306-321.

(Manuscript received November 14, 1997.)

s. POPA

Math Dept UCLA, Los Angeles, CA 90095-155505

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The relative dixmier property for inclusions of von Neuman algebras of finite index

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