CARTAN SUBALGEBRAS OF FINITE VON NEUMANN ALGEBRAS

CARTAN SUBALGEBRAS OF FINITE VON NEUMANN ALGEBRAS

ALLAN M. SINCLAIR and ROGER R. SMITH

1. Introduction

Sorin Popa [Po 1] [Po 2] has proved several results on the existence and properties of hyperfinite subfactorsRof a typeII₁factorMwith the relative commutantR⁰\MofRinMequal toC1. These theorems have been used in various cohomology calculations [CES, CS, PS, CPSS], as averaging over an amenable subgroup that generates the hyperfinite subfactor is a major step in showing that the continuous and completely bounded Hochschild coho- mology groups are equal. It has seemed reasonable that Popa's results could be extended from factors to general typeII1 von Neumann algebras by direct integral theory [KR, Chapter 4]. However, we do not know of such an at- tempt. Direct integral theory can be used directly to prove cohomology is zero and deduce results like [CPSS, Theorems 5.4 and 5.5] however these theorems on the continuous Hochschild cohomology for a von Neumann algebra with Cartan subalgebras are deduced from the theorems in this pa- per. This paper provides direct proofs of Popa's main two results in [Po 1] by modifying his proofs using an interpolation type result for projections in a maximal abelian selfadjoint sub-algebra (masa) of the typeII1 algebra.

This introduction contains a more detailed description of how our results extend Popa's, the basic definitions, and a brief reference to their use in the calculation of Hochschild cohomology groups in von Neumann algebras.

Though averaging plays an important role in calculating the Hochschild co- homology of von Neumann algebras for all types I; II₁; II₁; III of von Neumann algebras (see [Ri]), the results proved here are only used in the type II₁ situation. The reason is that the type I's are already trivially hy- perfinite, and the type II1 and III von Neumann algebras may be handled by their stability under tensoring withB…H†. This tensor factor B…H†ofM in the II1 and III cases gives a suitable hyperfinite algebra over which to average. Popa [Po 1] restricts his attention to type II …II₁ and II₁†factors.

Received February 17, 1993; in final revised form February 20, 1997.

Throughout this paper we restrict attention to typeII₁ von Neumann alge- bras. A maximal abelian selfadjoint subalgebra A of a von Neumann algebra M is a Cartan subalgebra if the von Neumann algebra generated by the unitary normaliser fu2u…M†:u A uˆAg of A in the unitary group u…M† of M is equal to M [Po 1]. A von Neumann subalgebra N of a von Neumann algebraMwith centreZis said to have trivial relative commutant ifN⁰\MˆZ:

The two results of [Po 1] that we generalise (Theorems 8 and 9) from type II1 factors to general typeII1von Neumann algebras with separable predual are the following.

Each type II₁ factor M with separable predual contains a hyperfinite sub- factor N such that N has trivial relative commutant in M:

For each Cartan subalgebra A in a type II₁factor M with separable predual there is a hyperfinite subfactor N with trivial relative commutant such that A is a Cartan subalgebra of N:

Note that the counter example in [Po 1, Section 4] shows that the hy- pothesis ``with separable predual is necessary.

The technical lemmas on the comparison of projections in a masa are stated and proved in Section 2. The proofs of these lemmas are modifica- tions of standard results on the comparison of projections in von Neumann algebras. Observe that though all the projections are in an abelian sub- algebra, the techniques are highly non-commutative as the equivalence is calculated in the whole algebra. ThroughoutM is taken to be a typeII₁ von Neumann algebra with centre Z and centre valued trace T [Ta 2]. The se- parable predual condition implies that there is a faithful normal state on Z, and ˆo T is a faithful normal tracial state on M. Let jjxjj₂ˆ…xx†¹⁼²for allx2M, and letL²…M†be the completion ofMin this norm. Since is a normal tracial state, a result of Takesaki [Ta 1] [St] im- plies that for each unital von Neumann subalgebraN ofMthere is a pre- serving conditional expectationEN fromMontoN:

Section 3 contains the statement and proof of Theorem 8. This result shows that for each type II₁ von Neumann algebra there is a hyperfinite subalgebra N with trivial relative commutant. This conclusion is the input required into the averaging arguments used in the applications of Grothen- diecks's inequality to cohomology computations (see [ES], [PS], [CPSS]).

Theorem 9, and a lemma on the construction of matrix units associated with a Cartan subalgebra, are stated and proved in Section 4. The result enables one to avoid the use of direct integral theory in the proof that H_c³…M;M† ˆ0 for a type II₁ von Neumann algebra M with Cartan sub- algebra [CPSS, Theorem 6.4]. One just needs to observe that the conclusion

of Theorem 9 can be used in place of Popa's result [Po 1] in the proof of [CPSS, Theorem 5.5].

The proofs of Theorems 8 and 9 are rather similar with one important difference which occurs in Popa's arguments [Po 1]. As the knkn matrix units are constructed at the n^th stage, they perturb the fine structure of the initial masa. In Theorem 8 this means the masa is changing from then^th to the …n‡1†^th stage of the construction. There is little control over the final masa. This difficulty is avoided in the Cartan algebra situation in Theorem 9 by ensuring that the partial isometries chosen leave the masa invariant (Lemma 10).

The authors were partially supported by a NATO collaborative research grant and the second author was partially supported by a National Science Foundation grant. The authors wish to thank the organisers of the Sympo- sium on Invariants in Operator Algebras at the Royal Danish Academy of Science and Letters in August 1992; participation in this symposium enabled the authors to discuss this research. The first author would like to thank Erik Christensen for many stimulating conversations about type II₁ von Neumann algebras. The authors wish to thank the referee for helping us correct certain errors and ambiguities.

2. Finite dimensional subalgebras of masas

This section contains the basic lemmas on constructing nice finite dimen- sional subalgebras of masas in typeII₁ von Neumann algebras. Throughout this section we shall assume thatMis a typeII₁ von Neumann algebra with separable predual and with centreZ, and that A is a masa inM. The centre valued tracial conditional expectation will be denoted byT (see [Ta 2], [Di], [SZ]). SinceMhas separable predual, so doesZ, and there is a faithful nor- mal stateon Z. Let ˆT be the resulting faithful tracial state onM, and letjjxjj₂ˆ ……xx††¹⁼²for all x2M. IfC is a von Neumann subalgebra ofM, letE_Cdenote the-preserving conditional expectation fromMontoC [Ta2 ] [ St ], and letP…C†denote the set of projections inC. Recall that the jj jj₂ topology on the closed unit sphere M1ˆ fx2M:jjxjj 1g in M is equivalent to the strong and ultrastrong topology onM1[Ta 1] [Di].

Throughout equivalence of projectionsef is equivalence within M, i.e.

there is av (a partial isometry) inM such that vvˆeand vv ˆf. Recall that the order 4on projections is defined by e4f if and only if there is a projectiong withegf:

Lemmas 1 to 4 are modifications of the standard comparison theory lem- mas for projections in von Neumann algebras but taking into account that the projections lie in the masa A. Lemma 3 provides the crucial interpolation

step that enables one to choose a projection in the masa between two other projections in a suitable way to continue the inductive construction of the increasing sequence of finite dimensional algebras.

Lemma1 (SZ, 4.5).If h…6ˆ0† 2P…A†, then there are f₁;f₂ in P…A†such that f₁h; f₂h; f₁f₂ˆ0and f₁f₂:

Proof. Choose ag2P…A†such thatgh; g6ˆ0; g6ˆhandg2= Zh. This choice ofgis possible, because hAis a masa inhMh;which is a typeII₁ von Neumann algebra so that

hA%Z…hMh† ˆthe centre of hMhhZ:

By the Comparison Theorem for projections in von Neumann algebras [SZ, Theorem 4.6] applied toM, there is ap2P…Z†such that

g p4…hÿg†pandg…1ÿp† …h< ÿg† …1ÿp†

Now either gp6ˆ0 or …hÿg†…1ÿp† 6ˆ0; because if gpˆ0 and hÿgÿhp‡gpˆ0, thengˆh…1ÿp† 2Z…hMh†contrary to the assumption gis not inZ…hMh†:

Letf₁ be eithergpor…hÿg†…1ÿp†, whichever is non zero. Let f₂gpwith f₂ …hÿg†piff₁ˆgp, or

f2 …hÿg†…1ÿp†withf2g…1ÿp†iff1ˆ …hÿg†…1ÿp†:

In the first case f1f2gp …hÿg†pˆ0 and in the second case f₁f₂ …hÿg†…1ÿp†g…1ÿp† ˆ0. Further f₁h and f₂h. (Recall in these last calculations that all our projections are in A andhgˆg:†

Lemma 2 (SZ, 4.11). If e2P…A†, then there are e₁, e₂2P…A† such that eˆe₁‡e₂, e₁e₂, e₁e₂ˆ0:

Proof. Letfˆ f…e1i;e2i†:i2Igbe a maximal family of pairs of projec- tions in Asuch thate_1ie,e_2ieand e_1ie_2i for all i2I, ande_1i:e_2j ˆ0 for all i, j2I; where I is a suitable index set. Let e₁ˆ _e_1i and e₂ˆ _e_2i. Then e1e2 and e1e2ˆ0 by standard von Neumann algebra projection theorems [SZ, Theorem 4.2]. If hˆeÿe1ÿe26ˆ0, then by Lemma 1 there exists f1;f22P…A† with 06ˆf1h, f2 h, f1f2 and f1f2ˆ0. Further f₁e_2i andf₂e_1iˆ0 for alli2I. Thus the pair…f₁;f₂†may be adjoined to the family f contradicting the maximality of f. Hence eˆe₁‡e₂ as re- quired.

The interpolation inA used is that of the following lemma. For a factor the following lemma is just the fact that given two projectionsef in the masaAand a real numberwith tr…e† tr…f†there is a projectiongin A with egf and tr…g† ˆ. If the hypothesis ``in A'' is dropped, then this fact is well known dating back to Murray and von Neumann.

Lemma 3 (SZ, 7.17). Let ef 2P…A†. If z2Z with TezTf , then there is a g2P…A†with egf and Tgˆz:

Proof. Let f be the family of all h2P…A† such that ehf and Thz, and let f have the order induced on it from the projection lattice P…A†. Let fo be a maximal totally ordered subfamily of f and let gˆ _fh:h2fog. By the total order of fo and the positivity and the ultra- weak continuity ofT; egf andTgz.

Thusg"f_o. Suppose thatzÿTg6ˆ0. By standard von Neumann algebra

techniques (the comparison theorem et al) there is an" >0 and a non-zero p2P…Z† such that …zÿT…g††p"p (if " and p do not exist, then

zÿTgˆ0†: Now …f ÿg†p "P…A†; …f ÿg†p: gˆ0 and T……f ÿg†p† ˆ

…Tf ÿTg†p …zÿTg†p"p, since p2Z. Hence …f ÿg†p6ˆ0. Let n2N

with 2^ÿn":

Repeated application of Lemma 2 to the projection …f ÿg†p in A shows that there are mutually orthogonal equivalent projections e1;e2; :::;e2ⁿ in A such that…f ÿg†pˆej:The equivalence implies that T…ej† ˆT…ek†for all j,k(by [SZ, Theorem 7.11], [Ta, Theorem 2.6]). Hence

T…e1† ˆ2^ÿnT……f ÿg†p† 2^ÿnp"p:

sinceT…f ÿg† 1. Note thate₁gˆ0 sog‡e₁2A. Further eg‡e1g‡ …fÿg†pˆg…1ÿp† ‡fp

f…1ÿp† ‡fpˆf;

T…g‡e₁† T…g† ‡"pT…g† ‡ …zÿT…g††p and

zp‡T…g†…1ÿp† z:

Thusg‡e12foand this contradicts the maximality ofginfo, sozˆTg as required.

Lemma 4. Let f 2P…A†, p2P…Z† and 0<r<m be integers. If Tf ˆrm^ÿ1p, then there are orthogonal equivalent projections e1; :::;erin A with f ˆ^r₁e_j(and Te_jˆm^ÿ1p†:

Proof. The projectionse1; :::;erare constructed inductively. By Lemma 3 there is e1f with Te1ˆm^ÿ1p. If orthogonal projections e1; :::;et…t<r†

have been constructed with Te_jˆm^ÿ1p, then T…f ÿ^t₁e_j† ˆ …rÿt†m^ÿ1p so there is a projection e_t‡1 with Te_t‡1ˆm^ÿ1p and e_t‡1f ÿ^t₁e_j. This gives orthogonal projections e1; :::;e_r with f ˆ^r₁e_j and Te_jˆm^ÿ1p; the equiva- lence ofej follows from them having equal central trace [SZ],[Ta1].

Note that although the next lemma involves approximation in a masaAin Mit is not a commutative result, because the equivalence is that inM:

Lemma5, Let B be a finite dimensional subalgebra of A such that there is an integer k with the property that if q is a minimal projection in B\Z, then there are exactly k minimal projections in qB and they are all equivalent (in M†. Let

" >0 and let g1; :::;gn be projections in A. Then there is a finite dimensional subalgebra Aoof A that contains B and has the property that there is a positive integer m so that

1. k divides m;

2. if p is a minimal projection in A_o\Z, then there are exactly m minimal projections in pAo and they are all equivalent (in M†, and

3. kgjÿEAogjk₂"for1jn:

Note that standard lemmas on approximating projections [Co] and [Ch]

imply that theyj could be chosen to be projections inAo, but we do not re- quire this subsequently so do not follow this up.

The condition on the minimal projections in the centre giving rise to equivalent minimal projections below may be written symbolically forB: ifq is a minimal projection inZ\B, then there are equivalent minimal projec- tionse1; :::;ek, in qB with e1 ej for 1jk. In the proof below no effort is made to control the values ofk. However, a little more care shows that ifk is a power of 2, thenmcan be chosen to be a power of 2.

Proof. The proof splits into a particular preliminary case with B\ZˆC1;which is the main part of the proof, and the general case, which is just a finite sum of the particular ones. For the first part of the proof (most of it) assume thatB\ZˆC1 and thatBis spanned by its equivalent minimal projections uj…1jk†; they are equivalent by hypothesis. Thus the centre valued traceT hasTujˆk^ÿ1:1;sinceTuj is a multiple of 1:

LetC₀ be the finite dimensional *-subalgebra of Agenerated by g₁; :::;g_n andB:Letm₀be a positive integer such thatm₀2^n‡1"^ÿ2and letmˆm₀k:

IfF is a finite set of projections, letNspan…F†denote the additive semi- group generated by F[ f0g; and if ris a positive integer, let r^ÿ1Nspan…F† denote this semigroup times r^ÿ1: The abelian von Neumann algebras with separable preduals are isomorphic toL¹‰0;1Š; `<sup>1</sup>…N†; `¹…r†forr2N, or to the direct sum of the first with one of the other two [Ta1, p.112]. Hence the real linear span of the set of projections inZis dense in the self-adjoint part ofZin the norm topology, and the setfxˆx2Z:x0gis covered by the open sets

fxˆx2Z:x0;jjxÿyjj<m^ÿ1 for some y2m^ÿ1Nspan…F†g asFruns over all finite sets of pairwise orthogonal projections inZthat add to 1:The set fw2TC0:0w1gis compact in the norm topology, since C0is finite dimensional. Hence there is a finite setF0of projections inZsuch

that for each w2TC₀ with 0w1 there is a v in m^ÿ1Nspan…F₀† with jjwÿvjj<m^ÿ1: If the projections in F₀ are multiplied by the minimal pro- jections inTC0;we obtain a finite setF of orthogonal projections inZadd- ing to 1 with the properties that

(1) if w2TC₀ with 0w1; then there is a v in m^ÿ1Nspan…F† with vÿw

j j

j j<m^ÿ1;and

(2) each projection inTC₀ is a sum of projections inF:

LetC be the finite dimensional subalgebra ofAgenerated by C₀ and F: Observe that each central projection inTC₀ is a sum of orthogonal projec- tions inF;soF is the set of minimal projections inZ\C:Fix a projectionp in F and a minimal projectionu…ˆuj†in B: Sincepu is a projection in the commutative finite dimensional algebra pC; there are minimal projections f₁; :::;f_{`</sub> in pC so thatpuˆP<sub>`}

1f_j: The dependence of ` and f<sub>j</sub> on u and p is omitted here to simplify notation. Note that pu is not zero, since T…pu† ˆpT…u† ˆk<sup>ÿ1</sup>p: Note that the number of minimal projections in C that add to each minimal central projection inC(i.e.pinF†is no more than the corresonding maximal number adding to a central projection in C0; be- cause C is generated by C<sub>0</sub> and F Z: This is because multiplying by a central projection does not increase this number. The projectionsg<sub>j</sub> may not be inZso each pair…gj;1ÿgj†can split a minimal projection in two. Hence the number of minimal projections in C below pu is no more than 2<sup>n</sup>; be- causeuis minimal inBandpis minimal central inC:Thus`2ⁿ:

Now T…pu† ˆpT…u† ˆk^ÿ1p: Also puf_jˆpf_j ˆf_j implies that 0f_j pu1 so that

0Tf_jˆpTf_jT…pu† ˆk^ÿ1p

forjˆ1; :::; `:Thus Tfj can be approximated in norm by a non-negative ra- tional multiple ofpwith denominatormby (1), the choice ofF andF0;and the minimality ofpin C\Z:Hence there is an integerrj with 0rjmso thatTf_jÿm^ÿ1r_jp<m^ÿ1:As Tf_j0;this is equivalent to

0m^ÿ1maxf0;…r_jÿ1†gpTf_jm^ÿ1…r_j‡1†p:

IfTf_jˆm^ÿ1r_jp;choosee_jˆf_j:IfTf_j6ˆm^ÿ1r_jp, then there is a projectione_j in A with ejfj and Tejˆm^ÿ1maxf0;…rjÿ1†gp by Lemma 3. Thus 0T…fjÿej† 2m^ÿ1pfor eachj:Lete0 ˆpuÿP_`

1ej:Note thate0; :::;e` are orthogonal projections, because f1; :::;f` are orthogonal projections with e_j f_j…1j`†;and that any of thee_j could be zero. The central trace of e₀ satisfies

Te0ˆk^ÿ1pÿX^`

1

T…ej† ˆr0m^ÿ1p

for some non-negative integerr0; sinceT…pu† ˆk^ÿ1pˆm0m^ÿ1p:By Lemma 4 each projectionej…0j`† may be subdivided into orthogonal equiva- lent projectionse_ji with T…e_ji† ˆm^ÿ1p for allj and i. The projections e_ji are orthogonal equivalent and add up topu:There arem₀of these projectionse_ji for eachpusinceT…pu† ˆk^ÿ1pˆm₀m^ÿ1p:

Note that pˆpu1‡:::‡puk; so that there are mˆkm0 orthogonal equivalent projectionse^p;1_ji ; :::;e^p;k_ji that add to p; denote these projections by h^p_s for 1sm:

LetA₀ be the linear span of fh^p_s :p2F;1smg so that A₀ is a com- mutative finite dimensional-subalgebra ofA:For eachpuabove, eache_j is inA0by construction ofh^p_s. Hence for eachujinB…1jk†and eachpin F;puj is inA0 as it is a sum ofei by construction of e0:Henceu is inA0 as uˆP

fpu:p2Fg:When constructedF was observed to be the set of mini- mal projections inC\Zand eachpinF was shown to be the sum ofmor- thogonal equivalent projectionsh^p_s…1sm†:The equivalence of h^p_s andh^p_t for 1s; tmimply that a sumP

h^p_j for fixedpover somejis in the cen- treZif and only if it is over all 1jm:This implies that F is the set of minimal projections inA0\Z and that eachpin F is the sum ofmequiva- lent (orthogonal) minimal projections inpA₀:This proves properties 1 and 2.

We now turn to proving property 3. Letgbe a projection inC:Sincegis a sum of minimal projections inC;there is a subsetW of

f…p;f_j^p†:p2F; f_j^p is a minimal projection inpCg;

such that

gˆX

ff_j^p:…p;f_j^p† 2Wg:

LethˆP

fe^p_j :…p;f_j^p† 2Wg where e^p_j are the projections constructed corre- sponding to the projectionsf_j^p:Note that the e^p_j are not minimal projections in A0 but are each sums of equivalent minimal projections in A0: By con- structione^p_j f_j^p for alljandp;sohis a projection in A0 withhg:Hence

jjgÿhjj²₂ˆ…gÿh† ˆX

…p…f_j^pÿe^p_j†† ˆX

…pT…f_j^pÿe^p_j††

by the properties of T and ˆT with the above sums over …p;f_j^p† 2W: For each j; 0pT…f_j^pÿe^p_j† 2m^ÿ1p and for each p there are at most 2ⁿk such elements because this is the maximal number of minimal projections in Cbelow a minimalpinC\Z:Hence using the minimality properties of the fact that E_A0 is the orthogonal projection onto A in the Hilbert space …M;jj jj₂†

jjgÿE_A0gjj²₂ jjgÿhjj²₂X

…p† 2 m^ÿ1 2ⁿk

where the sum is overp in F.

Thus

jjgÿE_A0gjj²₂m^ÿ1 2^n‡1kˆm^ÿ1₀ 2^n‡1"² by the choice ofm₀andmˆm₀k:

This proves the case whereB\ZˆC1:

In general B\Z is a finite dimensional commutative C-algebra with minimal projections q1; :::;qN:Cutting the projections g1; :::;gn by a central projection q_t gives n projections g₁q_t; :::;g_nq_t in the algebra Aq_t with Bq_t\Zq_t ˆCq_t andq_t the identity of the von Neumann algebraq_tM:Since

…1† ˆ1ˆP

…q_t†;the approximations obtained in the various factors add to the whole algebra providedis replaced onMqt by…qt†^ÿ1…†:Properties 1 and 2 follows directly as they concern elements below minimal projections in B\Z: Property 3 follows because of orthogonality ofjj jj₂-norm on the direct sum MˆL

Mq_t and the choice of state …q_t†^ÿ1…† on Mq_t: This completes the proof.

The following lemma is just the inductive version of Lemma 5; however, in the form given below it fits in better with the Popa's characterization of masa's in von Neumann algebras [Po1, Lemma 1.2].

Lemma6. Let A_o be a finite dimensional subalgebra of A so that there is a positive integer kowith the property that if p is a minimal projection in Ao\Z, then there are exactly kominimal projections in PAoand they are all equivalent (in M†. Then there is a sequence An of finite dimensional subalgebras of A for n2Nsuch that

1. A_nÿ1A_nfor n1;

2. knÿ1 divides kn for n1;

3. if q is a minimal projection in An\Z, then there are exactly kn minimal projections in q Anand they are all equivalent, and

4. the weak closure of the union of the A_nis A, i.e.…[A_n†^ÿwˆA:

Proof. The von Neumann algebraAhas a separable predual, sinceMhas a separable predual, so there is a sequence fg_n:n2Ng of projections in A such that the linear span offg_n:n2Ng is weakly dense inA.

By induction we choose an increasing sequence A_n of finite dimensional subalgebras of A satisfying (1), (2) and (3) of our conclusions above, and such that

5. jjg_jÿE_Ang_jjj₂2^ÿn

for 1jn. Observe that Lemma 5 gives the step from nÿ1 to n in the induction. Property (4) follows from inequality (5).

The next lemma is used in the proofs of Theorems 8 and 9 to show that the hyperfinite algebras constructed there have trivial relative commutants.

Lemma 7. LetC1ˆN_oN₂::: be an increasing sequence of finite di- mensional *-subalgebras of M such that the centre of Nnis equal to Nn\Z for all n. Let N be the weak closure of the union of Nn,i.e. Nˆ …[Nn†^ÿw. If N contains Z, then N\N⁰ˆZ.

Proof. Let xˆx2N\N⁰ with jjxjj 1. The strong topology and the jj:jj₂-topology coincide on the closed unit ballM₁ ofM. Hence, by the Ka- plansky density theorem [Ta 1], there is a n2N and y2N_n such that jjyjj 1 andjjxÿyjj₂< ". Sincex2N⁰N_n⁰; jjyuÿuyjj₂<2"for allu2N_n withjjujj 1. Thusjjyÿuyujj₂2 "for alluin the unitary groupu…Nn†of Nn. Let denote normalised Haar measure on the (compact) unitary group u…Nn†. Let zˆR

…yÿuyu†d…u†, with integration over the whole of the

unitary group u…N_n†. Then jjzjj₂2" and w…yÿz†wˆR

wuyuwd…u† ˆ

Rvyvd…v† ˆyÿz for all w2u…N_n†: Hence yÿz is in the centre of N_n,

which is equal to Z\Nn. Further jjxÿ …yz†jj₂3"; this shows thatx is in jj jj₂closure ofZ, so is inZ:

3. Injective subalgebras with trivial relative commutant

Theorem8. Let M be a type II1 von Neumann algebra with separable predual and centre Z. Then there is a masa A in M and an injective von Neumann subalgebra N of M containing A such that A is a Cartan subalgebra of N and N has trivial relative commutant in M:

Proof. Letfx_j:j2N[ f0ggbe a subsequence of the closed unit ballM₁ ofMthat is dense in this unit ball in the jj jj₂ norm; such a sequence exists by the separability of the predual ofM and the equivalence of thejj jj₂-to- pology and the strong topology onM1. We shall assume xoˆ1. By induc- tion onnwe shall construct sequences

A_oˆC1A₁A₂:::

of finite dimensional abelian subalgebras ofM;

NoˆC1N1N2:::

of finite dimensional *-subalgebras ofN, and kok1 k2:::

of positive integers such that 1. A_nN_n;

2. the centre ofNnisNn\Z, which is contained in A, 3. knÿ1 divideskn;

4. for each minimal projection q in Z\A_n, there are exactly k_n minimal projections inqA_n and they are all equivalent,

5. for each minimal projectionqin Z\An, the algebraqNn is isomorphic to Mkn…C† and has matrix units e^q_ij so that e^q_ii…1ikn† are the minimal projections ofqA_n;

6. for each 1jnÿ1 and each minimal projection p in A_j\Z, the natural unitary inN_j that interchangese^p₁₁ ande^p_iinormalisesA_n;

and

7. jj…EA⁰_n\MÿEAn†…xi†jj₂2^ÿn for 1in:

The induction starts withAoˆNoˆC1 andkoˆ1. Suppose thatAn,Nn

and kn have been constructed as above. Let B be a maximal abelian sub- algebra ofMcontainingAn. By Lemma 2.6 withAothere equal toAnchoose an increasing sequenceB_` of finite dimensional subalgebras ofBcontaining A_n with the properties of that lemma. As the algebra Bis a masa inM and Bˆ …[B†^ÿw;[Po 1, Lemma 1.2] implies that

jjE_B⁰

`\M…x† ÿE<sub>B`</sub>…x†jj₂!0

as `! 1 for all x2M. For each minimal projection p in Z\An, let e<sup>p</sup><sub>ij</sub>…1i;jk<sub>n</sub>†be the matrix units in the algebrapN<sub>n</sub>. Let<sub>n</sub>be the number of minimal projections inZ\A<sub>n</sub>. Choose`so large that

8. jj…E_B⁰

`\MÿE<sub>B`</sub>†…e^p_ijx_ie^p_ji†jj₂<2^ÿ…n‡1†ÿ_n¹²k^ÿ_n¹²

for 1in‡1, 1jk_n and all minimal projections p in Z\A_n. Let Q_n‡1 be the set of all minimal projections in this B_{`</sub>\Z. Now for each q2Qn‡1 the projectionqe<sup>p</sup><sub>11</sub>is a sum ofmnˆkn‡1=knminimal projections in qB` by Lemma 6, where kn‡1 is the integer corresponding toB` of that lem- ma. Number these m<sub>n</sub> minimal projections inqB<sub>`} as e^q_ii for I im_n. For eachq, the projectionse^q_ii…1im_n†are equivalent inM so form the diag- onal of a set of matrix unitse^q_ij…1i,jm_n†inM:

Let An‡1 be the linear subspace of M spanned by the set fe^p_j1e^q_tte^p_1j :1tmn, 1jkn, q2Qn‡1, p the minimal projection in Z\A_n withqˆqpg:

LetN_n‡1 be the linear subspace ofMspanned by the set

fe^p_i1e^q_ste^p_1j:1s;tm_n, 1i;jk_n,q2Q_n‡1;pthe minimal projection in Z\An withqˆqpg:

Then An‡1 is an abelian subalgebra of M with minimal projections ele- ments of its spanning set,AnAn‡1, andAn‡1Nn‡1. FurtherNn‡1 is a fi- nite dimensional subalgebra ofM which is a direct sum overq2Q_n‡1 of al- gebras each of which is isomorphic toM_kn‡1…C† with matrix units e^p_i1e^q_ste^p_1j , where 1s;tmn, 1i;jkn and p is the unique minimal projection in Z\Ansuch thatqˆqp. This shows that the centre ofNn‡1is the linear span

ofQ_n‡1so equalsN_n‡1\Z. Let 1snÿ1 and letrbe a minimal projec- tion inZ\A_s. Letube the natural unitary inN_sthat interchangese^r₁₁ande^r_ii in Ns. Then u normalises An by (6), and so normalises the minimal projec- tions e^p_j1e^q_tte^p_1j spanning An‡1. If u is the natural unitary in Nn that inter- changes e^p₁₁ and e^p_ii, then u normalises An‡1 by definition of the minimal projections inA_n‡1. We have checked conditions (1) to (6) of the induction, and only condition (7) remains to be proved from (8).

Firstly observe that for allx2M;

jj…E_A⁰

n‡1\MÿE_An‡1†…e^p_jjx e^p_jj†jj₂ˆ jj…E_A⁰

n‡1\MÿE_An‡1†…e^p_1jx e^p_j1†jj₂

because the mape^p_jjx e^p_jj!e^p_1jx e^p_j1 is an isometric isomorphism frome^p_jjM e^p_jj ontoe^p_1jM e^p_j1 that carriese^p_jjA⁰_n‡1\M e^p_jj onto e^p_1jA⁰_n‡1\M e^p_j1 and e^p_jjA_n‡1e^p_jj ontoe^p_1jAn‡1e^p_1j:

Now for each 1in‡1;

9. jj…E_A⁰

n‡1\MÿEAn‡1†…xi†jj²₂ ˆjje^p_jj…E_A⁰

n‡1\MÿEAn‡1†…xi†e^p_jjjj²₂;

where the summation is over 1jkn and all minimal projections p in An\Z, and equality holds because of the orthorgonality of e^p_jjM e^p_jj for dif- ferentjandp, ande^p_jjbeing in the commutants of the two algebrasA⁰_n‡1\M andA_n‡1. Furthere^p_jj is in these two algebras so (9) equals

jj…E_A⁰

n‡1\MÿE_An‡1†…e^p_jjx_ie^p_jj†jj²₂ˆjj…E_A⁰

n‡1\MÿE_An‡1†…e^p_1jpx_ie^p_j1†jj²₂ by the note above on the isometries. Now e^p₁₁A⁰_n‡1\M e^p₁₁ˆe^p₁₁B⁰<sub>`</sub>\M e<sup>p</sup><sub>11</sub> and e<sup>p</sup><sub>11</sub>An‡1e<sup>p</sup><sub>11</sub>ˆe<sup>p</sup><sub>11</sub>B`e^p₁₁ by the definition of An‡1 in terms of B` and its minimal projections. Thus (9) equals

jj…E_B⁰

`\MÿEB`†…e^p_1jxie^p_j1†jj²₂4^{ÿ…n‡1†ÿ1}_n k^ÿ1_n ˆ4^ÿ…n‡1†

as the summation extends over _n minimal projections p in Z\A_n and k_n subscripts since jje^p_1jye^p_j1jj₂ˆ jje^p_jjye^p_jjjj₂ for all p and j. This finishes the in- ductive construction.

Let Nˆ …[N_n†^ÿw and Aˆ …[A_n†^ÿw, where ^ÿw denotes weak closure.

Clearly A is self-adjoint abelian, N is injective and AN. By (7) and the jj jj₂-density of fx_i:i0g in the closed unit ball M₁, it follows that jj…E_A⁰_n\MÿEAn†…x†jj₂ tends to zero for all x2M. Hence A is a masa by [Po 1, Lemma 1.2]. NowN⁰\MA⁰\MˆA, sinceAis a masa in M, so N⁰\MN⁰\AN⁰\N, which equals Z by (2) and Lemma 7. This proves Theorem 8.

4. Injectives containing Cartan subalgebras

Theorem9. Let M be a type II₁von Neumann algebra with separable predual and let A be a Cartan subalgebra of M. Then there is an injective von Neumann subalgebra N of M with relative commutant the centre Z of M such that A is a Cartan subalgebra of N:

The hypothesis thatAis a Cartan subalgebra of a typeII₁ von Neumann algebraMapplies in the following discussion and definitions and in Lemma 10.

Letn…A†be the normalizer ofAin the unitary group ofM, sou2n…A†

if and only ifu A uˆA. Let

gˆ fv2M:vˆup;u2n…A†;p2P…A†g;

so elements ofgare partial isometrics withvvˆp2Aandvvˆupu2A.

Observe that g is a semigroup in M as vˆup and tˆwq implies that vtˆupwqˆuwwpwq with uw2n…A† and wpwq a projection in A since A is abelian and wpw2A. Further gˆg as vˆup implies v ˆuupu:

Lemma 10. Let f₁; :::;f_k be a set of orthogonal equivalent projections in A.

Then there are matrix units f_ijingsuch that f_iiˆf_ifor1ik:

Proof. The first stage is to show that ife₁ and e₂ are equivalent ortho- gonal projections in A then there is av2gwith vvˆe₁ andvvˆe₂. The construction ofvis done by a standard maximal trick; this part of the proof is exactly as in the first part of the proof of [Po 1, Prop 3.6], which is re- peated here for the readers convenience.

Let fˆ f…p;w†:p2P…A†, w2g, pe1, wwˆp and wwe2g: Note f is not empty since …0;0† 2f. Define a partial order on f by …p₁;w₁† …p₂;w₂†if and only ifp₁p₂ andw₁ˆw₂p₁. By Zorn's Lemma we obtain a maximal totally ordered subfamilyfooff(which has a countable cofinal subfamily since M has separable predual), and let…p;v† 2fbe the maximal element of fo. Suppose p6ˆe1. Let p1ˆe1ÿpˆe1ÿvv and q₁ˆe₂ÿvv. Note that p₁ and q₁ are equivalent projections - this can be seen easily here using the central traceT asT…p₁† ˆT…q₁†:

If p1uq1uˆ0 for each unitary u2n…A†, then the projection gˆ _fuq1u:u2n…A†g satisfies p1gˆ0; q1gˆq1 and g2P…Z†, because n…A† generates M as a von Neumann algebra. Now T…p1† ˆT…q1† ˆ T…q₁†gˆT…p₁†gˆT…p₁g† ˆ0 contrary top₁6ˆ0, where the first and third equalities hold because p₁q₁. This contradiction implies that p₁uq₁uˆp_o6ˆ0 for some u2n…A†. Now p_op₁ and up₁2g by con- struction ofg. Ifvˆwpwithw2n…A†(recall…p;v† 2f†, thenv‡upo2g

because p‡p_o2P…A†…pp_oˆ0†and there is a unitary u₁ 2n…A† such that u₁…p‡p_o† ˆv‡up_o. Hence…p‡p_o,v‡up_o† 2fand is strictly greater than …p;v†, contradicting the maximality of the element …p;v†. Hence v2g sa- tisfiesvvˆe1…ˆp†andvvˆe2:

Using the above choose f1j…1jk† in g such that f1jf1j ˆf1 and f_1jf_1jˆf_j for 1jk. Definef_ijˆf_1if_1j for 1i;jk. Thesef_ij are the re- quired matrix units ing:

Proof of Theorem9. Let fa_n:n2Ng be a sequence of projections in A whose linear span is dense in Ain the strong topology. Observe thatAhas such a sequence, because M and hence A has separable predual. Further note that as is a faithful normal tracial state onM, density in thejj jj₂ - topology implies strong density in the unit ball ofA.

By induction we construct an increasing sequence N_oˆC1CN₁ CN₂:::of finite dimensional *-subalgebras ofMwith matrix unitse^q_ij in each of the matrix direct summands satisfying

1. the centre ofNnis equal toNn\Z;

2. if Q…n† denotes the set of minimal idempotents in Nn\Z, then the matrix unitse^q_ij…1i;jk…q††constructed inq Nnare in gfor allq2Q…n†;

3. if A_n is the finite dimensional subalgebra generated by the set fe^q_ii:1ik…q†,q2Q…n†g;thenA_nA, and

4. jjajÿEAn…aj†jj₂2^ÿn for 1jn:

Recall thatE_An denotes the preserving conditional expectation from M onto An. Suppose that Nn has been constructed. If vˆue is in g with u2n…A†andea projection in A, thenv x vˆu e x e uˆu e x u is in A for allx2A. Thus the finite set

F ˆ fqe^q_1iate^q_i1:1tn‡1;1ik…q†g

is contained in qe^q₁₁A e^q₁₁, where all sums over q run over Q…n†. Now by Lemma 5 there is a finite dimensional subalgebra A_o of A and k2N such thatA_n A_o,

5. jj…IÿEAo†…e^q_1iate^q_i1†jj₂…q†¹²k^ÿ1_n 2^ÿ…n‡1† for allq2Q…n†, 1ikn and 1tn‡1, and

6. if Zn‡1ˆAo\Z, then for each minimal projection p in Zn‡1, the minimal projectionsf₁^p, ...,f_k^p inpA_o are all equivalent.

Let Q…n‡1† be the set of minimal projections in Zn‡1. For each p2Q…n‡1†and each q2Q…n†withpqˆp, the projectionp e^q₁₁is a sum of a finite number of f_i^p, which are the minimal projections in pA_o: By re- numbering the projections f_i^p (if necessary) we will assume that p e^q₁₁ˆ^k₁^0…p†f_i^p for some ko…p† and all pqˆp. As the projections

f_i^p…1ik† in A are orthogonal and equivalent, there is a set of matrix unitsf_ij^pingso thatf_ii^p ˆf_i^pfor 1ikby Lemma 10. The matrix unitse^p_`m are defined by a tensor operation usingf_ij^p,e^q_s1,e^q_1t forpqˆpand alli;j;s;t: 7. Let e^p_i;j;s;tˆe^q_s1f_ij^pe^q_1t for pˆqp…p2Q…n‡1†, q2Q…n††; 1i;jko…p†

and 1s;tk…q†:

LetNn‡1 be the linear span of alle^p_i;j;s;t defined in (7). Note thatNn‡1 is a

*-subalgebra ofMand that for eachp2Q…n‡1†,e^p_i;j;s;t are the matrix units of p N_n‡1 corresponding to the tensor product M_ko…p†…C† M_k…q†…C† with 1i;jk_o…p† and 1s;tk…q†. Clearly the centre of N_n‡1 is the linear span of Q…n‡1† so is Zn‡1. By construction e^q_i;j;s;t is in g, because g is a semigroup under multiplication. The matrix element e^p_i;i;s;s is of the form uefeu with u2n…A† and e, f 2P…A†; because e^q_ii2P…A†and e^q_1sˆ …e^q_s;1†; so is inA. LetA_n‡1 be the linear span ofe^p_i;i;s;sover all 1ik_n, 1sk, pˆpq, p2Q…n‡1† and q2Q…n†. Let k…p† ˆk_o…p†k…q†. Lastly we check inequality (4) as (1), (2) and (3) have been done for n‡1. Observe that e^q₁₁Aoe^q₁₁An‡1 for all q2Q…n† as the minimal projections in e^q₁₁Aoe^q₁₁ are the projections e^p_i;i;1;1 for 1iko…p† and pqˆp with p2Q…n‡1†. Now note that for eachq2Q…n†;

8. jj…IÿE_An‡1†…e^q_iix e^q_ii†jj₂ˆ jj…IÿE_Ao†…e^q_1ix e^q_i1†jj₂;

because dropping theq's, the mapL_ei1R_e1i…Lˆleft multiplication,Rˆright multiplication) frome_1iA e_i1 ontoe_iiA e_ii is an isometry since

jjL_ei1R_e1i…e_1ixe_i1†jj²₂ ˆ…e_i1e_1ixe_i1e_1ie_i1e_1ixe_i1e_1i† ˆ jje_iixe_iijj²₂:

Nowe^q_iie^r_jj ˆ0 ifq6ˆr2Q…n†, or qˆrandi6ˆj. The module property of the conditional expectation mapE_An‡1 with respect toe^q_ii2A_n‡1 implies that the spaces …IÿEAn‡1†…e^q_iiA e^q_ii† are orthogonal in jj jj₂ norm for different pairs…q;i†. Hence

jj…IÿEAn‡1†…at†jj²₂ˆjj…IÿEAn‡1†…e^q_iiate^q_ii†jj²₂;

with the sum running over allq2Q…n†and all 1ik_n. Equation (8) im- plies that

jj…IÿEAn‡1†…at†jj²₂

ˆ jj…IÿE_Ao†…e^q_1ia_te^q_i1†jj²₂ …q†k^ÿ1_n :4^ÿ…n‡1†

for 1tn‡1 by…5†

ˆ 4^ÿ…n‡1†

on recalling thatfq:q2Q…n†g ˆ1 and that 1ik_n:

The inductive construction ofZn‡1,An‡1,Nn‡1is done with properties (1)^

(4). LetNˆ …[Nn†^ÿw andBˆ …[An†^ÿw, where ^ÿwdenotes the weak closure.

ThenBN,BA, and by (4),BˆAsince thejj jj₂density of the unit ball ofBin that ofAimplies the weak density.

Now N⁰\MA⁰\MˆA, since A is maximal abelian in M; thus N⁰\MˆN⁰\AˆN⁰\N. By (1) of the inductive construction, the se- quence Nn satisfies the hypotheses of Lemma 7 so that N⁰\NˆZ and N has trivial relative commutant inMas required.

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