aa r X i v : . [ m a t h . OA ] A p r SINGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT
KUNAL MUKHERJEE
Abstract.
In this paper we study relations between the left-right-measure and properties of singularmasas. Part of the analysis is mainly concerned with masas for which the left-right-measure is theclass of product measure. We provide examples of Tauer masas in the hyperfinite II factor whose left-right-measure is the class of Lebesgue measure. We show that for each subset S ⊆ N , there existuncountably many pairwise non conjugate singular masas in the free group factors with Puk´anszkyinvariant S ∪ {∞} . Introduction and Preliminaries
Throughout the entire paper, M will denote a separable II factor equipped with its faithful normaltracial state τ . This trace gives rise to a Hilbert norm on M , given by k x k = τ ( x ∗ x ) , x ∈ M . TheHilbert space completion of M with respect to k·k is denoted by L ( M ). Let M act on L ( M ) vialeft multiplication. Let A ⊂ M be a maximal abelian self-adjoint subalgebra (masa). Dixmier in [5]defined the group of normalizing unitaries (or normalizer ) of A to be the set N ( A ) = { u ∈ U ( M ) : uAu ∗ = A } , where U ( M ) denotes the unitary group of M . He called( i ) A to be regular (also Cartan ) if N ( A ) ′′ = M ,( ii ) A to be semiregular if N ( A ) ′′ is a subfactor of M ,( iii ) A to be singular if N ( A ) ⊂ A .Two masas A, B of M are said to be conjugate , if there is an automorphism θ of M such that θ ( A ) = B . If there is an unitary u ∈ M such that uAu ∗ = B , then A and B are called unitarily ( inner ) conjugate . One of the most fundamental problem regarding masas is to decide the conjugacyof two masas. The most successful invariant so far in this regard is the Puk´anszky invariant [22].Nevertheless, it is not a complete invariant.The measure-multiplicity invariant of masas in II factors was studied in [8, 12, 14]. It was usedin [8] to distinguish two masas with the same Puk´anszky invariant . It is a stronger invariant thanthe
Puk´anszky invariant . It has two main components, a measure class ( left-right-measure ) anda multiplicity function, which together encode the structure of the standard Hilbert space as anassociated bimodule. In this paper, we study analytical relations between the left-right-measure andproperties of singular masas. We focus on the following question: To what extent does the standardHilbert space as a natural bimodule remember properties of the masa. In [12], we established that left-right-measure has all information to measure the size of N ( A ) (see Thm. 5.5 [12]).In this paper, we consider different kinds of singular masas. We introduce a condition on masaswhich forces vigorous mixing properties. Such masas are automatically strongly mixing [9] (for a proofsee Thm. 9.2 [1]) and consequently singular. We show that if A is such a masa in M with singletonmultiplicity, then the Hilbert space L ( M ) ⊖ L ( A ) as a natural A, A -bimodule is a direct sum ofcopies of L ( A ) ⊗ L ( A ), i.e., we show that its left-right-measure is the class of product measure. Wealso present a converse to the foresaid statement. The arguments required to prove this statementshow that, if B ⊂ A is diffuse, then L ( M ) ⊖ L ( B ′ ∩ M ) as a B, B -bimodule is a direct sum ofcopies of submodules of L ( B ) ⊗ L ( B ). There is an abundance of such masas in the hyperfinite II factor, but there are fewer examples of such masas, if in addition we demand that such a masa hasa bicyclic vector. We also study the left-right-measure of Γ and non-Γ singular masas. In particular,we show that under certain extra assumption on central sequences, the presence of central sequencesin a masa can be related to rigid measures . Examples of such masas come from Ergodic theory. Date : 30 April 2010.
Key words and phrases. von Neumann algebra; masa; measure-multiplicity invariant.
The following question asked by Banach is a long standing open problem in Ergodic theory. Doesthere exist a simple measure preserving (m.p.) automorphism with pure Lebesgue spectrum? It isimplicit in the above question that one is asking about an action of Z . Translated to operator algebrasthis means (see for instance [14]), whether there is a way to construct the hyperfinite II factor as L ∞ ( X, µ ) ⋊ Z , where L ( Z ) is a simple masa whose left-right-measure is the class of product measure.The term ‘simple’ of course means simple multiplicity or equivalently, the existence of a bicyclicvector of A .We provide an example of such a Tauer masa in the hyperfinite II factor. All Tauer masas aresimple [32]. We do not know if this example arises from an action of integers or any other groupaction. But quite surprisingly Banach’s problem has an easy and affirmative answer if we change thegroup. Using the methods developed in [8], we show that for each subset S ⊆ N (could be empty),there are uncountably many pairwise non conjugate singular masas in the free group factors with Puk´anszky invariant S ∪ {∞} .This paper is organized as follows. We provide the background material in this section itself. In §
2, we study masas for which the left-right-measure is the class of product measure. § § left-right-measure of Γ and non-Γ masas. In §
5, we exhibit examples of singular masas in free group factors.Let J denote the Tomita’s modular operator on L ( M ), obtained by extending the densely definedmap J : M 7→ M by J x = x ∗ . The image of a L vector ζ under J will be denoted by ζ ∗ . Let e A : L ( M ) L ( A ) be the Jones projection associated to A . Denote A = ( A ∪ J AJ ) ′′ . It isknown that e A ∈ A (Thm. 3.1 [21]). Let E A denote the unique, normal, trace preserving conditionalexpectation from M on to A . The conditional expectation E A and the trace extends to L ( M ) in acontinuous fashion (see § B.5 [27]). With abuse of notation, we will write e A ( ζ ) = E A ( ζ ) for L and L vectors. Similarly, we will use the same symbol τ to denote its extension. This will be clear fromthe context and will cause no confusion. This work relies on direct integrals. For standard resultson direct integrals we refer the reader to [6]. Throughout the entire paper N ∞ will denote the set N ∪ {∞} . For a set X , we will write ∆( X ) to denote the diagonal of X × X . Definition 1.1.
Given a type I von Neumann algebra B , we shall write Type( B ) for the set of allthose n ∈ N ∞ such that B has a nonzero component of type I n . Definition 1.2. [22] The
Puk´anszky invariant of A ⊂ M , denoted by P uk ( A ) (or P uk M ( A ) whenthe containing factor is ambiguous) is Type( A ′ (1 − e A )). Definition 1.3. [8, 12, 14] The measure-multiplicity invariant of A ⊂ M , denoted by m.m ( A ), isthe equivalence class of quadruples ( X, λ X , [ η | ∆( X ) c ] , m | ∆( X ) c ) under the equivalence relation ∼ m.m ,where,( i ) X is a compact Hausdorff space such that C ( X ) is an unital, norm separable, w.o.t dense subal-gebra of A ,( ii ) λ X is the Borel probability measure obtained by restricting the trace τ on C ( X ),and,( iii ) η | ∆( X ) c is the measure on X × X concentrated on ∆( X ) c , and( iv ) m | ∆( X ) c is the multiplicity function restricted to ∆( X ) c ,obtained from the direct integral decomposition of L ( M ) ⊖ L ( A ), so that A (1 − e A ) is the al-gebra of diagonalizable operators with respect to this decomposition, the equivalence ∼ m.m being,( X, λ X , [ η | ∆( X ) c ] , m | ∆( X ) c ) ∼ m.m ( Y, λ Y , [ η | ∆( Y ) c ] , m | ∆( Y ) c ), if and only if, there exists a Borel isomor-phism F : X Y such that, F ∗ λ X = λ Y , ( F × F ) ∗ [ η | ∆( X ) c ] = [ η | ∆( Y ) c ] ,m | ∆( X ) c ◦ ( F × F ) − = m | ∆( Y ) c , η | ∆( Y ) c a.e.It is easy to see that the Puk´anszky invariant of A ⊂ M is the set of essential values of themultiplicity function in Defn. 1.3. The measure class [ η | ∆( X ) c ] in Defn. 1.3 is said to be the left-right-measure of A . Both m.m ( · ) and P uk ( · ) are invariants of the masa under automorphisms of the factor M . Given a pair of masas, the mixed Puk´anszky invariant was introduced in [33]. Analogous to the mixed Puk´anszky invariant , one can define the joint-measure-multiplicity invariant for pair of masas A and B , by considering the direct integral decomposition of L ( M ) with respect to ( A ∪ J BJ ) ′′ . INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 3
For details check Ch. V [11]. Such invariants play a role in questions concerning unitary conjugacyof masas.In some cases, it is necessary to have a direct integral decomposition of L ( M ). These are situationswhen one considers tensors of masas. In these cases, the information on the diagonal ∆( X ) is to besupplied. Following § L ( M ) ∼ = Z ⊕ X × X H t,s d ( η | ∆( X ) c + ˜∆ ∗ λ X )( t, s ) , where ˜∆ : X X × X by t ( t, t ), H t,t = C for λ X almost all t , H t,s depends on the Puk´anszkyinvariant and A is diagonalizable with respect to this decomposition. In such cases, we will also call[ η | ∆( X ) c + ˜∆ ∗ λ X ] to be the left-right-measure of A . It is to be understood that, when we considerdirect integrals or make statements about diagonalizability, we need to complete the measures underconsideration.For a masa A ⊂ M , fix a compact Hausdorff space X such that C ( X ) ⊂ A is an unital, normseparable and w.o.t dense C ∗ subalgebra. Let λ denote the tracial measure on X . For ζ , ζ ∈ L ( M ),let κ ζ ,ζ : C ( X ) ⊗ C ( X ) C be the linear functional defined by, κ ζ ,ζ ( a ⊗ b ) = h aζ b, ζ i , a, b ∈ C ( X ) . Then κ ζ ,ζ induces an unique complex Radon measure η ζ ,ζ on X × X given by, κ ζ ,ζ ( a ⊗ b ) = Z X × X a ( t ) b ( s ) dη ζ ,ζ ( t, s ) , (1.1)and k η ζ ,ζ k t.v = k κ ζ ,ζ k , where k·k t.v denotes the total variation norm of measures.We will write η ζ,ζ = η ζ . Note that η ζ is a positive measure for all ζ ∈ L ( M ). It is easy to seethat the following polarization type identity holds:4 η ζ ,ζ = ( η ζ + ζ − η ζ − ζ ) + i ( η ζ + iζ − η ζ − iζ ) . (1.2)Note that the decomposition of η ζ ,ζ in Eq. (1.2) need not be its Hahn decomposition in general,but 4 | η ζ ,ζ | ≤ ( η ζ + ζ + η ζ − ζ ) + ( η ζ + iζ + η ζ − iζ ) = 4( η ζ + η ζ ) . (1.3)So | η ζ ,ζ | ≤ η ζ + η ζ . (1.4)To understand the relation between properties of masas and their left-right-measure , disintegrationof measures will be used, for which we refer the reader to § T be a measurable map from( X, σ X ) to ( Y, σ Y ), where σ X , σ Y are σ -algebras of subsets of X, Y respectively. Let β be a σ -finitemeasure on σ X and µ a σ -finite measure on σ Y . Here β is the measure to be disintegrated and µ isoften the push forward measure T ∗ β , although other possibilities for µ is allowed. Definition 1.4. [2] We say that β has a disintegration { β t } t ∈ Y with respect to T and µ or a ( T, µ )-disintegration if:( i ) β t is a σ -finite measure on σ X concentrated on { T = t } (or T − { t } ), i.e., β t ( { T = t } ) = 0, for µ -almost all t ,and, for each nonnegative σ X -measurable function f on X :( ii ) t β t ( f ) is σ Y -measurable.( iii ) β ( f ) = µ t ( β t ( f )) defn = R Y β t ( f ) dµ ( t ).If β in Defn. 1.4 is a complex measure, then the disintegration of β is obtained by decomposingit into a linear combination of four positive measures, using the Hahn decomposition of its real andimaginary parts. Notation:
The disintegrated measures are usually written with a subscript t β t in the litera-ture. But in this paper, we will use the superscript notation t β t to denote them. The ( π , λ )-disintegration of measures on X × X will be indexed by the variable t and the ( π , λ )-disintegrationwill be indexed by the variable s , where π i : X × X X , i = 1 ,
2, are the coordinate projections.We will only consider the ( π i , λ )-disintegrations of the measures η ζ , η ζ ,ζ defined in Eq. (1.1). Thesedisintegrations exist from Thm. 3.2 [12] (also see Thm. 1 [2]). The measure η tζ is concentrated on { t } × X and the measure η sζ is concentrated on X × { s } for λ almost all t, s respectively. We willdenote by ˜ η tζ the restriction of the measure η tζ on { t } × X . Similarly define ˜ η sζ . Thus, ˜ η tζ , ˜ η sζ can be KUNAL MUKHERJEE regarded as measures on X .The left-right-measure [ η ] of A has the following property. If θ : X × X X × X is the flip mapi.e., θ ( t, s ) = ( s, t ), then θ ∗ η ≪ η ≪ θ ∗ η (see Lemma 2.9 [12]). In fact, it is possible to obtain a choiceof η for which θ ∗ η = η . So in most of the analysis, we will only state or prove results with respect tothe ( π , λ )-disintegration. An analogous statement with respect to the ( π , λ )-disintegration is alsopossible. We will only work with finite or probability measures.2. The Product Class
It is not always easy to describe the properties of a singular masa based on its left-right-measure .However, we can write interesting properties of masas when the left-right-measure is the class ofproduct measure. Such masas are singular Thm. 5.5 [12]. Examples of such masas are easy to obtainin many situations and many known masas, for example, the single generator masas in the free groupfactors, the masas that arise out of Bernoulli shift actions of countable discrete abelian groups belongto this class. In this section, we shall give analytical conditions for the left-right-measure of a masato be the class of product measure.Let λ denote the Lebesgue measure on [0 ,
1] so that A ∼ = L ∞ ([0 , , λ ). Then λ is the tracialmeasure. Let [ η ] denote the left-right-measure of A . We assume that η is a probability measure on[0 , × [0 ,
1] and η (∆([0 , left-right-measure of any masa in the free group factors contains a part of λ ⊗ λ as a summand’.This statement of Voiculescu is one of the most important theorem in the subject (Cor. 7.6 [30] appliedto a system of free semicirulars does the job.) This is the precise reason for the absence of Cartansubalgebras in the free group factors. In many cases, the left-right-measures are difficult to calculate.So we need conditions in terms of operators that characterize the Lebesgue class. The following isthe main result of this section. It will be proved later in this section. Theorem 2.1.
The left-right-measure of a masa A ⊂ M is the class of product measure, if thereexists a set S ⊂ M such that E A ( x ) = 0 for all x ∈ S and ( i ) the linear span of S is dense in L ( M ) ⊖ L ( A ) , ( ii ) there is an orthonormal basis { v n } ∞ n =1 ⊂ A of L ( A ) such that ∞ X n =1 k E A ( xv n x ∗ ) k < ∞ for all x ∈ S, ( iii ) there is a nonzero vector ζ ∈ L ( M ) ⊖ L ( A ) such that E A ( ζu n ζ ∗ ) = 0 for all n = 0 , where u isa Haar unitary generator of A . We do not know whether the conditions in Thm. 2.1 are necessary for the same conclusion to hold.In Thm. 2.7, we provide an analogous condition which is necessary for the left-right-measure to be ofthe product class. In general, it is of interest to know whether there exist masas for which η ≪ λ ⊗ λ but [ η ] = [ λ ⊗ λ ]. Note that the sum in Thm. 2.1 is independent of the choice of the orthonormalbasis. This just follows by expanding elements of one orthonormal basis with respect to another.Hence by making similar arguments, ( iii ) in Thm. 2.1 holds for any Haar unitary generator of A , ifit holds for one Haar unitary generator. Conditions ( i ) and ( ii ) in Thm. 2.1 forces that η ≪ λ ⊗ λ .To assure λ ⊗ λ ≪ η we need condition ( iii ) (Cor. 2.8). As it will become clear, these conditionsare analogous to knowing a measure from the information of its Fourier coefficients. Condition ( iii )is an analogue of the fact that the Fourier coefficients of λ are 0 except for the zeroth coefficient.In this sense the operators E A ( xv n x ∗ ) can be thought of as the ‘Fourier coefficients of the bimodule AxA k·k ’.In order to motivate the conditions in Thm. 2.1, we cite some examples. Conditions ( i ) and ( ii )first appeared in the study of radial masas in the free group factors (see Thm. 3.1 [25]). In this case,the natural choice of the orthonormal basis is v n = χ n k χ n k , n ≥
0, where χ n = P w : | w | = n w . The set S consists of w − E A ( w ), 1 = w ∈ F k , k ≥
2. Single generator masas in the free group factors clearlysatisfies all the three conditions. In fact, for all masas exhibited in § iii ) in Thm. 2.1 is satisfied. However, in those examples ( i ) won’t be satisfied.The second class of examples comes from inclusion of groups. Suppose Γ is a countable discrete iccgroup with Λ < Γ, such that L (Λ) ⊂ L (Γ) is a singular masa. Assume further that Λ is malnormalin Γ. It is obvious that ( i ), ( ii ) and ( iii ) hold (Thm. 9.5, Cor. 9.8 [1]). Similar class of examplesconsists of freely complemented masas. INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 5
Next class of examples comes from ergodic theory. Consider a countable infinite abelian group Γacting on Q Γ ( X, µ ) by Bernoulli shift, where (
X, µ ) is a probability space with at least two points.Then L (Γ) ⊂ L ∞ ( Q Γ ( X, µ )) ⋊ Γ is a singular masa whose left-right-measure is the class of productHaar measure on b Γ × b Γ (Prop. 3.1 [14]). For any function f on the γ -th copy, γ ∈ Γ, such that µ ( f ) = 0 one has E L (Γ) ( f u γ ′ f ∗ ) = τ ( f α γ ′ ( f ∗ )) u γ ′ , γ ′ ∈ Γ , where u γ ′ are the canonical group unitaries in the crossed product, τ is the tracial state and α γ ′ denotes the automorphism corresponding to γ ′ . It follows that P γ ′ ∈ Γ (cid:13)(cid:13) E L (Γ) ( f u γ ′ f ∗ ) (cid:13)(cid:13) is finite. Itis now clear that ( i ) and ( ii ) of Thm. 2.1 hold. Since the maximal spectral type of the Bernoulliaction is the normalized Haar measure on b Γ, any function f ∈ L ( Q Γ ( X, µ )) for which the maximalspectral type is attained, satisfies E L (Γ) ( f u γ f ∗ ) = 0 for all γ = 1. The last statement is equivalentto ( iii ).In order to prove Thm. 2.1, we need to prove some auxiliary lemmas. Lemma 2.2.
Let ζ , ζ ∈ L ( M ) be such that E A ( ζ ) = 0 = E A ( ζ ) . Let η ζ ,ζ denote the Borelmeasure on [0 , × [0 , defined in Eq. (1.1) . ◦ . Then η ζ ,ζ admits ( π i , λ ) -disintegrations [0 , ∋ t η tζ ,ζ and [0 , ∋ s η sζ ,ζ , where π i , i = 1 , , denotes the coordinate projections. Moreover, η tζ ,ζ ([0 , × [0 , E A ( ζ ζ ∗ )( t ) , λ a.e. ◦ . Let f ∈ C [0 , . Then the functions [0 , ∋ t η tζ ,ζ (1 ⊗ f ) , [0 , ∋ s η sζ ,ζ ( f ⊗ are in L ([0 , , λ ) .If ζ i ∈ M for i = 1 , , then [0 , ∋ t η tζ ,ζ (1 ⊗ f ) , [0 , ∋ s η sζ ,ζ ( f ⊗ are in L ∞ ([0 , , λ ) . ◦ . Let b, w ∈ C [0 , . If E A ( ζ wζ ∗ ) ∈ L ( A ) , then k E A ( bζ wζ ∗ ) k = Z | b ( t ) | (cid:12)(cid:12) η tζ ,ζ (1 ⊗ w ) (cid:12)(cid:12) dλ ( t ) . Proof. ◦ . That η ζ ,ζ admits the stated disintegrations follows from Eq. (1.2), Lemma 5.7 [8] andLemma 2.9 [12]. The next statement in 1 ◦ follows from an argument similar to the proof of Lemma6.1 [12].2 ◦ . From Eq. (1.3), | η ζ ,ζ | admits ( π i , λ )-disintegrations. Use Hahn decomposition of measures andLemma 3.6 [12] to see that | η ζ ,ζ | t = (cid:12)(cid:12)(cid:12) η tζ ,ζ (cid:12)(cid:12)(cid:12) for λ almost all t . The function t η tζ ,ζ (1 ⊗ f ) isclearly measurable from Defn. 1.4, and from Eq. (1.4) we have Z (cid:12)(cid:12) η tζ ,ζ (1 ⊗ f ) (cid:12)(cid:12) dλ ( t ) ≤ k f k Z (cid:12)(cid:12) η tζ ,ζ (cid:12)(cid:12) ([0 , × [0 , dλ ( t ) ≤ k f k (cid:18)Z η tζ ([0 , × [0 , dλ ( t ) + Z η tζ ([0 , × [0 , dλ ( t ) (cid:19) = k f k ( k E A ( ζ ζ ∗ ) k + k E A ( ζ ζ ∗ ) k ) < ∞ . When ζ i ∈ M a similar argument shows that the stated functions are in L ∞ ([0 , , λ ).3 ◦ . Since ∞ > sup a ∈ C [0 , , k a k ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z a ( t ) b ( t ) E A ( ζ wζ ∗ )( t ) dλ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = sup a ∈ C [0 , , k a k ≤ | τ ( ab E A ( ζ wζ ∗ )) | = sup a ∈ C [0 , , k a k ≤ | τ ( abζ wζ ∗ ) | = sup a ∈ C [0 , k a k ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z a ( t ) b ( t ) η tζ ,ζ (1 ⊗ w ) dλ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) KUNAL MUKHERJEE and t g b ( t ) η tζ ,ζ (1 ⊗ w ) is in L ( λ ), so g is in L ( λ ) and k E A ( bζ wζ ∗ ) k = Z | b ( t ) | (cid:12)(cid:12) η tζ ,ζ (1 ⊗ w ) (cid:12)(cid:12) dλ ( t ) . (cid:3) Let w := { w n } ∞ n =1 ⊂ C [0 ,
1] be an orthonormal basis of L ( A ). Proposition 2.3.
Let x i ∈ M for i = 1 , , be such that E A ( x i ) = 0 . Let us suppose that ∞ X n =1 k E A ( x w n x ∗ ) k < ∞ . If w ′ := { w ′ n } ∞ n =1 be an orthonormal sequence in L ( A ) with w ′ n ∈ C [0 , for all n , then there is aset F ( w, w ′ ) ⊂ [0 , which depends on w, w ′ such that λ ( F ( w, w ′ )) = 0 and for all t ∈ F ( w, w ′ ) c , ∞ X n =1 (cid:12)(cid:12) η tx ,x (1 ⊗ w ′ n ) (cid:12)(cid:12) ≤ ∞ X n =1 (cid:12)(cid:12) η tx ,x (1 ⊗ w n ) (cid:12)(cid:12) < ∞ . Proof.
Note that the hypothesis implies that for any a ∈ C [0 , ∞ X n =1 k E A ( ax w n x ∗ ) k < ∞ and this sum is independent of the choice of the orthonormal basis. Therefore, for all a ∈ C [0 , ∞ X n =1 (cid:13)(cid:13) E A ( ax w ′ n x ∗ ) (cid:13)(cid:13) ≤ ∞ X n =1 k E A ( ax w n x ∗ ) k . Let r ∈ A be a nonzero projection. Identify r with a measurable subset E r of [0 , E r is a Borel set. We claim that Z E r ∞ X n =1 (cid:12)(cid:12) η tx ,x (1 ⊗ w ′ n ) (cid:12)(cid:12) dλ ( t ) ≤ Z E r ∞ X n =1 (cid:12)(cid:12) η tx ,x (1 ⊗ w n ) (cid:12)(cid:12) dλ ( t ) . (2.1)If the claim is true, then by standard measure theory arguments we are done.First assume E r is a compact set. Choose a sequence of continuous functions f l such that 0 ≤ f l ≤ f l ↓ χ E r pointwise as l → ∞ . Therefore by Lemma 2.2 and monotone convergence theorem, forall l we have, Z f l ( t ) ∞ X n =1 (cid:12)(cid:12) η tx ,x (1 ⊗ w ′ n ) (cid:12)(cid:12) dλ ( t ) = ∞ X n =1 (cid:13)(cid:13) E A ( f l x w ′ n x ∗ ) (cid:13)(cid:13) ≤ ∞ X n =1 k E A ( f l x w n x ∗ ) k = Z f l ( t ) ∞ X n =1 (cid:12)(cid:12) η tx ,x (1 ⊗ w n ) (cid:12)(cid:12) dλ ( t ) . Passing to limits, we see that Eq. (2.1) is true whenever E r is compact. Now use regularity of λ to see that Eq. (2.1) is true for all Borel sets of positive measure. (cid:3) Let X = Q ∞ n =1 C [0 , X with the product topology. Then X is known to be separable andmetrizable. Every f ∈ X is a infinite tuple f = ( f , f , · · · ). Also for a sequence f ( n ) ∈ X , f ( n ) → f as n → ∞ implies that f ( n ) k → f k in k·k ∞ for all k ∈ N .Let O = (cid:8) f ∈ X : { f k } ∞ k =1 is an orthonormal sequence in L ([0 , , λ ) (cid:9) . Then O ⊂ X is a closedset. Note that O is separable in the product topology. Proposition 2.4.
Let x ∈ M be such that E A ( x ) = 0 . Let us suppose that ∞ X k =1 k E A ( xw k x ∗ ) k < ∞ . Then η x ≪ λ ⊗ λ . INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 7
Proof.
Let { w ( m ) } ∞ m =1 ⊂ O be any countable dense set. From Prop. 2.3 and Lemma 2.2, it followsthat there is a set F ⊂ [0 ,
1] with λ ( F ) = 0 such that for t ∈ F c , η tx is a finite measure and ∞ X k =1 (cid:12)(cid:12)(cid:12) η tx (1 ⊗ w ( m ) k ) (cid:12)(cid:12)(cid:12) ≤ ∞ X k =1 (cid:12)(cid:12) η tx (1 ⊗ w k ) (cid:12)(cid:12) < ∞ for all m ∈ N . Let v = { v k } ∞ k =1 ∈ O . There exists a subsequence { w ( m j ) } ∞ j =1 such that w ( m j ) → v as j → ∞ . Therefore for t ∈ F c , ∞ X k =1 (cid:12)(cid:12) η tx (1 ⊗ v k ) (cid:12)(cid:12) = ∞ X k =1 lim j (cid:12)(cid:12)(cid:12) η tx (1 ⊗ w ( m j ) k ) (cid:12)(cid:12)(cid:12) (by Dominated convergence)= ∞ X k =1 lim inf j (cid:12)(cid:12)(cid:12) η tx (1 ⊗ w ( m j ) k ) (cid:12)(cid:12)(cid:12) ≤ lim inf j ∞ X k =1 (cid:12)(cid:12)(cid:12) η tx (1 ⊗ w ( m j ) k ) (cid:12)(cid:12)(cid:12) (by Fatou’s Lemma) ≤ ∞ X k =1 (cid:12)(cid:12) η tx (1 ⊗ w k ) (cid:12)(cid:12) < ∞ (as t ∈ F c ) . Therefore for each t ∈ F c ,sup f ∈O ∞ X k =1 (cid:12)(cid:12) η tx (1 ⊗ f k ) (cid:12)(cid:12) ≤ ∞ X k =1 (cid:12)(cid:12) η tx (1 ⊗ w k ) (cid:12)(cid:12) < ∞ . (2.2)Fix t ∈ F c . If ˜ η tx contains a nonzero part which is singular with respect to λ , then the supremum onthe left hand side of Eq. (2.2) is infinite. Indeed, for simplicity assume ˜ η tx ⊥ λ . Choose a compact set K ⊂ [0 ,
1] of almost full ˜ η tx measure such that λ ( K ) = 0. Fix a large positive number N . By regularityof λ , there is an open set U containing K such that λ ( U ) < N . Using compactness of K , we can finda finite number of open intervals ( a i , b i ) and small positive numbers δ i for i = 1 , , · · · , m , such thatthe open intervals { ( a i − δ i , b i + δ i ) } mi =1 are disjoint and K ⊂ ∪ mi =1 ( a i , b i ) ⊂ ∪ mi =1 ( a i − δ i , b i + δ i ) ⊂ U .Define f i ( s ) = N if a i ≤ s ≤ b i , Nδ i ( s − a i ) + N if a i − δ i ≤ s ≤ a i , − Nδ i ( s − b i ) + N if b i ≤ s ≤ b i + δ i , f = P mi =1 f i is continuous and k f k ,λ = O ( N ). Now consider g = f k f k ,λ . Inductivelyconstruct an orthonormal sequence in C [0 ,
1] with the first function as g , orthogonal with respect tothe λ measure. It is now clear that in this way the supremum in Eq. (2.2) can be made to exceedany large number.Consequently, it follows that for all t ∈ F c , ˜ η tx ≪ λ. (2.3)Finally from Lemma 3.6 of [12], it follows that η x ≪ λ ⊗ λ . (cid:3) Remark . Note that the proof of Prop. 2.4 actually shows that ˜ η tx ≪ λ with d ˜ η tx dλ ∈ L ([0 , , λ ) for λ almost all t .The set of finite signed measures on the measurable space ( X, σ X ) is a Banach space equippedwith the total variation norm k·k t.v , also called the L -norm, which is defined by k µ k t.v = | µ | ( X ),where | µ | denotes the variation measure of µ . It is well known that for probability measures µ and ν ,(2.4) k µ − ν k t.v = 2 sup B ∈ σ X | µ ( B ) − ν ( B ) | = Z X | f − g | dγ where f, g are density functions of µ, ν respectively with respect to any σ -finite measure γ dominatingboth µ, ν (see for instance Eq. (1 .
1) of [16]). We are now in a position to prove Thm. 2.1.
KUNAL MUKHERJEE
Proof of Thm. 2.1.
Fix a set S ⊂ M such that E A ( x ) = 0 for all x ∈ S , span S k·k = L ( A ) ⊥ and ∞ X k =1 k E A ( xu k x ∗ ) k < ∞ for all x ∈ S , where { u k } ∞ k =1 ⊂ C [0 ,
1] is an orthonormal basis of L ( A ). There is a vector ζ ∈ L ( A ) ⊥ such that k ζ k = 1 and η ζ = η . Choose a sequence x n ∈ span S such that k x n k = 1 and x n → ζ in k·k as n → ∞ . Then (see Lemma 3.10 [12]), we have η x n → η ζ = η in k·k t.v . Write x n = P k n i =1 c i,n y i,n with y i,n ∈ S , c i,n ∈ C for all 1 ≤ i ≤ k n and n ∈ N . As y i,n ∈ S , so for all n ∈ N and 1 ≤ i ≤ k n , ∞ X k =1 (cid:13)(cid:13) E A ( y i,n u k y ∗ i,n ) (cid:13)(cid:13) < ∞ . From Prop. 2.4 we have η y i,n ≪ λ ⊗ λ . But η x n = k n X i =1 | c i,n | η y i,n + k n X i = j =1 c i,n c j,n η y i,n ,y j,n . For 1 ≤ i = j ≤ k n , the measures η y i,n ,y j,n are possibly complex measures, but from Eq. (1.4), (cid:12)(cid:12) η y i,n ,y j,n (cid:12)(cid:12) ≤ η y i,n + η y j,n ≪ λ ⊗ λ . Therefore η x n ≪ λ ⊗ λ . Since η x n is a probability measure so fromEq. (2.4), 12 k η x n − η x m k t.v = Z [0 , × [0 , | f n ( t, s ) − f m ( t, s ) | d ( λ ⊗ λ )( t, s ) → n, m → ∞ , where f n = dη xn d ( λ ⊗ λ ) . Thus there is a function f ∈ L ([0 , × [0 , , λ ⊗ λ ) such that Z [0 , × [0 , | f n ( t, s ) − f ( t, s ) | d ( λ ⊗ λ )( t, s ) → n → ∞ . As η x n is a probability measure for each n , so k f n k L ( λ ⊗ λ ) = 1 for all n . Therefore k f k L ( λ ⊗ λ ) = 1 and η x n → f d ( λ ⊗ λ ) in k·k t.v . By uniqueness of limits η = f d ( λ ⊗ λ ).We will now use condition ( iii ) of Thm. 2.1 to show that λ ⊗ λ ≪ η . Let v ∈ A be the Haarunitary corresponding to the function t e πit . As noted after the statement of Thm. 2.1, we canassume u = v in statement ( iii ) of Thm. 2.1. There is a nonzero vector ξ ∈ L ( M ) ⊖ L ( A ) suchthat E A ( ξ v n ξ ∗ ) = 0 for all n = 0. By 3 ◦ of Lemma 2.2 we have η tξ (1 ⊗ v n ) = 0 for all n = 0 and for λ almost all t. Thus, the Fourier coefficients of the measure ˜ η tξ are ˜ η tξ ( n ) = 0, n = 0 for λ almost all t . Thus ˜ η tξ isequal to a multiple of λ , for λ almost all t . The scalar above is the total mass of the measure and inthis case is E A ( ξ ξ ∗ )( t ) for λ almost all t (see Lemma 2.2). A straight forward calculation shows that η ξ ( a ⊗ b ) = Z [0 , × [0 , a ( t ) b ( s ) E A ( ξ ξ ∗ )( t ) d ( λ ⊗ λ )( t, s ) , a, b ∈ C [0 , . Thus dη ξ d ( λ ⊗ λ ) = E A ( ξ ξ ∗ ) ⊗
1. Using Defn. 1.4 it is obvious that λ ⊗ λ ≪ η ξ as well. Thus[ λ ⊗ λ ] = [ η ξ ].Note that Aξ A k·k ⊆ L ( M ) ⊖ L ( A ) and Aξ A k·k ∼ = R ⊕ [0 , × [0 , C t,s dη ξ , where C t,s = C andassociated statement about diagonalizabilty of A holds. There are two cases to consider. If Aξ A k·k = L ( M ) ⊖ L ( A ), then η ξ is indeed the left-right measure of A . In this case there is nothing to prove.If there is a nonzero vector ξ ∈ L ( M ) ⊖ L ( A ) ⊖ Aξ A k·k , then by Lemma 5.7 [8]either Aξ A k·k ⊕ Aξ A k·k ∼ = Z ⊕ [0 , × [0 , H t,s d ( η ξ + ν ) , = ν ⊥ η ξ , or Aξ A k·k ⊕ Aξ A k·k ∼ = Z ⊕ [0 , × [0 , H t,s dη ξ , INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 9 for a η ξ + ν (or η ξ ) measurable field of Hilbert spaces H t,s , along with associated statement aboutthe diagonalizable algebra. In the first case, use direct integrals to conclude that there is a nonzerovector ˜ ξ ∈ L ( M ) (which is off course orthogonal to L ( A ) ⊕ Aξ A k·k ) such that η ˜ ξ = ν . Butby an argument analogous to the first part of the proof it follows that ν ≪ λ ⊗ λ . This forces that Aξ A k·k ⊕ Aξ A k·k ∼ = R ⊕ [0 , × [0 , H t,s dη ξ . Since we have to repeat this argument at most countablymany times to exhaust L ( M ) ⊖ L ( A ) (and in the process the multiplicity function will only changefrom Lemma 5.7 [8]), the proof is complete. (cid:3) Remark . The sum in ( ii ) of Thm. 2.1 is the square of the Hilbert Schmidt norm of the operator E A ( x · x ∗ ). This is precisely the reason the sum is independent of the choice of the orthonormal basis.Note that if P n ∈ Z k E A ( xv n x ∗ ) k < ∞ and P n ∈ Z k E A ( yv n y ∗ ) k < ∞ , where x, y ∈ M , E A ( x ) =0 , E A ( y ) = 0 and v is the standard Haar unitary generator of A , then P n ∈ Z k E A ( xv n y ∗ ) k < ∞ . Tosee this, one has to use the facts dη x,y d ( λ ⊗ λ ) , dη x d ( λ ⊗ λ ) , dη y d ( λ ⊗ λ ) ∈ L ( λ ⊗ λ ). Thus conditions ( i ) , ( ii ) in Thm.2.1 can be strengthened as: There exists a set D ⊂ M such that E A ( x ) = 0 for all x ∈ D , the set D is dense in L ( A ) ⊥ and ∞ X k =1 k E A ( x v k x ∗ ) k < ∞ for all x , x ∈ D, for some orthonormal basis { v k } ⊂ C [0 ,
1] of L ( A ). One choice of D is span S .When the left-right-measure of a masa is the class of product measure, the masa satisfies conditionsvery close to the ones described in Thm. 2.1. This is the content of the next theorem.For N ∞ ∋ n ∈ P uk ( A ), let E n ⊆ [0 , × [0 , \ ∆([0 , m.m ( A ) takes the value n . It is well known that E n is η -measurable. Then L ( M ) ⊖ L ( A ) ∼ = ⊕ n ∈ P uk ( A ) L ( E n , η | E n ) ⊗ C n ∼ = ⊕ n ∈ P uk ( A ) Z ⊕ [0 , × [0 , C nt,s dη | E n ( t, s ) , (2.5)where C nt,s = C n for ( t, s ) ∈ E n when n < ∞ , and C ∞ = C ∞ t,s = l ( N ). Under this decomposition onehas A ′ (1 − e A ) ∼ = ⊕ n ∈ P uk ( A ) L ∞ ( E n , η | E n ) ⊗M n ( C ) , where M ∞ ( C ) is to be interpreted as B ( l ( N )). Consequently, it follows that for N ∞ ∋ n ∈ P uk ( A )the projections χ E n ⊗ n lie in Z ( A ′ ) = A , where 1 n denotes the identity of M n ( C ) if n < ∞ and 1 ∞ = 1 B ( l ( N )) . For n ∈ P uk ( A ) choose vectors ζ ( n ) i , 1 ≤ i ≤ n (1 ≤ i < ∞ if n = ∞ ) sothat the projections P ( n ) i : L ( M ) Aζ ( n ) i A k·k are mutually orthogonal, equivalent in A ′ , and P ni =1 P ( n ) i = χ E n ⊗ n . Theorem 2.7.
Let A ⊂ M be a masa. Let the left-right-measure of A be the class of product measure.Then there is a set S ⊂ L ( M ) ⊖ L ( A ) such that span S is dense in L ( A ) ⊥ , ∞ X n =1 k E A ( ζw n ζ ∗ ) k < ∞ for all ζ ∈ S for some orthonormal basis { w n } ∞ n =1 ⊂ A of L ( A ) , and there is a nonzero ξ ∈ L ( M ) ⊖ L ( A ) suchthat E A ( ξv n ξ ∗ ) = 0 for all n = 0 , where v is a Haar unitary generator of A .Proof. We will first consider the case
P uk ( A ) = { } . In this case, L ( M ) ⊖ L ( A ) ∼ = L ([0 , × [0 , \ ∆([0 , , λ ⊗ λ ) , the left and the right actions of A being given by( af )( t, s ) = a ( t ) f ( t, s ) , ( f b )( t, s ) = b ( s ) f ( t, s ) where f ∈ L ( A ) ⊥ and a, b ∈ A .Let 0 = ζ ∈ L ( A ) ⊥ be a continuous function. Then for a, b ∈ C [0 , h aζb, ζ i L ( M ) = h aζb, ζ i L ( λ ⊗ λ ) = Z [0 , × [0 , a ( t ) b ( s ) ζ ( t, s ) ζ ( t, s ) dλ ( t ) dλ ( s )= Z [0 , × [0 , a ( t ) b ( s ) | ζ ( t, s ) | dλ ( t ) dλ ( s ) . Therefore dη ζ d ( λ ⊗ λ ) = | ζ | which is bounded, in particular in L ( λ ⊗ λ ). We claim that E A ( ζbζ ∗ ) ∈ L ( A ) for any b ∈ C [0 , a ∈ C [0 , τ extends to L , Z a ( t ) E A ( ζbζ ∗ )( t ) dλ ( t ) = τ ( a E A ( ζbζ ∗ )) = τ ( aζbζ ∗ ) = Z [0 , × [0 , a ( t ) b ( s ) dη ζ ( t, s )(2.6) = Z [0 , × [0 , a ( t ) b ( s ) | ζ | ( t, s ) dλ ( t ) dλ ( s )= Z a ( t ) λ ( | ζ | ( t, · ) b ) dλ ( t ) . Now consider the function [0 , ∋ t g λ ( | ζ | ( t, · ) b ). It is clearly λ -measurable and Z (cid:12)(cid:12)(cid:12) λ ( | ζ | ( t, · ) b ) (cid:12)(cid:12)(cid:12) dλ ( t ) = Z (cid:12)(cid:12)(cid:12)(cid:12)Z | ζ | ( t, s ) b ( s ) dλ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) dλ ( t ) ≤ k b k Z (cid:18)Z | ζ | ( t, s ) dλ ( s ) (cid:19) dλ ( t ) ≤ k b k Z Z | ζ | ( t, s ) dλ ( t ) dλ ( s ) < ∞ . Therefore from Eq. (2.6) we get,sup a ∈ C [0 , , k a k ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z a ( t ) E A ( ζbζ ∗ )( t ) dλ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = sup a ∈ C [0 , , k a k ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z a ( t ) λ ( | ζ | ( t, · ) b ) dλ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18)Z (cid:12)(cid:12)(cid:12) λ ( | ζ | ( t, · ) b ) (cid:12)(cid:12)(cid:12) dλ ( t ) (cid:19) < ∞ . Consequently, it follows that E A ( ζbζ ∗ ) ∈ L ( A ) and k E A ( ζbζ ∗ ) k = Z (cid:12)(cid:12)(cid:12) λ ( | ζ | ( t, · ) b ) (cid:12)(cid:12)(cid:12) dλ ( t ) . Let v ∈ A be the Haar unitary corresponding to the function t e πit . Then { v n } n ∈ Z is anorthonormal basis of L ( A ) and by Parseval’s theorem, X n ∈ Z k E A ( ζv n ζ ∗ ) k = X n ∈ Z Z (cid:12)(cid:12)(cid:12) λ ( | ζ | ( t, · ) v n ) (cid:12)(cid:12)(cid:12) dλ ( t ) = Z X n ∈ Z (cid:12)(cid:12)(cid:12) λ ( | ζ | ( t, · ) v n ) (cid:12)(cid:12)(cid:12) dλ ( t )= Z Z | ζ | ( t, s ) dλ ( s ) dλ ( t ) < ∞ . Thus { ζ ∈ L ( A ) ⊥ : P n ∈ Z k E A ( ζv n ζ ∗ ) k < ∞} is dense in L ( A ) ⊥ .In the general case, write L ( M ) ⊖ L ( A ) = ⊕ n ∈ P uk ( A ) (cid:18) n ⊕ i =1 Aζ ( n ) i A k·k (cid:19) , where ζ ( n ) i are vectors defined prior to the proof. For each n ∈ P uk ( A ) and 1 ≤ i ≤ n (or 1 ≤ i < n as the case may be), we consider the left and right actions of A on Aζ ( n ) i A k·k to reduce the problemto a case similar to having one bicyclic vector. In this case, one works with bounded measurablefunctions.Finally, let ζ ∈ L ( M ) correspond to the function χ { ( t,s ): t = s } . Then η ζ = λ ⊗ λ . By arguments INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 11 exactly similar to the first part of the proof, conclude that E A ( ζaζ ∗ ) ∈ L ( A ) for all a ∈ A . But by3 ◦ of Lemma 2.2 we get, k E A ( ζv n ζ ∗ ) k = Z (cid:12)(cid:12) η tζ (1 ⊗ v n ) (cid:12)(cid:12) dλ ( t ) = 0 for all n = 0 . (cid:3) The proof of Thm. 2.1 and Thm. 2.7 yield the following corollary.
Corollary 2.8.
For a masa A ⊂ M , the left-right-measure of A contains the product class as asummand, if and only if, there is a nonzero ξ ∈ L ( M ) ⊖ L ( A ) such that E A ( ξv n ξ ∗ ) = 0 for all n = 0 , where v is a Haar unitary generator of A .Proof. ⇒ By Lemma 5.7 [8], the left-right-measure of A is of the form [ λ ⊗ λ + ν ] where either ν = 0or ν ⊥ λ ⊗ λ . In any case, there is a nonzero vector ξ ∈ L ( A ) ⊥ such that η ξ = λ ⊗ λ . Now use theargument of last part of Thm. 2.7. The reverse direction follows from the proof of Thm. 2.1. (cid:3) Remark . The proof of the previous theorem shows that if A ⊂ M is a masa satisfying theconditions of Thm. 2.1 or the left-right-measure of A is the product class, then there is a measurablepartition { E n } n ∈ P uk ( A ) of ∆([0 , c such that A L ( M ) ⊖ L ( A ) A ∼ = ⊕ n ∈ P uk ( A ) ⊕ ni =1 A L ( E n , λ ⊗ λ ) A with the natural actions on the right hand side.Note that the measure-multiplicity invariant can be defined for any diffuse abelian subalgebra of M in exactly the similar way defined in Defn. 1.3. If the diffuse abelian algebra is not a masa, thenthe diagonal will correspond to the L completion of the relative commutant of the abelian algebra,so the multiplicity function along the diagonal will not be constantly 1. All other properties of theinvariant will remain the same. We will use this observation in the following result. Theorem 2.10.
Let A ⊂ M be a masa such that the left-right-measure of A is the class of productmeasure. Then for any diffuse algebra B ⊂ A , the left right-measure of B restricted to the off-diagonalis the class of product measure and N ( B ) ′′ = B ′ ∩ M = A .Proof. Since the left-right-measure of A is [ λ ⊗ λ ], so by Thm. 2.7, there is a set S orthogonal to L ( A ), such that span S is dense in L ( A ) ⊥ , P n ∈ Z k E A ( ζv n ζ ∗ ) k < ∞ for all ζ ∈ S , where v is thestandard Haar unitary generator of A . Moreover, the proof of Thm. 2.7 shows that we can assume dη ζ d ( λ ⊗ λ ) is bounded λ ⊗ λ almost everywhere.Arguments similar to the proof of Thm. 2.7 show that E A ( ζ · ζ ∗ ) defines a Hilbert Schmidt operatoron L ( A ). Fix a diffuse subalgebra B ⊂ A . Let w ∈ B be a Haar unitary generator of B . Since E A ( ζ · ζ ∗ ) is Hilbert Schmidt, so P n ∈ Z k E A ( ζw n ζ ∗ ) k < ∞ and since k E B ( · ) k ≤ k E A ( · ) k so X n ∈ Z k E B ( ζw n ζ ∗ ) k < ∞ , for all ζ ∈ S. Assuming B = L ∞ ([0 , , λ ) where λ is Lebesgue measure and using arguments required to proveProp. 2.4, one finds η ζ,B ≪ λ ⊗ λ for all ζ ∈ S . The extra suffix refers to the fact that we areconsidering measures with respect to B . (It should be noted that the proof of Prop. 2.4 nowhereuses the fact that A is a masa.) Thus E B ′ ∩M ( ζ ) = 0 for all ζ ∈ S . Indeed, write ζ = ζ + ζ with E B ′ ∩M ( ζ ) = ζ and E B ′ ∩M ( ζ ) = 0. For a, b ∈ B one has h aζ b, ζ i = τ ( aζ bζ ∗ ) = τ ( E B ′ ∩M ( aζ b ) ζ ∗ ) = 0 . Thus η ζ,B = η ζ ,B + η ζ ,B . But η ζ ,B ≪ ˜∆ ∗ λ (for ˜∆ see §
1) with the Radon-Nikodym derivative givenby E B ( ζ ζ ∗ ). Consequently, ζ = 0. Thus S ⊂ L ( B ′ ∩ M ) ⊥ and hence L ( A ) ⊥ ⊆ L ( B ′ ∩ M ) ⊥ . Itfollows that B ′ ∩ M = A .By arguments similar to the proof of Thm. 2.1, it follows that any member in left-right-measure of B restricted to the off-diagonal is dominated by λ ⊗ λ .There is a vector 0 = ξ ∈ L ( A ) ⊥ such that E A ( ξv n ξ ∗ ) = 0 for all n = 0. It follows that E A ( ξaξ ∗ ) = 0 for all a ∈ A with τ ( a ) = 0. Consequently, E A ( ξw n ξ ∗ ) = 0 and hence E B ( ξw n ξ ∗ ) = 0for all n = 0. By arguments made in the last part of the proof of Thm. 2.1, it follows that the left-right-measure of B restricted to the off diagonal is the class of product measure. Finally, if 0 = ζ ∈ L ( N ( B ) ′′ ), then Bζ B k·k ∈ C d ( B ) (Prop. 3.11 [12]). Thus by using Lemma5.7 [8], it follows η ζ ,B must be supported on the diagonal; equivalently E B ′ ∩M ( ζ ) = ζ . Thus ζ ∈ L ( A ). This completes the proof. (cid:3) Tauer Masas in the Hyperfinite II Factor
In this section, we will calculate the left-right-measures of certain Tauer masas in the hyperfiniteII factor R . The examples of Tauer masas in which we are interested are directly taken from [23]. Definition 3.1. (White) A masa A in R is said to be a Tauer masa , if there exists a sequence offinite type I subfactors {N n } ∞ n =1 such that,( i ) N n ⊂ N n +1 for all n ,( ii )( ∪ ∞ n =1 N n ) ′′ = R ,( iii ) A n = A ∩ N n is a masa in N n for every n .This allows one to write the structure of every Tauer masa A in R with respect to the chain {N n } ∞ n =1 as follows. Switching to the notation of tensor products, the above definition means thatwe can find finite type I subfactors {M n } ∞ n =1 such that, N n = n ⊗ r =1 M r for every n . For m > n , the m -th finite dimensional approximation of A can be written in terms of the n -th one as,(3.1) A m = M e ∈P ( A n ) e ⊗ A ( e ) m,n , where the direct sum is over the set of minimal projections P ( A n ) in A n and A ( e ) m,n is a masa in m ⊗ r = n +1 M r . Note that the Cartan masa arising from the infinite tensor product of diagonal matricesinside the hyperfinite II factor is a Tauer masa. In Thm. 4.1 [32], White had shown that the Puk´anszky invariant of every Tauer masa is { } . In fact, it follows from his proof that the bicyclicvector for any Tauer masa can be chosen to be an operator from R itself.Sinclair and White [23] has exhibited a continuous path of singular masas in R , no two of whichcan be connected by automorphisms of R . We are interested in two masas that correspond to theend points of this path. For all Tauer masas, it is clear that the Cantor set is the natural spacewhere we have to build the measures. For ease of calculation, we need to index the minimal projec-tions in the approximating stages in an appropriate fashion. It is now time to introduce some notation.1 ◦ Notation : If N n = n ⊗ r =1 M k r ( C ), then the minimal projections of A n will be denoted by ( n ) f t ( n ) ,where t ( n ) = ( t , t , · · · , t n ) with 1 ≤ t i ≤ k i , 1 ≤ i ≤ n . The convention that we follow is ( n ) f ( t ,t , ··· ,t n ) = ( n − f ( t ,t , ··· ,t n − ) ⊗ ( n ) e ( t ,t , ··· ,t n − ) t n , where ( n ) e ( t ,t , ··· ,t n − ) t n are the minimal projections of the algebra A ( t ,t , ··· ,t n − ) n,n − , in accordance withEq. (3.1). The matrix units corresponding to this family of minimal projections will be denotedby ( n ) f t ( n ) , s ( n ) and we will understand ( n ) f t ( n ) , t ( n ) = ( n ) f t ( n ) . For two tuples ( t , t , · · · , t n ) and( s , s , · · · , s n ) such that t i = s i for 1 ≤ i ≤ n − t n = s n , we will write ( n ) f t ( n ) , s ( n ) = ( n ) f ( · ,t n ) , ( · ,s n ) .2 ◦ Notation:
For any two subsets
S, T ⊆ M , we will denote by S · T the set span { ab : a ∈ S,b ∈ T } . The normalized trace of M n ( C ) will be denoted by tr n . The unique normal tracial state ofthe hyperfinite factor R will be denoted by τ R . This trace τ R when restricted to A gives rise to ameasure on a Cantor set which will also be denoted by τ R .Recall from [19] that two subalgebras B, C in a finite factor N are called orthogonal with respectto the unique normal tracial state τ N , if τ N ( bc ) = τ N ( b ) τ N ( c ) for all b ∈ B , c ∈ C . The next lemmais very well known but we record it for convenience. Lemma 3.2.
If A, B are two masas in M n ( C ) orthogonal with respect to the normalized trace tr n ,then A · B = M n ( C ) . INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 13
Tauer Masa of Product Class.
Following Sinclair and White [23], we shall calculate the measure-multiplicity invariant of a Tauermasa A , whose description is elaborated below. The Γ invariant of this Tauer masa is 0 (A is totallynon-Γ [23]). We will show that its left-right-measure belongs to the product class. This example isimportant, as, it is an example of a masa in R with simple multiplicity whose left-right-measure isthe class of product measure. Such masas are rare in R . We do not know whether it arises from adynamical system.Let k = 2, and for each r ≥
2, let k r be a prime exceeding k k · · · k r − . Set M r to be the algebraof k r × k r matrices. By Thm. 3.2 [19], there is a family { ( r ) D t ( r − } t ( r − of pairwise orthogonalmasas in M r . Let N n = n N r =1 M r . There is a natural inclusion x x ⊗ N n inside N n +1 andone works in the hyperfinite II factor R , obtained as a direct limit of these N n with respect to thenormalized trace. With respect to the chain {N n } ∞ n =1 of finite type I subfactors of R , the masa A isconstructed as follows.Let A = D ( C ) ⊂ M be the diagonal masa. Having constructed A n , one constructs A n +1 as,(3.2) A n +1 = M t ( n )( n ) f t ( n ) ⊗ ( n +1) D t ( n ) . That ( ∪ ∞ n =1 A n ) ′′ is a masa in R , follows from a theorem of Tauer (see Thm. 2.5 [29]). This Tauermasa is singular from Prop. 2.1 [23].We denote by P ( n ) t ( n ) ,s ( n ) the orthogonal projection from L ( R ) onto the subspace ( n ) f t ( n ) L ( R ) ( n ) f s ( n ) ,and let,(3.3) P = ∞ X n =1 X t ( n ) , s ( n ): t = s , ··· ,t n − = s n − ,t n = s n P ( n )( t , ··· ,t n ) , ( s , ··· ,s n ) . Clearly, P ( n ) t ( n ) ,s ( n ) = ( n ) f t ( n ) J ( n ) f s ( n ) J and is in A . At the first sight, it might not be clear that thesum in Eq. (3.3) makes sense, but, the projections involved in the sum are orthogonal and sums to1 − e A . Indeed, since e A is the limit in strong operator topology of e A ′ n ∩R = P t ( n ) P ( n )( t ,t , ··· ,t n ) , ( t ,t , ··· ,t n ) ( § P = 1 − e A follows by rearranging terms in Eq. (3.3).The following lemma, part of which was recorded by Sinclair and White [23], will be crucial forour calculations. Lemma 3.3.
For each n ∈ N , let R = N n N R n , where R n = ( ∞ N r = n +1 M k r ( C )) ′′ . Then (3.4) A = M t ( n )( n ) f t ( n ) ⊗ A t ( n ) ∞ ,n +1 , where A t ( n ) ∞ ,n +1 are Tauer masas in R n and whenever t ( n ) = s ( n ) we have ( i ) A t ( n ) ∞ ,n +1 and A s ( n ) ∞ ,n +1 are orthogonal in R n , ( ii ) ( A t ( n ) ∞ ,n +1 · A s ( n ) ∞ ,n +1 ) −k . k = L ( R n ) . Moreover, for each t ( n ) if { A t ( n ) m,n +1 } ∞ m =1 denote the m -th approximation of A t ( n ) ∞ ,n +1 in R n , then (3.5) A t ( n )1 ,n +1 = ( n +1) D t ( n ) , and, (3.6) A t ( n ) m +1 ,n +1 = M e ∈P ( A t ( n ) m,n +1 ) e ⊗ ( m +1) D t ( n ) e,n +1 , where for each fixed m and t ( n ) , the family { ( m +1) D t ( n ) e,n +1 } e are pairwise orthogonal masas in M k n + m +1 ( C ) .Proof. It should be understood that in ( ii ) of the statement, the closure is taken with respect to thefaithful normal tracial state of R n . We only have to prove ( ii ). The rest of the statements are just rephrasing of Lemma 5.6 of [23].Use Lemma 3.2, ( i ) and Eq. (3.5) to conclude that M k n +1 ⊆ ( A t ( n ) ∞ ,n +1 · A s ( n ) ∞ ,n +1 ) −k·k . Since A t ( n ) ∞ ,n +1 and A s ( n ) ∞ ,n +1 are orthogonal, so is A t ( n ) m,n +1 and A s ( n ) m,n +1 for all m ≥ n + 1. Use Lemma3.2 to conclude that m N r = n +1 M k r ( C ) ⊆ ( A t ( n ) ∞ ,n +1 · A s ( n ) ∞ ,n +1 ) −k·k for all m ≥ n + 1. Now use density ofalgebraic tensor product of matrix algebras in L ( R n ) to finish the proof. (cid:3) For each n , let X n = { x ( n )1 , x ( n )2 , · · · , x ( n ) k n } denote a set of k n points. Let Y ( n ) = n Q k =1 X k , X ( n ) = ∞ Q k = n +1 X k and X = ∞ Q k =1 X k , so that for each n , X = Y ( n ) × X ( n ) . Therefore, X = lim ∞←− n Y ( n ) and C ( X )is norm separable and w.o.t dense in A . The identification is a standard one and we omit the details.Write B = C ( X ). Therefore,(3.7) B = M t ( n )( n ) f t ( n ) ⊗ B t ( n ) ∞ ,n +1 ,B t ( n ) ∞ ,n +1 ∼ = C ( X ( n +1) ) and is a w.o.t dense, norm separable C ∗ subalgebra of A t ( n ) ∞ ,n +1 . Lemma 3.4.
For each n and t ( n ) = s ( n ) , ( i )( A ( n ) f t ( n ) ,s ( n ) A ) −k . k = ( n ) f t ( n ) L ( R ) ( n ) f s ( n ) , ( ii ) for a , b ∈ B , h a ( n ) f t ( n ) , s ( n ) b, ( n ) f t ( n ) , s ( n ) i τ R = k k · · · k n Z X Z X ( n ) f t ( n ) ( t ) ( n ) f s ( n ) ( s ) a ( t ) b ( s ) d ( τ R ⊗ τ R )( t, s ) . Moreover, ( A ( n ) f t ( n ) ,s ( n ) A ) −k . k is orthogonal to ( A ( n ) f t ′ ( n ) ,s ′ ( n ) A ) −k . k whenever t ( n ) = s ( n ) , t ′ ( n ) = s ′ ( n ) and ( t ( n ) , s ( n )) = ( t ′ ( n ) , s ′ ( n )) .Proof. For a, b ∈ A , using Eq. (3.4) write a = ⊕ q ( n )( n ) f q ( n ) ⊗ a q ( n ) and b = ⊕ p ( n )( n ) f p ( n ) ⊗ b p ( n ) for a q ( n ) ∈ A q ( n ) ∞ ,n +1 , and b p ( n ) ∈ A p ( n ) ∞ ,n +1 . By direct multiplication, we get a ( ( n ) f t ( n ) , s ( n ) ⊗ R n ) b = ( n ) f t ( n ) , s ( n ) ⊗ a t ( n ) b s ( n ) . Therefore ( i ) follows from ( ii ) of Lemma 3.3. Moreover, for a, b ∈ B , h a ( ( n ) f t ( n ) , s ( n ) ⊗ R n ) b, ( n ) f t ( n ) , s ( n ) ⊗ R n i τ R = tr Q ni =1 k i ( ( n ) f t ( n ) ) τ R n ( a t ( n ) b s ( n ) )= 1 k k · · · k n τ R n ( a t ( n ) b s ( n ) )= k k · · · k n τ R ( a ( ( n ) f t ( n ) ⊗ τ R ( b ( ( n ) f s ( n ) ⊗ ii ))= k k · · · k n Z X Z X a ( t )( ( n ) f t ( n ) ⊗ t ) b ( s )( ( n ) f s ( n ) ⊗ s ) d ( τ R ⊗ τ R )( t, s )= k k · · · k n Z x (1) t ×···× x ( n ) tn × X ( n ) Z x (1) s ×···× x ( n ) sn × X ( n ) a t ( n ) ( t ) b s ( n ) ( s ) d ( τ R ⊗ τ R )( t, s )= k k · · · k n Z X × X ( n ) f t ( n ) ( t ) ( n ) f s ( n ) ( s ) a ( t ) b ( s ) d ( τ R ⊗ τ R )( t, s ) , where the indicators of ( x (1) t × · · · × x ( n ) t n ) × X ( n ) and ( x (1) s × · · · × x ( n ) s n ) × X ( n ) corresponds to ( n ) f t ( n ) and ( n ) f s ( n ) respectively. This proves ( ii ). Clearly the final statement follows from ( i ) and the fact INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 15 that ( n ) f t ( n ) J ( n ) f s ( n ) J and ( n ) f t ′ ( n ) J ( n ) f s ′ ( n ) J are orthogonal projections in L ( R ) if ( t ( n ) , s ( n )) =( t ′ ( n ) , s ′ ( n )). (cid:3) Remark . The following observation will be used in the next proof. On every occasion below,where we add direct integrals, Lemma 5.7 [8] is invoked. For t i = s i , 1 ≤ i ≤ n − t n = s n , theprojection P ( n )( t , ··· ,t n ) , ( s , ··· ,s n ) ∈ A ′ and hence is in A , as A is maximal abelian in B ( L ( R )). Therefore, P ( n )( t , ··· ,t n ) , ( s , ··· ,s n ) is decomposable (see Ch. 14 [10]). Denote E ( · ,t n ) , ( · ,s n ) = ( x (1) t × · · · × x ( n − t n − × x ( n ) t n × X ( n ) ) × ( x (1) t × · · · × x ( n − t n − × x ( n ) s n × X ( n ) ) . ¿From Lemma 3.4, it follows that the range P ( n )( t , ··· ,t n ) , ( s , ··· ,s n ) ( L ( R )) is the direct integral of complexnumbers over the set E ( · ,t n ) , ( · ,s n ) with respect to τ R ⊗ τ R , and, A P ( n )( t , ··· ,t n ) , ( s , ··· ,s n ) is the diagonal-izable algebra with respect to this decomposition. For t ( n − = t ′ ( n − P ( n )( t ( n − ,t n ) , ( t ( n − ,s n ) and P ( n )( t ′ ( n − ,t ′ n ) , ( t ′ ( n − ,s ′ n ) with t n = s n and t ′ n = s ′ n rest over disjoint subsets of X × X . Therefore, the range of P ( n ) = P t ( n ) , s ( n ): t = s , ··· ,t n − = s n − ,t n = s n P ( n )( t , ··· ,t n ) , ( s , ··· ,s n ) is the direct integral ofcomplex numbers with respect to τ R ⊗ τ R over the set E n = ∪ k t =1 · · · ∪ k n − t n − =1 ∪ k n t n = s n =1 E ( · ,t n ) , ( · ,s n ) , andassociated statements about diagonalizability of A P ( n ) hold. It is important to note that E n ∩ E m = ∅ for all n = m .Let c n = n Q r =1 k r for n ≥ c = 1. Proposition 3.6.
The vector ∞ P n =1 P t ( n ) 1 √ c n ( n ) f ( · ,t n ) , ( · ,s n ) is a cyclic vector of A (1 − e A ) and (1 − e A )( L ( R )) ∼ = Z ⊕ X × X C t,s d ( τ R ⊗ τ R )( t, s ) , where C t,s = C . Moreover, A (1 − e A ) is the algebra of diagonalizable operators with respect to this decomposition.Proof. Fix n ∈ N . For each 1 ≤ t i ≤ k i , 1 ≤ i ≤ n −
1, and 1 ≤ t n = s n ≤ k n , working with vectors √ c n ( n ) f ( · ,t n ) , ( · ,s n ) , one finds (using Lemma 3.4) a positive measure η ( n )( t , ··· ,t n ) , ( s , ··· ,s n ) supported on E ( · ,t n ) , ( · ,s n ) such that dη ( n )( t , ··· ,t n ) , ( s , ··· ,s n ) = ( n ) f ( t , ··· ,t n ) ⊗ ( n ) f ( s , ··· ,s n ) d ( τ R ⊗ τ R ) . By making arguments similar to Rem. 3.5, for each n find a positive measure η ( n ) on E n such that η ( n ) = χ E n d ( τ R ⊗ τ R ) and P ( n ) ( L ( R )) = X t ( n ) , s ( n ): t = s , ··· ,t n − = s n − ,t n = s n P ( n )( t , ··· ,t n ) , ( s , ··· ,s n ) ( L ( R )) ∼ = Z X × X ⊕ C t,s dη ( n ) ( t, s ) , where C t,s = C and A P ( n ) is diagonalizable with respect to this decomposition. Note that(3.8) η ( n ) ( X × X ) = c n − ( k n − k n ) c n = 1 c n − − c n . From Rem. 3.5, note that the measures η ( n ) are supported on disjoint sets. Hence by Lemma 5.7[8], (1 − e A )( L ( R )) ∼ = Z ⊕ X × X C t,s dη ( t, s ) , where C t,s = C , η = ∞ X n =1 η ( n ) . (3.9) Moreover, A (1 − e A ) is diagonalizable with respect to the decomposition in Eq. (3.9). Clearly, η ( X × X ) = lim N →∞ N X n =1 η ( n ) ( X × X ) = lim N →∞ c − c N = 1 . Finally, η = τ R ⊗ τ R . Indeed, for a , b ∈ C ( X ), Z X × X a ( t ) b ( s ) dη ( t, s )= ∞ X n =1 Z X × X a ( t ) b ( s ) dη ( n ) ( t, s )= ∞ X n =1 X t ( n ) , s ( n ): t = s , ··· ,t n − = s n − ,t n = s n Z X × X a ( t ) b ( s ) ( n ) f ( t , ··· ,t n ) ( t ) ( n ) f ( s , ··· ,s n ) ( s ) d ( τ R ⊗ τ R )( t, s ) . But N P n =1 P t ( n ) , s ( n ): t = s , ··· ,t n − = s n − ,t n = s n ( n ) f ( t , ··· ,t n ) ( t ) ( n ) f ( s , ··· ,s n ) ( s ) ↑ χ ∆( X ) c pointwise τ R ⊗ τ R almost ev-erywhere. Use dominated convergence theorem and the fact ( τ R ⊗ τ R )(∆( X )) = 0 to conclude η = τ R ⊗ τ R . This completes the proof. (cid:3) For A , the operator x = ∞ P n =1 P t ( n ) 1 √ c n ( n ) f ( · ,t n ) , ( · ,s n ) gives rise to a choice of a vector in ( iii ) ofThm. 2.1. In order to get an appropriate vector one has to apply an appropriate transformationbetween the Cantor set and [0 , B ( L ( R )) preserving the bimodulestructure. Since the Puk´anszky invariant of every
Tauer masa is { } , we have computed the measure-multiplicity invariant of A . Note that AxA is dense in L ( R ) ⊖ L ( A ). For a ∈ A and any orthonormalbasis { v n } ∞ n =1 ⊂ A of L ( A ), one has P n k E A ( xav n x ∗ ) k = P n R X (cid:12)(cid:12) η tx (1 ⊗ av n ) (cid:12)(cid:12) dτ R ( t ) = k a k , as η x = τ R ⊗ τ R (see Lemma 3.6 [12]). This shows that the Tauer masa above satisfy conditions ( i ) and( ii ) of Thm. 2.1 with S = AxA . The above Tauer masa was denoted by A (0) in [23]. There is aTauer masa of exactly opposite flavor, which we call the alternating Tauer masa .3.2. Alternating Tauer Masa.
The alternating Tauer masa A (1) is a singular Tauer masa in the hyperfinite II factor R , con-structed by White and Sinclair [23]. It contains nontrivial centralizing sequences of R . In fact, itsΓ-invariant is 1. This masa will play a role in §
5. In §
4, we will describe its left-right-measure .The chain for this masa is exactly similar to the masa of the product class described before. Let A (1) = D ( C ) ⊂ M be the diagonal masa. Having constructed A (1) n ⊂ N n , one constructs A (1) n +1 as,(3.10) A (1) n +1 = A (1) n ⊗ ( n +1) D n +1 , n even, ( n +1) D n +1 is the diagonal masa in M k n +1 ( C ) , L t ( n )( n ) f t ( n ) ⊗ ( n +1) D t ( n ) , n odd , ( n +1) D t ( n ) pairwise orthogonal in M k n +1 ( C ) . We will prove that the left-right-measure of A (1) is singular with respect to the product measure.Having understood the left-right-measures of A (0) and A (1), we can describe the same for the entirepath of masas exhibited in [23]. 4. Γ and Non Γ Masas
In this section, we study properties of left-right-measures of masas that possess nontrivial central-izing sequences of the factor. We also study properties of left-right-measures that prevent a masa tocontain nontrivial centralizing sequences. This section contains partial answers. Some results in thissection can be proved by bringing in the notion of strongly mixing masas [9]. To keep this paper ina reasonable size, we postpone the notion of strong mixing to a future paper.
INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 17
Definition 4.1.
A centralizing sequence in a II factor M is a bounded sequence { x n } ⊂ M suchthat k x n y − yx n k → n → ∞ for all y ∈ M . The centralizing sequence { x n } is trivial, if thereexists a sequence λ n ∈ C such that k x n − λ n k → n → ∞ .For a masa A ⊂ M , the Γ invariant of A is defined byΓ( A ) = sup { τ ( p ) : p ∈ A is a projection and Ap contains nontrivial centralizing sequences of p M p } . It is immediate that Γ( A ) = Γ( θ ( A )), where θ is an automorphism of M [23]. If Γ( A ) = 0, then A is said to be totally non-Γ. We continue to assume that A = L ∞ ([0 , , λ ), where λ is the Lebesguemeasure. Proposition 4.2.
Let A ⊂ M be a masa. Let the left-right-measure of A be [( λ ⊗ λ ) + µ ] , where µ ⊥ λ ⊗ λ and µ is finite. Then A cannot contain non trivial centralizing sequences of M . Moreover, Γ( A ) = 0 .Proof. Write [0 , × [0 , \ ∆([0 , E ∪ F , where ( λ ⊗ λ )( E ) = 0 and µ ( F ) = 0. There exists anonzero vector ζ ∈ L ( M ) ⊖ L ( A ) such that for a, b ∈ C [0 , η ζ ( a ⊗ b ) = λ ( a ) λ ( b ) . The direct integral of ζ is supported on F .If possible, let { a n } ⊂ A be a non trivial centralizing sequence. By making a density argument, wecan assume that a n = a ∗ n ∈ C [0 ,
1] and τ ( a n ) = 0 for all n . Also assume that lim sup n k a n k = α > k a n ζ − ζa n k → n → ∞ . However, k a n ζ − ζa n k = h a n ζ, a n ζ i − h ζa n , a n ζ i − h a n ζ, ζa n i + h ζa n , ζa n i (4.1) =2 λ ( a ∗ n a n ) . Eq. (4.1) shows that k a n ζ − ζa n k n → ∞ , which is a contradiction.The last statement follows from the above argument by considering compressions of M by projec-tions in A , because, for any nonzero projection p ∈ A , identifying p as the indicator of a measurableset E p , it follows that the left-right-measure of the inclusion Ap ⊂ p M p will be the class of therestriction of λ ⊗ λ + µ to E p × E p . (cid:3) The next result is a generalization of Prop. 4.2. We skip its proof, as the proof is similar to theproof of Prop. 4.2.
Proposition 4.3.
Let A ⊂ M be a masa. Let the left-right-measure of A restricted to the projection pJ qJ contain the product measure as a summand, where p and q are nonzero projections in A . Then: ( i ) Γ( A ) < . ( ii ) If r ≥ p, q is any projection in A , then Ar cannot contain nontrivial centralizing sequences of r M r . Proposition 4.4.
Let A ⊂ M be a masa. Let the left-right-measure of A be [ ν + µ ] , where µ ⊥ λ ⊗ λ , ν ≪ λ ⊗ λ , ν and µ are finite and ν = 0 . Then A cannot contain any centralizing sequence of M consisting of weakly null unitaries.Proof. Without loss of generality, we can assume that f = dνd ( λ ⊗ λ ) ∈ L ( λ ⊗ λ ). Write [0 , × [0 , \ ∆([0 , E ∪ F , where ν ( E ) = 0 and µ ( F ) = 0. There exists a nonzero vector ζ ∈ L ( M ) ⊖ L ( A )such that for a, b ∈ C [0 , η ζ ( a ⊗ b ) = Z [0 , × [0 , a ( t ) b ( s ) f ( t, s ) dλ ( t ) dλ ( s ) . The direct integral of ζ is supported on F . Arguing as in the proof of Thm. 2.7, we conclude that E A ( ζ bζ ∗ ) ∈ L ( A ) for all b ∈ C [0 ,
1] and P k ∈ Z (cid:13)(cid:13) E A ( ζ v k ζ ∗ ) (cid:13)(cid:13) < ∞ , where v ∈ A is the Haar unitarygenerator corresponding to the function t e πit .Suppose to the contrary, there is a sequence { a n } ⊂ C [0 , ⊂ A of weakly null unitaries thatcentralize M . Given ǫ >
0, choose k ∈ N such that P | k |≥ k (cid:13)(cid:13) E A ( ζ v k ζ ∗ ) (cid:13)(cid:13) < ǫ . Therefore on onehand, k E A ( ζ a n ζ ∗ ) k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ∈ Z τ ( a n v − k ) E A ( ζ v k ζ ∗ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
18 KUNAL MUKHERJEE ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | k | 0. On the other hand, k a ∗ n ζ a n ζ ∗ − ζ ζ ∗ k → n → ∞ and consequently k a ∗ n E A ( ζ a n ζ ∗ ) − E A ( ζ ζ ∗ ) k → 0. So k E A ( ζ a n ζ ∗ ) k → k E A ( ζ ζ ∗ ) k .This is a contradiction as ζ is nonzero. (cid:3) Corollary 4.5. The left-right-measure of A (1) is singular with respect to the product class.Proof. By construction A (1) contains a centralizing sequence of weakly null unitaries. (cid:3) Corollary 4.6. Every strongly stable ( McDuff ) factor contains a singular masa whose left-right-measure is singular with respect to the product class.Proof. For existence of a singular masa in a II factor see [20]. The statement follows by tensoringany singular masa in the factor by the alternating Tauer masa in R (see [20, 28], Prop. 5.2 [8] andLemma 3.5 [12]). (cid:3) It is now natural to ask the following question. If M = M ⊗ M , where both M , M are II factors then does M contain a (singular) masa whose left-right-measure is singular with respect tothe product class? Definition 4.7. A finite measure µ on [0 , 1] (or S ) is called α - rigid for | α | = 1, if and only if, thereis a subsequence b µ n k of b µ n = R e − πint dµ ( t ) (or b µ n = R S z − n dµ ( z )) that converges to αµ ([0 , αµ ( S )) as k → ∞ . A 1-rigid measure is called rigid or a Dirichlet measure.We now recall some properties of α -rigid measures. For details check Ch.7 [13]. Let µ be a α -rigidmeasure on [0 , n k along which b µ n k converges to αµ ([0 , µ . It is easy to see that, µ is α -rigid, if and only if, the sequence of functions[0 , ∋ t e − πin k t converges to α in µ -measure. Thus ν is α -rigid with associated sequence n k forany ν ≪ µ . So α -rigidity is a property of equivalence class of measures, and hence can be thought ofas a property of unitary operators, by considering appropriate Koopman operators. Atomic measuresare always rigid.To motivate what follows, we consider rigid m.p. transformations. Let T be a m.p. automorphismof a standard probability space ( X, µ ). Let U T denote the associated Koopman operator on L ( X, µ ).The transformation T is said to be rigid if 1 ∈ { U nT } n ∈ Z \{ } s.o.t [14].Assume further that T is weakly mixing. Then L ( Z ) ⊂ L ∞ ( X, µ ) ⋊ T Z is a singular masa [15](also see Thm. 2.1 [14]). Let U n k T s.o.t → k → ∞ . A simple calculation shows that L ( Z ) contains acentralizing sequence of the crossed product factor consisting of powers of the standard Haar unitarygenerator. It is not known whether this is always the case for Γ-masas. Let ν (which is a measure on b Z = S ) denote the maximal spectral type of the action T . Then there is a unit vector f ∈ L ( X, µ )such that b ν n = h U nT f, f i for all n ∈ Z . It follows that ν is a Dirichlet measure. The relationshipbetween the maximal spectral type of an action and the left-right measure of the associated masaappeared in Prop. 3.1 [14]. Thus by general theory of α -rigid measures (see Ch. 7 [13]), it followsthat for λ almost all t ( λ is Haar measure), the measure ˜ η t is α -rigid for all α ∈ S .In the general case, when A contains a nontrivial centralizing sequence of M , one can choose a INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 19 central sequence consisting of trigonometric polynomials without constant term. We do not knowwhether we can choose a central sequence of the form t e πin k t . In case we can, results analogousto the crossed product situation hold.Making appropriate changes to the proof of Lemma 2.2, we get the following result. Its proof usesbasic facts about L spaces associated to finite von Neumann algebras. We omit its proof. Lemma 4.8. Let ζ ∈ L ( M ) be such that E A ( ζ ) = 0 . Let η ζ denote the measure on [0 , × [0 , defined in Eq. (1.1) . Let b, w ∈ C [0 , . Then k E A ( bζwζ ∗ ) k = Z | b ( t ) | (cid:12)(cid:12) η tζ (1 ⊗ w ) (cid:12)(cid:12) dλ ( t ) . Theorem 4.9. Let A ⊂ M be a singular masa. Let v ∈ A be a Haar unitary generator of A . Supposethere exists a subsequence n k ( n k < n k +1 for all k ) such that for all y ∈ M , k v n k y − yv n k k → as k → ∞ . Then the measure ˜ η t is β -rigid for all β ∈ S , λ almost all t , where [ η ] is the left-right-measure of A .Proof. Let w be the Haar unitary generator of A that corresponds to the function [0 , ∋ t e πit .The map from L ∞ ([0 , , λ ) to itself, which sends v n to w n for n ∈ Z , implements a m.p. Borelisomorphism T : [0 , [0 , T × T implements an unitary U : L ( M ) L ( M ), whichpreserves the structure of L ( M ) as the natural A, A -bimodule (see Defn. 1.3). Standard densityarguments show that if ξ ∈ L ( M ), then k v n k ξ − ξv n k k → k → ∞ . So, we can assume v = w .We know that there is a nonzero vector ζ ∈ L ( M ) ⊖ L ( A ) such that η = η ζ . Therefore k E A ( v − n k ζv n k ζ ∗ ) − E A ( ζζ ∗ ) k → k → ∞ . Consequently, using similar arguments that areneeded to prove Lemma 4.8, we have, (cid:13)(cid:13) E A ( v − n k ζv n k ζ ∗ ) − E A ( ζζ ∗ ) (cid:13)(cid:13) = Z (cid:12)(cid:12) e − πin k t η t (1 ⊗ v n k ) − E A ( ζζ ∗ )( t ) (cid:12)(cid:12) dλ ( t ) → k → ∞ . Hence, there exists a further subsequence n k l and a subset E ⊂ [0 , 1] such that λ ( E ) = 0,and for t ∈ E c , e − πin kl t η t (1 ⊗ v n kl ) − E A ( ζζ ∗ )( t ) → l → ∞ . (4.2)Note that E A ( ζζ ∗ )( t ) = ˜ η t ([0 , < ∞ (see Lemma 2.2) almost everywhere λ .It is known that for almost every β ∈ [0 , 1] (with respect to λ ), the set of limit points of thesequence e − πin kl β contains a point of the form e πiα with α irrational (see Ch. 7 [13]). Thus byenlarging the null set E and renaming it to be E again, we conclude that e − πin klm t → e πiα t for t ∈ E c , α t irrational. The subsequence in the last statement depends on t . By a diagonal argument,it follows that for t ∈ E c , the measure ˜ η t is β -rigid for all β ∈ S (see Ch. 7 [13]). (cid:3) Remark . Examples of singular masas in the hyperfinite II factor can be constructed that satisfythe hypothesis of Thm. 4.9. There exist weakly mixing actions of a stationary Gaussian process thathas the desired properties (check § A is Cartan in R , then there is a centralizingsequence in A consisting of powers of some Haar unitary generator. This follows from Thm. 5.5 [12],Prop. 3.1 [14], Thm. 4 [31] and [4]. For example, consider the Cartan masa in the hyperfinite II factor which arises from a irrational rotation along the direction of the group.5. Examples of Singular Masas in the Free Group Factors In this section, we show that given any subset S of N , there are uncountably many pairwise nonconjugate singular masas in L ( F k ), k ≥ 2, with Puk´anszky invariant S ∪ {∞} . All examples exhibitedin this section are constructed from examples appearing in [8, 26]. For any masa A considered in thissection, we assume A = L ∞ ([0 , , λ ), where λ is the Lebesgue measure. If A ⊂ M is a masa and[ η ] is its left-right-measure , then we will most of the time assume that η (∆[0 , A ⊂ M ⊂ M ∗ N ( M, N are diffuse) as deduced in this sectionfrom results in [12], can also be deduced from Thm. 2.3 [8] or [19]. Corollary 5.1. Let k ∈ N ∞ and k ≥ . Let A ⊂ L ( F k ) be a masa. If A is freely complementedthen P uk ( A ) = {∞} and its left-right-measure is the class of product measure. In particular, A issingular.Proof. Follows directly from Lemma 5 . . 10 [8]. Singularity follows from Cor. 3.2 in[21]. (cid:3) Corollary 5.2. Let k ∈ N ∞ and k ≥ . Let A ⊂ L ( F k ) be a masa. Let A B L ( F k ) , where B isa subalgebra and B is freely complemented. ( i ) If the left-right-measure [ η B ] of the inclusion A ⊂ B is singular with respect to λ ⊗ λ , then P uk L ( F k ) ( A ) = P uk B ( A ) ∪ {∞} and the left-right-measure of the inclusion A ⊂ L ( F k ) is [ η B + λ ⊗ λ ] . ( ii ) If the left-right-measure [ η B ] of the inclusion A ⊂ B is [ λ ⊗ λ ] , then P uk L ( F k ) ( A ) = {∞} and theleft-right-measure of the inclusion A ⊂ L ( F k ) is [ λ ⊗ λ ] .Proof. Follows from Lemma 5 . . 10 [8]. (cid:3) Let T be a nonempty subset of N . Let T = { n k } with n < n < · · · . Define P T = α = { α n k } | T | k =1 : α n k > α n k +1 , < α n k < k, | T | X k =1 α n k = 1 . (5.1)For α, β ∈ P T , we say α = β if α n k = β n k for some k . Theorem 5.3. Let B ⊂ R be a singular masa such that the left-right-measure [ η R ] of the inclusion B ⊂ R is singular with respect to λ ⊗ λ . For each α ∈ P N , there exists a singular masa B α ⊂ L ( F ) with P uk L ( F ) ( B α ) = P uk R ( B ) ∪ {∞} . If α = β are any two elements of P N , then B α and B β arenot conjugate.Proof. Fix α ∈ P N . Let R α = ⊕ ∞ n =1 R . Equip R α with the faithful trace τ R α ( · ) = ∞ X n =1 α n τ R ( · ) , where τ R denotes the unique normal tracial state of R .Then B α = ⊕ ∞ n =1 B is a singular masa in the hyperfinite algebra R α and the latter is separable.The projections (0 ⊕ · · · ⊕ ⊕ ⊕ ⊕ · · · ), where 1 appears at the n -th coordinate, is a centralprojection p n of R α and it belongs to B α . The projections p n correspond to the indicator function ofmeasurable subsets F n ⊂ ([0 , , λ ) respectively, so that F n ∩ F m is a set of λ measure 0 for all n = m .By applying appropriate transformations, the left-right-measure of B ⊂ R can be transported toeach F n × F n , which we denote by [ η n ]. We also assume η n ( F n × F n ) = 1 for all n . Note that byfactoriality of R it follows that η n ( E × F ) > E × F ⊂ F n × F n suchthat λ ( E ) > , λ ( F ) > M , τ M ) = ( R α , τ R α ) ∗ ( R , τ R ). Then M is isomorphic to L ( F ) by a well known theoremof Dykema [7]. Then B α ⊂ L ( F ) is a singular masa by Thm. 2.3 [8]. The left-right-measure of theinclusion B α ⊂ L ( F ) is [ λ ⊗ λ + ∞ X n =1 n η n ]and P uk L ( F ) ( B α ) = ∪ n P uk ( R∗R ) ( B ) ∪ {∞} = P uk R ( B ) ∪ {∞} from Cor. 5.2 and Thm. 3.2 [8].Since automorphisms of II factors preserve the trace and orthogonal projections, the non conjugacyof B α and B β for α = β follows by considering the left-right-measures . Indeed, if B β = φ ( B α ) for someautomorphism φ of L ( F ) and α, β ∈ P N , then B α and B β would have identical bimodule structure.Therefore, there is a Borel isomorphism ˜ φ : [0 , [0 , 1] such that ˜ φ ∗ λ = λ and ( ˜ φ × ˜ φ ) ∗ [ η B α ] = [ η B β ],where η B α , η B β denote the left-right-measures of B α and B β respectively.Let F n , E n , n = 1 , , · · · , be the measurable partitions of [0 , 1] associated to the left-right-measures of B α and B β (as described above) respectively. Clearly, the class of the singular part of η B α will be INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 21 pushed forward to the class of the singular part of η B β . If we let χ E ′ n = φ ( χ F n ), then E ′ n = E k n mod λ for some k n . But the λ -measure of F n and hence E k n are strictly decreasing as n increases. Thus k n = n for all n . This completes the proof. (cid:3) Now we construct non conjugate singular masas in the free group factors which have the samemultiplicity. We will give a case by case argument. Case: { , ∞} : In Thm. 5.3, let B be the alternating Tauer masa A (1). Case: { , n, ∞} , n = 1: Consider the matrix groups G n = (cid:8) ( f x (cid:1) | f ∈ P n , x ∈ Q (cid:9) , H n = (cid:8) ( f 00 1 (cid:1) | f ∈ P n (cid:9) ⊂ G n , where(5.2) P ∞ = (cid:26) pq | p, q ∈ Z , p, q odd (cid:27) and P n = n f kn | f ∈ P ∞ , k ∈ Z o if n < ∞ , are subgroups of the multiplicative group of nonzero rational numbers. Then L ( G n ) is the hyperfiniteII factor R and L ( H n ) ⊂ L ( G n ) is a singular masa with Puk´anszky invariant { n } . The left-right-measure of the inclusion L ( H n ) ⊂ L ( G n ) is the class of product Haar measure λ b H n ⊗ λ b H n on b H n × b H n ,where b H n denotes the character group of H n (see Example 5.1 [26] and Example 6.2 [8]).As R⊗R ∼ = R , so L ( H n ) ⊗ A (1) is a singular masa in R from [28] (see also [3] and Thm. 5.15 [12]).Let A (1) = L ∞ ([0 , , λ ). The Puk´anszky invariant of the inclusion L ( H n ) ⊗ A (1) ⊂ R is { , n } fromThm. 2 . left-right-measure of the inclusion L ( H n ) ⊗ A (1) ⊂ R is the class of λ b H n ⊗ λ b H n ⊗ η + ˜∆ ∗ λ b H n ⊗ η + λ b H n ⊗ λ b H n ⊗ ˜∆ ∗ λ, on b H n × b H n × [0 , × [0 , η ] is the left-right-measure of the alternating Tauer masa restrictedto the off diagonal and ˜∆ is the map, that maps a set to its square by sending x ( x, x ) (see Prop.5.2 [8]). In this case, we need to specify the measures on the diagonals as they are necessary. Given α ∈ P N , replace the role of B in Thm. 5.3 by L ( H n ) ⊗ A (1) to construct a masa A α,n ⊂ L ( F ). Case: { n, ∞} , n = 1: Let H n ⊂ G n and H ∞ ⊂ G ∞ be as in the previous case. Then L ( H n × H ∞ )is a singular masa in L ( G n × G ∞ ) whose measure-multiplicity invariant is the equivalence class of( b H n × b H ∞ , λ b H n ⊗ λ b H ∞ , [ η ] , m ) , where η is the sum of( i ) Haar measure on ( b H n × b H ∞ ) × ( b H n × b H ∞ );( ii ) Haar measure on the subgroup D ∞ = { ( α, β , α, β ) | α ∈ b H n , β , β ∈ b H ∞ } ;( iii ) Haar measure on the subgroup D n = { ( α , β, α , β ) | α , α ∈ b H n , β ∈ b H ∞ } ;and where the multiplicity function on the off-diagonal is given by m ( γ ) = (cid:26) n, γ ∈ D n \ ∆( b H n × b H ∞ ) , ∞ , otherwise . This was calculated in § 6, [8]. Note that η contain singular summands, singular with respectto product Haar measure off the diagonal ∆( b H n × b H ∞ ). For each α ∈ P N , make a constructionanalogous to Thm. 5.3 with B replaced by L ( H n ) ⊗ L ( H ∞ ), to construct a masa A α,n ⊂ L ( F ). Notethat P uk L ( F ) ( A α,n ) = { n, ∞} from Thm. 3.2 and Lemma 5.7 [8]. The left-right-measure of theinclusion A α,n ⊂ L ( F ) is of the same form as discussed in the previous cases. Use Lemma 3.6, Thm.3.8 of [12] to decide non conjugacy of A α,n , A β,n whenever α = β ∈ P N . Case: S ∪ {∞} , S ⊆ N , 1 ∈ S and | S | ≥ 2: Write S = { n k : 1 = n < n < · · · } . Let P n and P ∞ be the subgroups of the multiplicative group of rational numbers as before. Let G n , n ≥ 1, be thematrix group G n = (cid:26)(cid:18) x y f 00 0 g (cid:19) : x, y ∈ Q , f ∈ P n , g ∈ P ∞ (cid:27) and H n the subgroup consisting of the diagonal matrices in G n . Then as noted in Example 5.2 of[26], G n is amenable and L ( G n ) ∼ = R . It is also true that L ( H n ) is a singular masa in L ( G n ) with Puk´anszky invariant { n, ∞} (see Prop. 2.5 [8]). Consider M n = L ( G n ) ⊗R ∼ = R and consider themasa A n = L ( H n ) ⊗ A (1). Then A n ⊂ M n is a singular masa with P uk M n ( A n ) = { , n, ∞} (see Thm.2.1 [26]). Fix α ∈ P S . Now consider M α = ⊕ n ∈ S M n and A α = ⊕ n ∈ S A n , where τ M α ( · ) = X n ∈ S α n τ M n ( · ) , where τ M n denotes the faithful normal tracial state of M n . Then M α ∗ L ( Z ) = M α ∗ L ( F ) ∼ = L ( F )[7], A α is a singular masa in L ( F ) and P uk L ( F ) ( A α ) = S ∪ {∞} from Thm. 3 . { p n } n ∈ S ⊂ A α with the property that P n ∈ S p n = 1 and τ L ( F ) ( p n ) = α n , such that the left-right-measure of the inclusion A α ⊂ L ( F ) has λ ⊗ λ as asummand and measures singular with respect to λ ⊗ λ on the squares p n × p n (here by abuse ofnotation we think of p n as a measurable set which corresponds to the projection p n ). The singularpart on p n × p n has the property that its ( π , λ ) , ( π , λ ) disintegrations are non zero almost every-where on p n . Non conjugacy of A α and A β for α = β follows easily from Lemma 3.6, Thm. 3.8 of [12]. Case: S ∪ {∞} , S ⊆ N , 1 S and | S | ≥ 2: Let G n , H n for n ∈ N ∞ be the groups defined in Eq.(5.2). Let M n = L ( G n × G ∞ ) and A n = L ( H n × H ∞ ) for n ∈ S . Fix α ∈ P S . Let M α,S = ⊕ n ∈ S M n and A α,S = ⊕ n ∈ S A n , where M α,S is equipped with the trace τ M α,S ( · ) = P n ∈ S α n τ M n ( · ), where τ M n denotes the faithful normal tracial state of M n . Replace the role of the masa A α in the previous caseby A α,S . We omit the details. Case: {∞} : Consider the hyperfinite II factor R with a singular masa A such that P uk R ( A ) = {∞} .Consider the inclusion B = ⊗ ∞ n =1 A ⊂ ⊗ ∞ n =1 R . Since up to isomorphism, there is one hyperfinite II factor, so B ⊂ R is a masa from Tomita’s theorem on commutants. Since P uk ( B ) = {∞} fromLemma 2.4 [33], so B is singular from Cor. 3.2 [21]. Clearly, Γ( B ) = 1. The left-right-measure of theinclusion B ⊂ R is singular to the product class from Thm. 4.4. Now apply Thm. 5.3.The above constructions lead to the following theorem. Theorem 5.4. Let S be an arbitrary ( could be empty ) subset of N . Let k ∈ { , , · · · , ∞} and let Γ be any countable discrete group. There exist uncountably many pairwise non conjugate singularmasas in L ( F k ∗ Γ) whose Puk´anszky invariant is S ∪ {∞} .Proof. We have already proved that there exist uncountably many pairwise non conjugate singularmasas { A α } α ∈ I , where I is some indexing set, in L ( F ) whose Puk´anszky invariant is S ∪ {∞} . Onehas isomorphisms [7] L ( F ) ∗ L ( F k − ∗ Γ) ∼ = L ( F k ∗ Γ) for k ≥ . For k ≥ 2, each A α is a singular masa in L ( F k ∗ Γ) [8]. Use Lemma 5.7, Prop. 5.10 [8] to decide thenon conjugacy of A α and A β when α = β , in the free product. (cid:3) Theorem 5.5. There exist non conjugate singular masas A, B in L ( F k ) , ≤ k ≤ ∞ , with samemeasure-multiplicity invariant.Proof. Let R = L ∞ ( Q n ∈ Z ( { , } , µ )) ⋊ Z , where µ ( { } ) = µ ( { } ) = and the action is Bernoullishift. Then the copy of Z gives rise to a masa A ⊂ R whose multiplicity is {∞} and whose left-right-measure is the class of product measure. This follows from the fact that the maximal spectraltype of Bernoulli action is Lebesgue measure and its multiplicity is infinite on the orthocomplementof constant functions and Prop. 3.1 [14]. Consequently, for k ≥ A ⊂ R ∗ ( k − ∗ r =1 R ) ∼ = L ( F k ) [7]is a singular masa whose left-right-measure is the class of product measure and whose multiplicityfunction is m ≡ ∞ , off the diagonal. Let B be any single generator masa of L ( F k ). The same holdsfor the single generator masas as well due to malnormality of group inclusions. A is not conjugate to B , as the former is not maximally injective, while the single generator masas are maximally injectivefrom Cor. 3.3 [18]. (cid:3) Remark . We end this paper with the following observation. The following example was con-structed by Smith and Sinclair in Example 5.1 of [26]. It produces an example of a m.p. dynamicalsystem which solves Banach’s problem with the group under consideration being Q × . Consider the INGULAR MASAS AND MEASURE-MULTIPLICITY INVARIANT 23 matrix groups G = (cid:8) ( f x ) : f ∈ Q × , x ∈ Q (cid:9) and H the subgroup of G consisting of diagonal matri-ces. Then L ( G ) is isomorphic to the hyperfinite II factor, and, L ( H ) is a singular masa in L ( G ).Note that H is a malnormal subgroup of G and G = N ⋊ H , where N = { ( x ) : x ∈ Q } . Matrixmultiplication shows that ( ) is a bicyclic vector of L ( H ). The left-right-measure of L ( H ) ⊂ L ( G )is the class of product measure (see Example 6.2 [8]). 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