Bimodules in crossed products and regular inclusions of finite factors
aa r X i v : . [ m a t h . OA ] J a n BIMODULES IN CROSSED PRODUCTS ANDREGULAR INCLUSIONS OF FINITE FACTORS
Jan Cameron ( ∗ ) Roger R. Smith ( ∗∗ ) Department of Mathematics Department of MathematicsVassar College Texas A&M UniversityPoughkeepsie, NY 12604 College Station, TX [email protected] [email protected]
Abstract
In this paper, we study bimodules over a von Neumann algebra M in two relatedcontexts. The first is an inclusion M ⊆ M ⋊ α G , where G is a discrete group acting ona factor M by outer automorphisms. The second is a regular inclusion M ⊆ N of finitefactors. In the case of crossed products, we characterize the M -bimodules X that liebetween M and M ⋊ α G and are closed in the Bures topology, in terms of the subsetsof G . We show that this characterization also holds for w ∗ -closed bimodules when G has the approximation property ( AP ), a class of groups that includes all amenable andweakly amenable ones. As an application, we prove a version of Mercer’s extensiontheorem for certain w ∗ -continuous isometric maps on X . We establish a similar theoremfor bimodules arising from regular inclusions of finite factors, which generalizes thecrossed product situation when G acts on a finite factor. In the final section we applythese ideas to provide new examples of singly generated finite factors. Key Words: von Neumann algebra, crossed product, bimoduleAMS Classification: 46L10, 46L06( ∗ ) JC was partially supported by an AMS-Simons research travel grant.( ∗∗ ) RS was partially supported by NSF grant DMS-1101403. Corresponding author.1 Introduction
The starting point for this paper is a theorem due to H. Choda [7] which describes the vonNeumann algebras which lie between a factor M and its crossed product M ⋊ α G by a discretegroup G acting on M by outer automorphisms. Choda proved that an intermediate vonNeumann algebra N for which there exists a normal conditional expectation E : M ⋊ α G → N must have the form M ⋊ α H for a subgroup H of G . When M is type II , the existenceof E is immediate and so the intermediate von Neumann algebras are precisely of the form M ⋊ α H for subgroups H of G . The result for other types of factors was improved in [22],where it was shown that such conditional expectations always exist, at least when M hasseparable predual. It is natural to ask whether the intermediate w ∗ -closed M -bimodulescan be characterized in a similar way. Each unital subset S of G gives rise to a w ∗ -closed M -bimodule as X S = span w ∗ { mg : m ∈ M, g ∈ S } , so the question is whether all w ∗ -closed M -bimodules occur in this way. The first part of the paper is devoted to this problem, and weare able to provide a complete answer when we work with a different topology introduced byBures in [4]. Our techniques are largely based on Fourier series in the crossed product and, asnoted by Mercer [27], the Bures topology is the correct one for understanding the convergenceof such series. We establish a correspondence between the Bures closed M -bimodules andthe subsets of G , and recapture the results on intermediate von Neumann algebras abovewithout separability restrictions. The Bures closed M -bimodules are all w ∗ -closed, but thereverse statement is open. However, for groups with a finite Haagerup constant (a class ofgroups known as the weakly amenable groups, which contains all amenable groups) the twoclasses of bimodules coincide.Bimodules over subalgebras of von Neumann algebras have been studied previously invarious contexts. For example, the case of an inclusion A ⊆ M for a Cartan subalgebra A wasexamined in [29], but unfortunately the arguments for the Spectral Theorem for Bimodulescontained a gap, as pointed out in [1]. A proof for the case when the containing von Neumannalgebra M is amenable was given in [18]. A full rehabilitation of the Spectral Theorem forBimodules was then attempted in [28], but this regrettably also contained an error. Aslightly different formulation of the result was found recently in [6] which characterized theBures closed A -bimodules in M rather than the w ∗ -closed ones, but the status of the originalformulation of the Spectral Theorem for Bimodules remains uncertain. There are certainly w ∗ -closed subspaces that are not Bures closed, suggesting that the same phenomenon mightoccur for bimodules, but no such examples are known. We also note that bimodules havebeen studied in the related context of tensor products of von Neumann algebras by Kraus[26]. If N is a factor satisfying a technical condition called the weak ∗ operator approximationproperty and R is a von Neumann algebra, then he was able to characterize the σ -weaklyclosed N -bimodules of N ⊗ R as those subspaces of the form N ⊗ T where T is a σ -weaklyclosed subspace of R .If X is a w ∗ -closed unital subspace of a von Neumann algebra which it generates, thenwe may consider the inclusion X ⊆ W ∗ ( X ). An old question, dating back at least to earlywork of Arveson [2, 3], asks whether w ∗ -continuous unital isometries on X extend to ∗ -automorphisms of W ∗ ( X ). This has been studied in [11, 6], the latter paper concentratingon the situation A ⊆ X ⊆ M where A is Cartan in M and X is a w ∗ -closed A -bimodulethat generates M . Mercer [28] claimed that isometric unital maps on X that respected the2imodule structure extended to ∗ -automorphisms of M , but the proof was flawed. If sucha result were true then the original map would have to be w ∗ -continuous, so the correctformulation of this theorem is to assume w ∗ -continuity, and the extension result was provedin [6] under this hypothesis. In this paper we prove a version of Mercer’s theorem in thecontext of bimodules over a von Neumann factor.The structure of the paper is as follows. Section 2 recalls the properties of crossedproducts by discrete groups, and also some facts about weakly amenable groups. Section 3is devoted to a brief discussion of the Bures topology and its relationship to the w ∗ -topology.Section 4 deals with M -bimodules X for an inclusion M ⊆ M ⋊ α G , and characterizes thosethat are closed in the Bures topology as being in bijective correspondence with the unitalsubsets S of G via the map S span B { mg : m ∈ M, g ∈ S } . In the special case of a weakly amenable group, the map S span w ∗ { mg : m ∈ M, g ∈ S } also identifies subsets of G with the w ∗ -closed M -bimodules. Our techniques for provingthese and subsequent results depend crucially on a theorem of Christensen and Sinclair [9]which, in the form used here, constructs a projection from the space of completely boundedmaps on a von Neumann algebra M to the space of right M -module maps. Not only theexistence of this projection but also the method of constructing it will be important for thispaper (see Theorem 4.1).Section 5 examines M -bimodules Y inside the reduced C ∗ -algebra crossed product M ⋊ α,r G and which generate this C ∗ -algebra. We show that M ⋊ α,r G is the C ∗ -envelope of suchbimodules Y , after showing that a nontrivial norm closed ideal in M ⋊ α,r G must havenontrivial intersection with M . These are the preliminary results needed for a version ofMercer’s theorem for M -bimodules X satisfying M ⊆ X ⊆ M ⋊ α G , proved in Theorem 6.4.Inclusions M ⊆ M ⋊ α G are particular examples of regular inclusions M ⊆ N of factors.Section 7 takes our earlier results on crossed products and extends them to regular inclusionsof finite factors. Formally, the results are the same but require somewhat different methodsin this more general context. The last section applies our results to a different problem, thequestion of single generation of von Neumann algebras. Here we establish some new classesof finite factors for which a positive answer can be given, based on an invariant introducedby Shen [34]. A ∗ -automorphism θ of a von Neumann algebra M is said to be properly outer if there doesnot exist a nonzero projection p ∈ M so that θ restricts to an inner ∗ -automorphism of pM p . In the case of a factor, this is equivalent to θ being outer , meaning that it is notinner. It is well known that proper outerness is equivalent to freeness [37, §
17] which isdefined by the condition that no nonzero element t ∈ M can satisfy the equation tx = θ ( x ) t for x ∈ M . We note that Lemma 5.1 below gives a useful variant on freeness that also3haracterizes proper outerness of ∗ -automorphisms. When we say that a group G acts by(properly) outer automorphisms on a von Neumann algebra M , we of course intend this toapply to each automorphism α g for g = e since α e is always the identity which is triviallyinner. The following discussion of crossed products and Lemma 2.1 do not require properlyouter actions, but such an assumption is necessary from Section 4 onwards.Let G be a discrete group that acts on a von Neumann algebra M ⊆ B ( H ) by automor-phisms α g , g ∈ G . We may define a faithful normal representation π : M → B ( H ⊗ ℓ ( G ))by π ( x )( ξ ⊗ δ h ) = α h − ( x ) ξ ⊗ δ h , x ∈ M, ξ ∈ H, h ∈ G, (2.1)and a unitary representation of the group G by λ g ( ξ ⊗ δ h ) = ξ ⊗ δ gh , ξ ∈ H, g, h ∈ G, (2.2)which is I ⊗ ℓ g for the left regular representation g ℓ g of G on ℓ ( G ). These operators arechosen so that λ g π ( x ) λ g − = π ( α g ( x )) , x ∈ M, g ∈ G, (2.3)and so there is no loss of generality in assuming that M , in its original representation, has itsautomorphisms α g spatially implemented by a unitary representation g a g , g ∈ G . Thisis standard theory, and we follow the presentation and notation of [40]. The crossed product M ⋊ α G is then the von Neumann algebra in B ( H ⊗ ℓ ( G )) generated by the operators π ( x )and λ g for x ∈ M and g ∈ G .If we define a unitary operator W ∈ B ( H ⊗ ℓ ( G )) by W ( ξ ⊗ δ h ) = a h ξ ⊗ δ h , ξ ∈ H, h ∈ G, (2.4)then W π ( x ) W ∗ = x ⊗ I, x ∈ M, (2.5)and W λ g W ∗ = a g ⊗ ℓ g , g ∈ G, (2.6)(see [40, Prop. 2.12]). Note that (2.5) and (2.6) imply that W ( M ⋊ α G ) W ∗ ⊆ B ( H ) ⊗ L ( G ),where L ( G ) is the group von Neumann algebra. The predual of L ( G ) is the Fourier algebra A ( G ) consisting of the functions g
7→ h ℓ g ξ, η i where ξ, η ∈ ℓ ( G ). A completely boundedmultiplier is a function f on G such that f · A ( G ) ⊆ A ( G ) and the associated multiplicationoperator M f is completely bounded. The space of completely bounded multipliers is denotedby M ( A ( G )) and is a Banach space in the cb -norm. In [20] it is shown that M ( A ( G )) is adual space with predual denoted by Q ( G ), and G is said to have the approximation property ( AP ) if the constant function 1 is in the w ∗ -closure of the space of finitely supported functionson G . This is a large class of groups that contains all amenable discrete groups and also theweakly amenable groups of [10].The operators in part (vi) of the following lemma are only given a concrete description onthe generators of M ⋊ α G . The extension to general elements will be established in equation(4.21).In an earlier version of the paper, this lemma was formulated for weakly amenable discretegroups. We thank Jon Kraus for pointing out that essentially the same proof works for groupswith the AP . 4 emma 2.1. Let G be a discrete group with the AP acting by automorphisms α g , g ∈ G ,on a von Neumann algebra M ⊆ B ( H ) . Then there exist a net ( f γ ) γ ∈ Γ of finitely supportedfunctions on G and a net ( T γ : M ⋊ α G → M ⋊ α G ) γ ∈ Γ of normal maps with the followingproperties: (i) For each γ ∈ Γ , M f γ is completely bounded. (ii) For each g ∈ G , lim γ f γ ( g ) = 1 . (iii) For each function h ∈ A ( G ) , lim γ k M f γ h − h k = 0 . (iv) For each γ ∈ Γ , T γ is completely bounded. (v) For each y ∈ M ⋊ α G , lim γ T γ ( y ) = y in the w ∗ -topology. (vi) For each x ∈ M and g ∈ G , T γ ( π ( x ) λ g ) = f γ ( g ) π ( x ) λ g .Proof. The existence of such a net of functions satisfying (i)-(iii) is [20, Theorem 1.9]. Thepredual of B ( H ) ⊗ L ( G ) is the operator space projective tensor product B ( H ) ∗ ˆ ⊗ op A ( G ), [14].Thus the operators I ⊗ M f γ on B ( H ) ∗ ˆ ⊗ op A ( G ) are completely bounded and converge to I ⊗ I in the point norm topology. Let S γ denote the normal map ( I ⊗ M f γ ) ∗ : B ( H ) ⊗ L ( G ) → B ( H ) ⊗ L ( G ). Then lim γ S γ = I in the point w ∗ -topology and each S γ is completely bounded.Using the operator W of (2.4), we define T γ : M ⋊ α G → W ∗ ( B ( H ) ⊗ L ( G )) W by T γ ( y ) = W ∗ S γ ( W yW ∗ ) W, γ ∈ Γ , y ∈ M ⋊ α G. (2.7)Clearly each T γ is a normal map. From the duality between A ( G ) and L ( G ), we see that M f γ ( ℓ g ) = f γ ( g ) ℓ g , and so S γ ( x ⊗ ℓ g ) = x ⊗ f γ ( g ) ℓ g , x ∈ B ( H ) , g ∈ G. (2.8)Thus T γ ( π ( x ) λ g ) = W ∗ S γ ( xa g ⊗ ℓ g ) W = W ∗ ( xa g ⊗ f γ ( g ) ℓ g ) W = f γ ( g ) π ( x ) λ g , γ ∈ Γ , x ∈ M, g ∈ G, (2.9)using (2.5)-(2.8). Then normality shows that T γ maps M ⋊ α G to itself and establishes (vi).The remaining properties (iv) and (v) follow from the corresponding properties of S γ .Henceforth we will ease notation and write generating elements of M ⋊ α G as xg , x ∈ M , g ∈ G , subject to the relation gx = α g ( x ) g . 5 The Bures topology
In this section we investigate a topology, introduced by Bures [4], which was shown byMercer [28] to be the appropriate one for questions of convergence of Fourier series in crossedproducts. The setting is an inclusion M ⊆ N of von Neumann algebras for which there existsa faithful, normal conditional expectation E : N → M . Such a conditional expectationalways exists when N is the crossed product M ⋊ α G of M by a discrete group G and this isthe primary situation where we will use this topology. The Bures topology , or B - topology aswe will call it, is defined by the seminorms x φ ( E ( x ∗ x )) / , x ∈ N , where φ ranges overthe normal states of M . Thus a set U ⊆ N is B -open if, and only if, given x ∈ U , thereexist normal states φ , . . . , φ n ∈ M ∗ and ε > ∩ ni =1 (cid:8) y : φ i ( E (( y − x ) ∗ ( y − x ))) < ε (cid:9) ⊆ U. (3.1)The objective of the following two lemmas is to establish the relationship of the B -topologyto the w ∗ -topology. Lemma 3.1.
Let M ⊆ N be von Neumann algebras with a faithful, normal conditionalexpectation E : N → M . Let K be a convex subset of N . If K is B -closed, then it is w ∗ -closed.Proof. Suppose not. Then there exists x ∈ K w ∗ \ K . By hypothesis, the complement K c is B -open and so there exist normal states φ , . . . , φ n ∈ M ∗ and ε > ∩ ni =1 (cid:8) y : φ i ( E (( y − x ) ∗ ( y − x ))) < ε (cid:9) ⊆ K c . (3.2)Define a normal state φ ∈ M ∗ as the average of the φ i ’s, and let δ = ε/ √ n . Then (3.2)implies that (cid:8) y : φ ( E (( y − x ) ∗ ( y − x ))) < δ (cid:9) ⊆ K c . (3.3)On N , define a semi-inner product by h x, y i = φ ( E ( y ∗ x )) , x, y ∈ N, (3.4)and let X ⊆ N be the subspace of null vectors. Then the quotient space N/X has an inducedinner product and we write H for the Hilbert space completion. Let ˆ K be the image of K in H and note that (3.3) gives d ( x + X, ˆ K ) ≥ δ. (3.5)By Hahn-Banach separation there exist real constants α and β with β > ξ ∈ H so that Re h x + X, ξ i ≥ α + β > α ≥ Re h k + X, ξ i , k ∈ K. (3.6)Define a linear functional ψ : N → C by ψ ( x ) = h x + X, ξ i , x ∈ N. (3.7)Then (3.6) becomes Re ψ ( x ) ≥ α + β > α ≥ Re ψ ( k ) , k ∈ K, (3.8)6nd so it suffices to show that ψ ∈ N ∗ in order to reach a contradiction. Choose a sequence y n ∈ N , n ≥
1, so that y n + X → ξ in H and k y n + X k H ≤ k ξ k H . Define ψ n ∈ N ∗ by ψ n ( x ) = φ ( E ( y ∗ n x )), x ∈ N . Then, for x ∈ N , | ψ n ( x ) − ψ ( x ) | = |h x + X, ( y n + X ) − ξ i| ≤ k x k H k ( y n + X ) − ξ k H ≤ k x k k ( y n + X ) − ξ k H , (3.9)showing that k ψ n − ψ k →
0. Thus ψ ∈ N ∗ , completing the proof. Lemma 3.2.
Let M ⊆ N be von Neumann algebras with a faithful, normal conditionalexpectation E : N → M . Let Y ⊆ N be a norm bounded subset. If Y is w ∗ -closed, then it is B -closed.Proof. Suppose that x ∈ N lies in the B -closure of Y . Then there is a net ( x λ ) λ ∈ Λ in Y converging to x in the B -topology, and w ∗ -compactness allows us to drop to a subnet whichalso converges in the w ∗ -topology to some y ∈ Y. We will show that x = y . Given an arbitrary normal state φ ∈ M ∗ , and z ∈ N , define φ z ∈ N ∗ by φ z ( x ) = φ ( E ( zx )) , x ∈ N, (3.10)and denote by L the span of these functionals in N ∗ as φ and z vary. We claim that L isnorm dense in N ∗ . Indeed, if it were not, then there would exist a nonzero x ∈ N such that φ ( E ( zx )) = 0 (3.11)for every normal state φ ∈ M ∗ and z ∈ N. Taking z = x ∗ and letting φ vary over M ∗ , thefaithfulness of E would then imply that x = 0 , a contradiction. Thus, L is dense in N ∗ .If φ is a normal state in M ∗ and z ∈ N , then applying the Cauchy-Schwarz inequality tothe state φ ◦ E gives | φ ( E ( z ( x λ − x ))) | ≤ φ ( E ( zz ∗ )) / φ ( E (( x λ − x ) ∗ ( x λ − x ))) / ≤ k z k φ ( E (( x λ − x ) ∗ ( x λ − x ))) . (3.12)Since x λ → x in the B -topology, this giveslim λ ψ ( x λ − x ) = 0 , ψ ∈ L, (3.13)and the uniform bound on k x λ k implies that x λ → x in the w ∗ -topology from (3.13) andthe norm density of L in N ∗ . Thus x = y , as desired. Remark . (i) Lemmas 3.1 and 3.2 jointly imply that the w ∗ - and B -closures of normbounded convex sets are equal, so differences in these topologies, at least for convex sets,appear only for unbounded sets. In principle, the B -topology on N is dependent on thechoice of the von Neumann subalgebra M and of the normal conditional expectation ontoit. These lemmas show, however, that for norm bounded convex sets the various possible B -topologies give the same closure. In the two situations where we will need the B -topology,the choices of subalgebras and conditional expectations will be canonical.7ii) The converse of Lemma 3.1 and the conclusion of Lemma 3.2 for general w ∗ -closedsets are both false, as the following example shows. Let N = L ∞ [0 ,
1] and M = C I , anddefine E : N → M by E ( f ) = (cid:18)Z f ( t ) dt (cid:19) I, f ∈ L ∞ [0 , . (3.14)In this case, the B -topology on L ∞ [0 ,
1] is the restriction of the k · k -norm topology on L [0 , w ∗ -continuous functional φ on L ∞ [0 ,
1] by integration against t − / ∈ L [0 , \ L [0 ,
1] and let K be its kernel. Then K is k · k -dense in L [0 ,
1] since any nonzerovector orthogonal to K would have to be a multiple of t − / which is impossible. Thus, forexample, the constant function 1 is not in the w ∗ -closed set K but lies in its B -closure. ∗ -closed bimodules In this section, we study the structure of certain w ∗ -closed bimodules arising in the crossedproduct setting. In particular, given an outer action of a countable, discrete group G on afactor M , any subset S of G gives rise to a B -closed M -bimodule X S = span B { xg : x ∈ M, g ∈ S } ⊆ M ⋊ α G. It is natural to ask whether these are all the B -closed M -bimodules in the crossed product;that this is indeed the case is the main result of this section.An important tool for our investigation here is a result of Christensen and Sinclair [9]which we state for the reader’s convenience after establishing some notation. For a vonNeumann algebra M , let L cb ( M, M ) denote the space of completely bounded linear maps φ : M → M , while L cb ( M, M ) M is the subspace of right M -module maps, those that satisfy φ ( xm ) = φ ( x ) m for x, m ∈ M . These of course have a very simple form, φ ( x ) = tx for x ∈ M and an element t = φ ( I ) ∈ M . Each collection β = ( m j ) j ∈ J from M satisfying P j ∈ J m ∗ j m j = I induces a map φ φ β on L cb ( M, M ), where φ β ( x ) = P j ∈ J φ ( xm ∗ j ) m j , x ∈ M. (4.1)The complete boundedness of φ guarantees that φ β is well defined and satisfies k φ β k cb ≤k φ k cb . For a fixed element c ∈ M , we denote by φ c the map φ c ( x ) = φ ( cx ) for x ∈ M .In the following theorem, the first two parts constitute a special case of [9, Theorem 3.3],(see also [35, Theorem 1.7.4]), while the third part is just an observation that we will needsubsequently. Theorem 4.1.
Let M be a von Neumann algebra. (i) There exists a contractive projection ρ : L cb ( M, M ) → L cb ( M, M ) M . (ii) There is a net ( β ) so that ρφ ( x ) = w ∗ − lim β φ β ( x ) , φ ∈ L cb ( M, M ) , x ∈ M. (4.2)8iii) If c, d ∈ M and φ ∈ L cb ( M, M ) satisfy φ ( cx ) = dφ ( x ) , x ∈ M, (4.3) then ρφ c = dρφ. (4.4) Proof.
As noted above, only the last part requires proof, so suppose that φ ∈ L cb ( M, M )satisfies (4.3). For each β = ( m j ) j ∈ J , P j ∈ J φ c ( xm ∗ j ) m j = P j ∈ J dφ ( xm ∗ j ) m j , x ∈ M, (4.5)so using (ii) and taking w ∗ -limits in (4.5) gives (4.4) as required.We now apply this result to specific maps that will occur subsequently. Lemma 4.2.
Let M be a von Neumann algebra, let α be a properly outer automorphism of M and let a ∈ M be fixed. If φ : M → M is the completely bounded map defined by φ ( x ) = α ( x ) a, x ∈ M, (4.6) then ρφ = 0 .Proof. Since ρφ is a right M -module map, there exists t ∈ M so that ρφ ( x ) = tx, x ∈ M. (4.7)For each y ∈ M , φ ( yx ) = α ( y ) α ( x ) a = α ( y ) φ ( x ) , x ∈ M, (4.8)so from Theorem 4.1 (iii) we obtain ρφ y ( x ) = α ( y ) ρφ ( x ) = α ( y ) tx, x ∈ M. (4.9)From Theorem 4.1 (ii), ρφ y ( x ) = ρφ ( yx ) , x ∈ M, (4.10)so α ( y ) tx = tyx, x, y ∈ M. (4.11)Put x = 1 in (4.11) to obtain that α ( y ) t = ty for y ∈ M . This implies that t = 0 since α isproperly outer, and so ρφ = 0 from (4.7).If G is a discrete group acting on a factor M by outer automorphisms, then each x ∈ M ⋊ α G has a Fourier series x = P g ∈ G x g g , where convergence takes place in the B -topologyfor the standard normal conditional expectation E : M ⋊ α G → M [28]. The g -coefficients x g ∈ M are given by x g = E ( xg − ). Theorem 4.3.
Let a discrete group G act on a factor M by outer automorphisms α g , g ∈ G ,and let X ⊆ M ⋊ α G be an intermediate M -bimodule which is either closed in the B -topologyor the w ∗ -topology. If g ∈ G is such that there exists x ∈ X with a nonzero g -coefficient,then g ∈ X . roof. By Lemma 3.1, it suffices to consider the case when X is w ∗ -closed. Suppose that x ∈ X has a nonzero g -coefficient. Multiplying on the left by x ∗ g , we may assume that x g ≥
0, and a further left multiplication by a suitable element of M allows us to assumethat x g is a nonzero projection p . Since M is a factor, there exist partial isometries ( v k ) k ∈ K so that v ∗ k v k ≤ p and P k ∈ K v k v ∗ k = I , so P k ∈ K v k xα g − ( v ∗ k ) ∈ X (since α g − is completelybounded), and the g -coefficient of this element is I . Thus we may assume that x g = I .Let β = ( m j ) j ∈ J be the net from Theorem 4.1, pre- and post-multiply x by m ∗ j and α g − ( m j ) and sum over j to obtain elements x β ∈ X given by x β = P j ∈ J m ∗ j xα g − ( m j ) . (4.12)The g -coefficient of x β is P j ∈ J m ∗ j x g α gg − ( m j ). For g = g , this is P j ∈ J m ∗ j m j = I , whilefor g = g we may write this as α gg − (cid:16)P j ∈ J α g g − ( m ∗ j ) α g g − ( x g ) m j (cid:17) . (4.13)If we define φ g ∈ L cb ( M, M ) by φ g ( t ) = α g g − ( t ) α g g − ( x g ) , t ∈ M, (4.14)then the w ∗ -limit over β in (4.13) is α gg − ( ρφ g ( I )), using the w ∗ -continuity of α gg − , andthis is 0 by applying Lemma 4.2 to the outer automorphism α g g − and the fixed element a = α g g − ( x g ). Since k x β k ≤ k x k , we may drop to a subnet and assume in addition that x β → y ∈ X in the w ∗ -topology. If y has Fourier series P g ∈ G y g g , then y g = E ( yg − ) = lim β E ( x β g − ) (4.15)by w ∗ -continuity of E and so, from above, y g = I for g = g and is 0 otherwise. Thus g = y ∈ X as required. Theorem 4.4.
Let G be a discrete group acting by outer automorphisms α g , g ∈ G , on afactor M . (i) There is a bijective correspondence between subsets S of G and B -closed M -bimodulesof M ⋊ α G given by S X S := span B { xg : x ∈ M, g ∈ S } . (4.16)(ii) If G has the AP , then the collections of B -closed and w ∗ -closed bimodules coincide.Proof. (i) Clearly the empty set corresponds to the M -bimodule { } . Now let S and S be distinct nonempty subsets of G and suppose without loss of generality that g ∈ S butthat g / ∈ S . For a finite subset F ⊆ S , consider a sum P g ∈ F x g g . Then E (cid:16)(cid:16) g − P g ∈ F x g g (cid:17) ∗ (cid:16) g − P g ∈ F x g g (cid:17)(cid:17) = I + P g ∈ F α g − ( x ∗ g x g ) ≥ I (4.17)and so g / ∈ span B { xg : x ∈ M, g ∈ S } . Thus the map of (4.16) is injective.10uppose that X is a B -closed M -bimodule. Then, by Theorem 4.3, X contains eachgroup element that appears with a nonzero coefficient in the Fourier series of some elementof X . Let S denote the set of such elements. Then S X ⊆ X . If x ∈ X has Fourier series P g ∈ G x g g then, restricting to the terms with nonzero coefficients, we obtain that the finitepartial sums lie in X S and converge to x in the B -topology by [28]. This shows that X ⊆ X S ,proving equality.(ii) Suppose now that G has the AP . By Lemma 3.1, each B -closed M -bimodule is w ∗ -closed so it is only necessary to prove the converse. Let X ⊆ M ⋊ α G be a w ∗ -closed M -bimodule. Let S be the set of group elements that appear with a nonzero coefficient inthe Fourier series of some element of X and note that S ⊆ X by Theorem 4.3. Consideran element y in the B -closure of X with Fourier series P g ∈ G y g g , and suppose that y g = 0for some g ∈ G \ S . Choose a normal state φ ∈ M ∗ such that φ ( α − g ( y ∗ g y g )) > ε >
0. Let x = P g ∈ G x g g be an arbitrary element of X . By construction of S , x g = 0, and so the e -coefficient of ( x − y ) ∗ ( x − y ) is at least g − y ∗ g y g g = α − g ( y ∗ g y g ) . (4.18)Thus φ ( E (( x − y ) ∗ ( x − y ))) ≥ ε > . (4.19)Since x ∈ X was arbitrary, this gives the contradiction that y is not in the B -closure of X and we conclude that the nonzero coefficients of y can only occur for g ∈ S .Now let ( f γ ) γ ∈ Γ and ( T γ ) γ ∈ Γ be the functions and operators of Lemma 2.1, let z ∈ M ⋊ α G be arbitrary, and let x β = P g ∈ G x βg g be a net of finitely supported elements converging to z in the w ∗ -topology. Then, for each g ∈ G , w ∗ − lim β x βg = z g since x βg = E ( x β g − ). For each γ ∈ Γ, T γ ( x β ) = P g ∈ G f γ ( g ) x βg g (4.20)from Lemma 2.1 (vi), and since T γ ( x β ) → T γ ( z ) in the w ∗ -topology, we also obtain that T γ ( z ) = P g ∈ G f γ ( g ) z g g, z ∈ M ⋊ α G. (4.21)Applying this to y which is supported on S , we see that each T γ ( y ) is finitely supported on S and thus lies in X . Since T γ ( y ) → y in the w ∗ -topology, we conclude that y ∈ X , completingthe proof.As an immediate consequence we have the following. Corollary 4.5.
Let G be a discrete group acting by outer automorphisms α g , g ∈ G , on afactor M . The B -closed subalgebras of M ⋊ α G that contain M are in bijective correspondencewith the unital semigroups S ⊆ G by the map S span B { xg : x ∈ M, g ∈ G } . (4.22) If, in addition, G has the AP , then the sets of B -closed and w ∗ -closed subalgebras thatcontain M coincide. H ⊆ G is a subgroup, then the C ∗ -subalgebra of M ⋊ α G generated by M and H isdenoted by M ⋊ α,r H . This is the reduced C ∗ -algebra crossed product of M by H . Remark . If N is a von Neumann algebra satisfying M ⊆ N ⊆ M ⋊ α G then, by Corollary4.5, there is a subgroup H of G so that M ⋊ α,r H ⊆ N ⊆ N B = M ⋊ α H. (4.23)Taking w ∗ -closures gives N = M ⋊ α H . This was first established in [7] under the hypothesisthat there is a normal conditional expectation of M ⋊ α G onto N , and this assumption wasremoved in [22], although M was required to have a separable predual in order to make useof special hyperfinite subfactors which may not exist in the general case. (cid:3) ∗ -envelopes In this section, G is a discrete group acting by outer automorphisms α g , g ∈ G , on a factor M . Note that M is not type I since such factors have no outer automorphisms. We do notwish to place unnecessary restrictions on M , but some of the facts that we will require areonly valid for σ -finite factors so we begin with a brief discussion of such algebras.Recall that a factor M is σ - finite if every set of orthogonal projections in M is at mostcountable. This is equivalent to the existence of a faithful normal state φ on M , [38, Prop.II.3.19]. More generally, a projection p ∈ M is σ -finite if pM p is σ -finite. Using the charac-terization by faithful normal states, it is clear that any projection dominated by a σ -finiteprojection is itself σ -finite. The same characterization also makes it obvious that countableorthogonal sums of σ -finite projections are again σ -finite. If p and q are σ -finite projections,then σ -finiteness of p ∨ q follows from the general equivalence (see [38, Prop. V.1.6])( p ∨ q ) − q ∼ p − ( p ∧ q ) (5.1)showing that p ∨ q = ( p ∨ q − q ) + q is an orthogonal sum of two σ -finite projections. Moregenerally, if ( p n ) n ≥ is a countable set of σ -finite projections with lattice supremum p , then σ -finiteness of p follows by writing this projection as an orthogonal sum of σ -finite projections( q n ) n ≥ where q = p and, for n ≥ q n = ∨ { p i : 1 ≤ i ≤ n } − ∨ { p i : 1 ≤ i ≤ n − } . (5.2)If p is a σ -finite projection and G is countable, then it is dominated by a G -invariant σ -finiteprojection, namely ∨ { α g ( p ) : g ∈ G } . Our final observation is that any nonzero projection q dominates a nonzero σ -finite projection p : choose a normal state φ with φ ( q ) >
0, choose amaximal set of orthogonal subprojections ( q λ ) λ ∈ Λ with φ ( q λ ) = 0, and set p = q − P λ ∈ Λ q λ .We will be interested in the structure of norm closed ideals in factors and this can varywith cardinality. For example, the norm closed span of the σ -finite projections in a type IIIfactor M is a norm closed ideal which will be proper if the cardinality is sufficiently large,in contrast to the simplicity of such factors on separable Hilbert spaces.Our first objective is Lemma 5.2, a technical lemma that constitutes part of the proof ofthe subsequent theorem and which will be used again in Section 7.12 emma 5.1. Let α be a ∗ -automorphism of a von Neumann algebra Q and let a, b, c, d ∈ Q satisfy axb = cα ( x ) d, x ∈ Q. (5.3) If there exists x ∈ Q such that ax b = 0 , then α is not properly outer.Proof. Since ( ax b ) ∗ ( ax b ) = b ∗ x ∗ ( a ∗ ax b ), we see that a ∗ ax b = 0. This allows us tomultiply on the left in (5.3) by a ∗ so that we may assume that a ≥
0. Let p ε be the spectralprojection of a for the interval [ ε, ∞ ). For a sufficiently small choice of ε , p ε x b = 0 and sowe may multiply on the left in (5.3) by a suitable element which allows us to replace a by aprojection p ∈ Q . Thus we work with an equation of the form pxb = c α ( x ) d , x ∈ Q, (5.4)where px b = 0. As in [36, p.61], we may find a family of partial isometries v i ∈ Q so that v ∗ i v i ≤ p and the projections v i pv ∗ i are pairwise orthogonal and sum strongly to the centralsupport c ( p ) of p . Then p ≤ c ( p ) so c ( p ) x b = 0 and xc ( p ) b = c ( p ) xb = X i v i pv ∗ i xb = X i v i c α ( v ∗ i ) α ( x ) d , x ∈ Q, (5.5)the sums converging strongly due to the complete boundedness of α . This allows us toreplace p by 1 in (5.4) and to assume that we have an equation of the form xb = c α ( x ) d , x ∈ Q, (5.6)where x b = 0. Then c α ( x ) d = 0.Repeat this argument on the right on both sides of (5.6) to replace d by 1. The element b will change, but we will have reduced to an equation of the form xb = c α ( x ) , x ∈ M, (5.7)where x b = 0. Setting x = 1 gives c = b = 0, and we have shown that α is not free andhence not properly outer. Lemma 5.2.
Let Q be a von Neumann algebra with a faithful normal semifinite trace T r andlet α be a properly outer ∗ -automorphism of Q . Let p ∈ Q be a projection, let N = { p } ′ ∩ Q ,and denote the unitary group of N by U . If y ∈ Q ∩ L ( Q, T r ) and K := conv { uyα ( u ∗ ) : u ∈ U } , then ∈ K k·k .Proof. Since the elements of K are bounded in the operator norm by k y k and in the k · k -norm by k y k , [36, Lemma 9.2.1 (v)] shows that K k·k ⊆ Q ∩ L ( Q, T r ). Let t ∈ Q ∩ L ( Q, T r )be the element of minimal k · k -norm in K k·k . Then utα ( u ∗ ) = t, u ∈ U , (5.8)13o, by taking linear combinations, we see that xt = tα ( x ) , x ∈ N. (5.9)To derive a contradiction, suppose that t = 0. Then either pt = 0 or p ⊥ t = 0, and sincethe argument is the same in both cases we assume without loss of generality that the firstof these possibilities holds. Now pQp, p ⊥ Qp ⊥ ⊆ N , so substitution of elements of the firstof these subalgebras into (5.9) gives pxpt = tα ( p ) α ( x ) α ( p ) , x ∈ Q. (5.10)(We would use the second subalgebra if p ⊥ t were nonzero). Since pt = 0, the hypotheses ofLemma 5.1 hold with a = p , b = pt , c = tα ( p ), d = α ( p ), and x = 1. Thus α is not properlyouter, and this contradiction shows that t = 0 as required. Lemma 5.3.
Let Q ⊆ P be an inclusion of von Neumann algebras with a faithful normalcontractive conditional expectation E : P → Q , and let Q have a faithful normal semifinitetrace T r . Let Λ be a set with a distinguished element λ and let { v λ : λ ∈ Λ } ⊆ N ( Q ⊆ P } bea collection of normalizing unitaries such that v λ = 1 and the ∗ -automorphisms α λ ∈ Aut( Q ) given by α λ ( x ) = v λ xv ∗ λ , λ ∈ Λ , x ∈ Q, (5.11) are properly outer for λ = λ . Let A be a C ∗ -algebra with the following properties: (i) Q ⊆ A ⊆ P . (ii) For each λ ∈ Λ , there is an algebraic ideal I λ ⊆ Q with I λ = Q such that A is thenorm closure of span { I λ v λ : λ ∈ Λ } .Let J be a nonzero norm closed ideal in A . Then Q ∩ J = { } .Proof. An application of E to the equation α λ ( x ) v λ = v λ x, x ∈ Q, (5.12)shows that E ( v λ ) = 0 for λ = λ , otherwise one of these automorphisms would fail to beproperly outer.Let j ∈ J + be a nonzero positive element. By faithfulness, E ( j ) = 0, so multiplyingby a suitable element of Q allows us to assume that E ( j ) is a nonzero finite projection p ∈ Q . Since finite sums of terms from I λ v λ are norm dense in A , we may choose a finitesum P ′ λ ∈ Λ x λ v λ with x λ ∈ I λ (where P ′ denotes a finite sum) so that (cid:13)(cid:13) j − P ′ λ ∈ Λ x λ v λ (cid:13)(cid:13) < / . (5.13)Applying the expectation E , we obtain k p − x λ k < /
4, so if we add ( p − x λ ) to the finitesum then we have a new approximation (cid:13)(cid:13) j − P ′ λ ∈ Λ x λ v λ (cid:13)(cid:13) < / x λ is now p . Multiplying on the left by p , we mayfurther assume that each x λ lies in L ( M, T r ).Consider the set of all pairs ( j, P ′ λ ∈ Λ x λ v λ ), where j ∈ J , E ( j ) is a finite nonzero projec-tion p , x λ = p , x λ ∈ I λ ∩ L ( M, T r ) for λ ∈ Λ and the inequality (5.14) is satisfied. We havejust seen that there is at least one such choice, so from all the possibilities, select one forwhich the sum has minimal length, and call it ( j, P ′ λ ∈ Λ x λ v λ ). Our objective is to show thatthe length is 1, so to obtain a contradiction suppose that there is a λ = λ with x λ = 0.Fix ε > ε < T r ( p ) and (cid:13)(cid:13) j − P ′ λ ∈ Λ x λ v λ (cid:13)(cid:13) + √ ε < / . (5.15)Since α λ is an outer automorphism, we may apply Lemma 5.2 to find a finite set of unitaries (cid:8) u i ∈ N := { p } ′ ∩ Q : 1 ≤ i ≤ k (cid:9) and positive constants µ i summing to 1 so that (cid:13)(cid:13)(cid:13)P ki =1 µ i u i x λ α λ ( u ∗ i ) (cid:13)(cid:13)(cid:13) < ε. (5.16)Define a complete contraction T : A → A by T ( y ) = P ki =1 µ i pu i yu ∗ i , y ∈ A. (5.17)Then T ( j ) ∈ J , E ( T ( j )) is still p , and applying T to (5.15) gives a new approximation (cid:13)(cid:13) T ( j ) − P ′ λ ∈ Λ z λ v λ (cid:13)(cid:13) + √ ε < / . (5.18)The sum in (5.18) is still of minimal length, z λ = p since the u i ’s commute with p and wehave the additional features that k z λ k < ε and pz λ = z λ . Let q be the spectral projectionof z λ z ∗ λ for the interval [ ε, ∞ ). Then εq ≤ z λ z ∗ λ so εT r ( q ) ≤ T r ( z λ z ∗ λ ) = k z λ k < ε , (5.19)and thus T r ( q ) < ε < T r ( p ) . (5.20)Note also that q ≤ p , and that the projection q = p − q is nonzero since T r ( q ) > q is orthogonal to q , we have k q z λ z ∗ λ q k ≤ ε , so k q z λ k ≤ √ ε . Returningto (5.18) and multiplying on the left by q , we obtain (cid:13)(cid:13) q T ( j ) − P ′ λ ∈ Λ q z λ v λ (cid:13)(cid:13) < / − √ ε, (5.21)and it follows that (cid:13)(cid:13)(cid:13) q T ( j ) − P ′ λ = λ q z λ v λ (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13) q T ( j ) − P ′ λ ∈ Λ q z λ v λ (cid:13)(cid:13) + k q z λ k < / . (5.22)Note that q T ( j ) ∈ J has E ( q T ( j )) = q p = q and q z λ = q . Consequently (5.22) gives astrictly shorter sum with the same properties, a contradiction. Thus the shortest such sumhas length 1, so there exists j ∈ J and a nonzero projection p ∈ Q such that k j − p k < / Q of the quotient map onto A/J is not isometric, and so J ∩ Q = { } as desired. 15ecall that M ⋊ α,r G denotes the reduced C ∗ -algebra crossed product, a subalgebra of M ⋊ α G which is the norm closed span of the elements mg for m ∈ M and g ∈ G . Theorem 5.4.
Let G be a discrete group acting on a factor M by outer automorphisms α g , g ∈ G . Let J be a norm closed nonzero ideal in M ⋊ α,r G . Then J ∩ M = { } .Proof. We will prove this for countable groups and σ -finite factors and deduce the generalcase from this. We begin by handling the reduction, which we accomplish in two stages.Suppose that the result is true for countable groups and general factors.Let J be a nonzero norm closed ideal in M ⋊ α,r G . We may write G as the union ofan increasing net ( G λ ) λ ∈ Λ of countable groups, and M ⋊ α,r G is then the norm closure of ∪ λ ∈ Λ M ⋊ α,r G λ . Since J = { } , it must have nontrivial intersection with M ⋊ α,r G λ for λ sufficiently large and so J ∩ M = { } since each G λ is countable.Now suppose that the result is true for countable groups and σ -finite factors, and considera countable group G , a general factor M , and a nonzero norm closed ideal J in M ⋊ α,r G .Let j ∈ J be a nonzero element and write j = P g ∈ G x g g for the Fourier series in M ⋊ α G .At least one coefficient x g is nonzero, so multiplying by x ∗ g on the left and by g − onthe right allows us to assume that x e ≥ M allows us to assume that x e is a nonzero projection q . Choosea nonzero σ -finite subprojection p and a G -invariant σ -finite projection p that dominates p . Then pjp has Fourier series P g ∈ G px g p g and is nonzero since the e -coefficient is p .Moreover, all coefficients lie in the σ -finite factor p M p . A simple approximation argumentgives ( M ⋊ α,r G ) ∩ ( p M p ⋊ α G ) = p M p ⋊ α,r G , and so J ∩ M = { } since we are assumingthis for σ -finite factors. This completes the reduction and so it suffices to prove the resultwhen G is countable and M is σ -finite and we henceforth assume that we are in this case,although this is only necessary for the type III situation.If M is type III and K ⊆ M is a nonzero norm closed ideal, then spectral theory givesus a nonzero projection p ∈ K and this is equivalent to I − p since M is σ -finite, [38, Prop.V.1.39]. Then there is a partial isometry v ∈ M so that I − p = vpv ∗ , and we obtain I ∈ K and K = M . Thus σ -finite type III factors are simple, as are all type II factors. In bothcases M ⋊ α,r G is simple, [25], and thus J certainly contains M .It remains to consider the case of a σ -finite type II ∞ factor M acted upon by a group G .Such a factor has the form M ⊗ B ( H ) for a type II factor M , and H is separable otherwise M would not be σ -finite. Thus M has a faithful normal semifinite trace and Lemma 5.3applies to the inclusion M ⊆ M ⋊ α G . The result is then immediate from this lemma bytaking Q = M , P = M ⋊ α G , Λ = G , λ = e , v λ = g , I λ = M , and A = M ⋊ α,r G .We now use these results to show that M ⋊ α,r G is the C ∗ -envelope of certain operatorsubspaces of this C ∗ -algebra. Recall that a C ∗ -algebra A is said to be the C ∗ - envelope of aunital operator space X if there is a completely isometric unital embedding ι : X → A sothat ι ( X ) generates A , and if B is another C ∗ -algebra with a completely isometric unitalembedding ι ′ : X → B whose range generates B , then there is a *-homomorphism π : B → A so that π ◦ ι ′ = ι (which entails surjectivity of π ). Every unital operator space has a uniqueC ∗ -envelope denoted C ∗ env ( X ), [2, 3, 21]. 16 heorem 5.5. Let G be a discrete group acting on a factor M by outer automorphisms α g , g ∈ G , let S be a unital subset of G which generates G , and let X ⊆ M ⋊ α,r G be an operatorspace that contains { mg : m ∈ M, g ∈ S } . Then C ∗ env ( X ) = M ⋊ α,r G .Proof. Let ι : X → C ∗ env ( X ) be a completely isometric unital embedding and let ι ′ : X → M ⋊ α,r G be the identity embedding, whose range generates M ⋊ α,r G . From the definition,there exists a surjective *-homomorphism π : M ⋊ α,r G → C ∗ env ( X ) so that π ◦ ι ′ = ι . If π were not a ∗ -isomorphism, then it would have a nontrivial kernel J . By Theorem 5.4, J ∩ M would contain a nonzero element m , giving the contradiction ι ( m ) = π ( m ) = 0. Thus π is a ∗ -isomorphism, proving the result. In this section we will consider w ∗ -closed M -bimodules M ⊆ X ⊆ M ⋊ α G and isometric w ∗ -continuous maps θ : X → M ⋊ α G which respect the modular structure in the sense thatthe restriction of θ to M is a ∗ -automorphism and θ ( m xm ) = θ ( m ) θ ( x ) θ ( m ) , m , m ∈ M, x ∈ X. (6.1)Our objective is to show that such maps extend to ∗ -automorphisms of M ⋊ α G when X generates M ⋊ α G . We will require some preliminary lemmas. The first is well known toexperts, but we include it for completeness. Lemma 6.1.
Let G be a discrete group acting on a factor M by outer automorphisms α g , g ∈ G . Let w ∈ M ⋊ α G be a unitary that normalizes M . Then there exist g ∈ G and aunitary u ∈ M so that w = ug .Proof. Let θ be the ∗ -automorphism of M defined by θ ( x ) = w ∗ xw , x ∈ M , and let w = P g ∈ G w g g be the Fourier series of w . Then X g ∈ G xw g g = X g ∈ G w g α g ( θ ( x )) g, x ∈ M. (6.2)Thus xw g = w g α g ( θ ( x )) , x ∈ M, (6.3)so α g ◦ θ is inner whenever w g = 0. If g = g are elements of G for which w g and w g areboth nonzero, then α g ◦ θ and α g ◦ θ are both inner, implying that α g g − = α g ◦ θ ◦ θ − ◦ α g − is also inner, contradicting the hypothesis that this is outer. Thus there exists precisely one g ∈ G so that w g = 0, and so w = w g o g and w g is then seen to be a unitary, completingthe proof. Lemma 6.2.
Let G be a discrete group acting on a factor M by outer automorphisms α g , g ∈ G . Let θ be a ∗ -automorphism of M ⋊ α,r G satisfying θ ( M ) = M . Then θ extends to a ∗ -automorphism of M ⋊ α G . roof. For each g ∈ G , θ ( g ) is a unitary normalizer of M and so, by Lemma 6.1, has theform w g ρ ( g ) where w g is a unitary in M and ρ is a permutation of G with ρ ( e ) = e . Consideran element x = P g ∈ G x g g ∈ M ⋊ α G and choose, by the Kaplansky density theorem, auniformly bounded net ( x β ) β ∈ B from span { yg : y ∈ M, g ∈ G } converging to x in the w ∗ -topology. Drop to a subnet if necessary to assume that the net ( θ ( x β )) β ∈ B converges in the w ∗ -topology to an element y ∈ M ⋊ α G . If we write x β = P g ∈ G x βg g , a finite sum, then θ ( x β ) = P g ∈ G θ ( x βg ) w g ρ ( g ) and so the ρ ( g )-coefficient of y is w ∗ − lim β θ ( x βg ) w g = θ ( x g ) w g (6.4)since θ | M is w ∗ -continuous. In particular, y is independent of the choice of approximatingnet. Thus there is a well-defined map φ : M ⋊ α G → M ⋊ α G given by φ (cid:16)P g ∈ G x g g (cid:17) = P g ∈ G θ ( x g ) θ ( g ) , x = P g ∈ G x g g ∈ M ⋊ α G. (6.5)It is clear from its definition that φ is linear. If k x k ≤
1, then the net ( x β ) β ∈ B could have beenchosen to be a net of contractions, showing that k φ ( x ) k ≤
1. Thus φ is a linear contraction.By applying the same argument to θ − , we find that φ has a contractive inverse and isthus a surjective isometry.We now turn to the w ∗ -continuity of φ . By the Krein-Smulian theorem, in proving w ∗ -continuity of a linear map, it suffices to consider a uniformly bounded net ( x β ) β ∈ B convergingto 0 in the w ∗ -topology. Since the net ( φ ( x β )) β ∈ B is uniformly bounded, let y be any w ∗ -limitof a subnet ( φ ( x γ )) γ ∈ Γ . Then, writing x γ = P g ∈ G x γg g , we see that w ∗ − lim γ x γg = 0 for each g ∈ G , so the w ∗ -limit of the ρ ( g )-coefficients of φ ( x β ) is 0. Thus y = 0 and it followsthat φ ( x β ) → w ∗ -topology. Having shown that φ is w ∗ -continuous, we immediatelyconclude that it is a ∗ -automorphism of M ⋊ α G . Lemma 6.3.
Let G be a discrete group acting on a factor M by outer automorphisms α g , g ∈ G . Let M ⊆ X ⊆ M ⋊ α G be a w ∗ -closed M -bimodule and let θ : X → X be a w ∗ -continuous isometric surjective isomorphism so that θ is a ∗ -automorphism of M and θ ( m xm ) = θ ( m ) θ ( x ) θ ( m ) , m , m ∈ M, x ∈ X. (6.6) If x ∈ X has Fourier series P g ∈ G x g g , then θ ( x ) has Fourier series P g ∈ G θ ( x g ) θ ( g ) .Proof. Let S be the set of group elements that appear in the Fourier series of elements of X with a nonzero coefficient. By Theorem 4.3, { mg : m ∈ M, g ∈ S } ⊆ X . If g ∈ S , then gm = α g ( m ) g, m ∈ M, (6.7)so θ ( g ) θ ( m ) = θ ( α g ( m )) θ ( g ) , m ∈ M. (6.8)This implies that θ ( g ) θ ( g ) ∗ and θ ( g ) ∗ θ ( g ) lie in M ′ ∩ ( M ⋊ α G ) = C I , and since these elementshave norm 1 we conclude that θ ( g ) is a unitary. Moreover, by (6.8), θ ( g ) normalizes M soby Lemma 6.1, it has the form u g ρ ( g ) for a unitary u g ∈ M and a permutation ρ of S , with18 ( e ) = e and u e = I . Consequently P g ∈ G θ ( x g ) θ ( g ) is indeed a Fourier series although it isnot expressed in the customary way.We argue by contradiction, so suppose that there are elements x ∈ X and g ∈ S so that x has Fourier series P g ∈ G x g g but the ρ ( g )-term in the Fourier series of θ ( x ) is not θ ( x g ) θ ( g ).We first consider the special case where g = e . By subtracting x e from x , we may assumethat x has e -coefficient 0 but θ ( x ) has a nonzero e -coefficient. Multiplying on the left bya suitably chosen element of M , this e -coefficient may be taken to be a nonzero projection p ∈ M . Choose partial isometries ( v λ ) λ ∈ Λ so that P λ ∈ Λ v λ pv ∗ λ = I and let F ⊆ Λ be a finitesubset. For each z ∈ M ⋊ α G , define a contraction T F by T F ( z ) = P λ ∈ F θ − ( v λ ) zθ − ( v ∗ λ ).The net ( T F ( x )), indexed by the finite subsets of Λ, is uniformly bounded so, by dropping toa subnet if necessary, we may assume convergence in the w ∗ -topology to an element y ∈ X whose e -coefficient is 0. By w ∗ -continuity of θ , the net ( θ ( T F ( x ))) converges in the w ∗ -topology to θ ( y ) whose e -coefficient is P λ ∈ Λ v λ pv ∗ λ = I . Replacing x by y if necessary, wemay make the further assumption that the e -coefficient of θ ( x ) is 1.From Section 4, let β = ( m j ) j ∈ J be the set of collections of operators satisfying P j ∈ J m ∗ j m j = I and such that Theorem 4.1 holds. Define a complete contraction R β : M ⋊ α G → M ⋊ α G by R β ( z ) = P j ∈ J m ∗ j zm j , z ∈ M ⋊ α G, (6.9)where the sum converges in the w ∗ -topology. Then R β maps X to itself since X is w ∗ -closed,and the g -coefficient of R β ( z ) for z = P g ∈ G z g g is P j ∈ J m ∗ j z g α g ( m j ). Thus R β ( x ) has e -coefficient 0 while the g -coefficients for g = e tend to 0 in the w ∗ -topology over the net ( β ) asin the proof of Theorem 4.3. By dropping to a subnet, we may assume that w ∗ − lim β R β ( x )exists, and this must be 0. Thus w ∗ − lim β θ ( R β ( x )) = 0. From (6.9), θ ( R β ( x )) = P j ∈ J θ ( m ∗ j ) θ ( x ) θ ( m j ) (6.10)and the e -coefficient in this sum is 1. Thus θ ( R β ( x )) cannot converge to 0 in the w ∗ -topology,a contradiction.Returning to the general case, suppose that the desired formula fails at the ρ ( g )-term.Define a w ∗ -closed M -bimodule by Y = Xg − and define θ ′ : Y → Y by θ ′ ( xg − ) = θ ( x ) θ ( g ) ∗ , x ∈ X. (6.11)This map is isometric since θ ( g ) is a unitary. Now, for m ∈ M , θ ′ ( m ) = θ ′ ( mg g − ) = θ ( mg ) θ ( g ) ∗ = θ ( m ) θ ( g ) θ ( g ) ∗ = θ ( m ) , (6.12)while, for x ∈ X and m ∈ M , θ ′ ( mxg − ) = θ ( mx ) θ ( g ) ∗ = θ ( m ) θ ( x ) θ ( g ) ∗ = θ ′ ( m ) θ ′ ( xg − ) . (6.13)For multiplication on the right, first note that g α − g ( m ) = mg , m ∈ M, (6.14)yielding θ ( α − g ( m )) = θ ( g ) ∗ θ ( m ) θ ( g ) , m ∈ M. (6.15)19hus θ ′ ( xg − m ) = θ ′ ( xα − g ( m ) g − ) = θ ( xα − g ( m )) θ ( g ) ∗ = θ ( x ) θ ( α − g ( m )) θ ( g ) ∗ = θ ( x ) θ ( g ) ∗ θ ( m ) = θ ′ ( xg − ) θ ′ ( m ) , m ∈ M. (6.16)This puts us into the case of failure at g = e for the pair ( Y, θ ′ ) and we have already shownthat this cannot happen.We come now to the main result of the section for which we will need the concept of anorming subalgebra [32]. Let A ⊆ B be an inclusion of C ∗ -algebras. If X is an n × n matrixover B and R and C are respectively rows and columns of length n over A of unit norm,then k RXC k ≤ k X k . (6.17)If the supremum of the left hand side over rows and columns of unit norm equals k X k forevery matrix X of any size, then we say that A norms B . This has proved useful in settlingissues of automatic complete boundedness of bounded maps, and will be used below. Theorem 6.4.
Let M ⊆ X ⊆ M ⋊ α G be a w ∗ -closed M -bimodule that generates M ⋊ α G ,let θ : X → X be a w ∗ -continuous surjective isometry satisfying θ ( m xm ) = θ ( m ) θ ( x ) θ ( m ) , m , m ∈ M, x ∈ X, (6.18) and suppose that the restriction of θ to M is a ∗ -automorphism. Then θ extends uniquely toa ∗ -automorphism of M ⋊ α G .Proof. Let S be the set of g ∈ G so that g appears with a nonzero coefficient in the Fourierseries of an element of X . Let Y = span k·k { mg : m ∈ M, g ∈ S } , (6.19)which is contained in X by Theorem 4.3. For each g ∈ S , θ ( g ) is a normalizer of M andconsequently θ maps Y to itself. When M is type II ∞ or III, it norms M ⋊ α G , [32], whilethe same conclusion is reached for type II factors in [33]. Thus X and Y are normed by M and so θ is a complete isometry. Since X generates M ⋊ α G , S must generate G and so Y must generate M ⋊ α,r G as a C ∗ -algebra. By Theorem 5.5, the C ∗ -envelope of Y is M ⋊ α,r G and so, from the theory of C ∗ -envelopes, the embedding θ of Y into M ⋊ α,r G extends to a ∗ -automorphism φ of M ⋊ α,r G , and thence to a ∗ -automorphism of M ⋊ α G , also denotedby φ (see Lemma 6.2). For g ∈ G and m ∈ M , we have φ ( mg ) = θ ( mg ) = θ ( m ) θ ( g ) . (6.20)For each x ∈ X with Fourier series P g ∈ G x g g , Lemma 6.3 gives θ ( x ) = P g ∈ G θ ( x g ) θ ( g ). Ifwe apply this lemma again with M ⋊ α G in place of X and φ replacing θ , then we obtain φ ( z ) = P g ∈ G φ ( z g ) φ ( g ) for each z = P g ∈ G z g g ∈ M ⋊ α G . Since θ and φ agree on Y , wenow see that they agree on X . Thus we have found an extension to M ⋊ α G , and uniquenessis immediate since any two extensions will agree on X which contains a generating set for M ⋊ α G . Remark . Under the notation and hypotheses of Theorem 6.4, if X happens to be anintermediate subalgebra, then θ is the restriction of a ∗ -automorphism of M ⋊ α G and so isautomatically an algebraic isomorphism of X . It is perhaps surprising that such a conclusionfollows from the assumption of just being an isometry.20 Regular subfactors
In this section we investigate the situation of a regular inclusion M ⊆ N of II factors, where regular means that N is generated by the group N ( M ⊆ N ) of unitaries in N that normalize M . In particular, we will consider whether analogous results to those obtained in Section 6hold for a w ∗ -closed Q -bimodule X with Q ⊆ X ⊆ N , where Q is the von Neumann algebragenerated by M and its relative commutant C := M ′ ∩ N . We will state below a structuralresult for such inclusions, but first we establish some notation and prove two preliminarylemmas.If we let L denote the group generated by the commuting groups U ( M ) and U ( C ), then L is a normal subgroup of N ( M ⊆ N ) and there is a short exact sequence1 → L → N ( M ⊆ N ) → G → G is the quotient group N ( M ⊆ N ) /L which we view as a discrete group. Thequotient map has a cross section g u g , g ∈ G , which is defined pointwise by choosingrepresentatives of the cosets, and we always make the choice that u e = 1. For each pair g, h ∈ G , the unitaries u g u h and u gh differ by an element of L , and so there is an L -valued2-cocycle ω ( g, h ) on G × G such that u g u h = ω ( g, h ) u gh , g, h ∈ G. (7.2)In general, it is only possible to choose a homomorphic cross section when ω is a 2-coboundary.Each u g induces a ∗ -automorphism Ad u g on Q which we denote by α g , and (7.2) gives α g α h = Ad ( ω ( g, h )) α gh , g, h ∈ G, (7.3)so that the map g α g is not generally a homomorphism of G into Aut( Q ). The next twolemmas set out the properties of the u g ’s and α g ’s. The first of these is due to Kallman [24].We include a short proof for the reader’s convenience. We use the notation E P to denotethe trace preserving conditional expectation of N onto a von Neumann subalgebra P . Notethat E P is a P -bimodule map. Lemma 7.1.
Let M ⊆ N be an inclusion of II factors and let τ be the trace on N . Let C = M ′ ∩ N and let Q = W ∗ ( M, C ) . If θ is a ∗ -automorphism of Q which restricts to anouter ∗ -automorphism of M , then θ is properly outer on Q .Proof. Suppose that θ is not properly outer on Q and choose a nonzero element q ∈ Q suchthat qx = θ ( x ) q, x ∈ Q. (7.4)Since products of elements from M and C span a w ∗ -dense subspace of Q , we may choose m ∈ M and c ∈ C so that τ ( mcq ) = 0. Then E M ( mcq ) = 0 and the same is true for E M ( cq )since mE M ( cq ) = E M ( mcq ). Noting that c commutes with θ ( x ) for x ∈ M , we may multiply(7.4) by c on the left and apply E M to reach E M ( cq ) x = θ ( x ) E M ( cq ) , x ∈ M. (7.5)This contradicts the outerness of θ on M , so θ is properly outer on Q .21 emma 7.2. Let M ⊆ N be an inclusion of II factors, let C = M ′ ∩ N , let Q = W ∗ ( M, C ) and let L = U ( M ) U ( C ) . Let G = N ( M ⊆ N ) /L and let g u g ∈ N ( M ⊆ N ) be a crosssection for G with u e = 1 . (i) For g ∈ G \ { e } , α g := Ad u g is a properly outer ∗ -automorphism of Q and an outer ∗ -automorphism of M . (ii) α g α − h is properly outer on Q if and only if g = h . (iii) For g ∈ G \ { e } , E M ( u g ) = 0 and E Q ( u g ) = 0 .Proof. (i) Suppose that g = e . If α g is inner on M then there exists v ∈ U ( M ) such that α g = Ad v on M . Then v ∗ u g ∈ M ′ ∩ N = C , so v ∗ u g can be written as a unitary w ∈ U ( C ).Thus u g = vw ∈ L . This is a contradiction, and so α g is outer on M . It then follows fromLemma 7.1 that it is also properly outer on Q .(ii) Since there are unitaries v, w ∈ L such that u − h = vu h − and u g u h − = wu gh − , α g α − h differs from α gh − by an inner automorphism on Q , and the result follows from (i).(iii) If we apply E M to the equation u g x = α g ( x ) u g for x ∈ M , the result is E M ( u g ) x = α g ( x ) E M ( u g ). It follows from (i) that E M ( u g ) = 0 for g = e , otherwise α g would be inner on M . Similarly E Q ( u g ) = 0, otherwise the equation E Q ( u g ) x = α g ( x ) E Q ( u g ) for x ∈ Q wouldcontradict the proper outerness of α g on Q , by part (i).The twisted crossed product Q ⋊ α,ω G is defined to be the von Neumann algebra generatedby Q and the set of normalizers { u g : g ∈ G } . Such algebras were studied in [8]. Theconnection to regular inclusions of subfactors is exhibited by the following basic structuralresult from [5], in the spirit of the Feldman-Moore theory of Cartan masas [15, 16]. Lemma 7.3 (Theorem 4.6 of [5]) . Let M ⊆ N be a regular inclusion of II factors, and let Q denote the von Neumann algebra generated by M and M ′ ∩ N. Then there exists a discretegroup G , and a 2-cocycle ω : G × G → U ( Q ) such that N = Q ⋊ α,ω G . This result was used in [5] to show that, if M ⊆ N is a regular inclusion of II factors,then M norms N . Note that in this situation the von Neumann subalgebra Q is spatiallyisomorphic to the tensor product M ⊗ ( M ′ ∩ N ) . Throughout the remainder of this section, we maintain the following notation: M ⊆ N is a regular inclusion of II factors, C = M ′ ∩ N, and Q = M ⊗ C . Then, as in Lemma 7.3we have that N = Q ⋊ α,ω G for some discrete group G . We use X to denote a w ∗ -closed Q -bimodule with Q ⊆ X ⊆ N .It follows from Lemma 7.2 and the regularity of the inclusion M ⊆ N that the subspaces { Qu g : g ∈ G } are mutually orthogonal in L ( N ) and that this space is the direct sum ofthe k · k -closures of these subspaces. Thus each x ∈ N has a Fourier series x = P g ∈ G x g u g where x g ∈ Q is given by x g = E Q ( xu ∗ g ), just as in the crossed product case. This seriesconverges in k · k -norm. Lemma 7.4.
For each g ∈ G, the set J g = Q ∩ Xu ∗ g is a w ∗ -closed ideal in Q , and if x ∈ X has Fourier series P g ∈ G x g u g , then x g ∈ J g for each g ∈ G . roof. It is clear that J g is a w ∗ -closed ideal in Q . Fix g ∈ G and consider x = P g ∈ G x g u g ∈ X. Multiply by x ∗ g , so that x ∗ g x = P g ∈ G x ∗ g x g u g ∈ X . If x ∗ g x g ∈ J g then so is x g , so itsuffices to assume that x g ≥ . For each ε > p ε be the spectral projection of x g for [ ε, ∞ ) . Then x g p ε is invertiblein p ε Qp ε , so multiplying by ( x g p ε ) − , we may further replace x g by p ε . Thus we mayassume that x g is a projection p . If z is its central support then there are partial isometries v i so that z = P i v i pv ∗ i , so multiplying on the left by v i and on the right by α − g ( v ∗ i ) , and summing, we can then make x g the central projection z . Indeed if z ∈ J g , thenalso p = pz ∈ J g , so it suffices to take x g = z. Now for each unitary u ∈ Q, we have uxα − g ( u ∗ ) = P g ∈ G ux g α g ( α − g ( u ∗ )) u g ∈ X . Averaging over the unitary group of Q , weobtain an element h ∈ X whose u g -coefficient is the element h g of minimal k·k in the weakclosure of conv (cid:8) ux g α g ( α − g ( u ∗ )) : u ∈ U ( Q ) (cid:9) , which satisfies uh g = h g α g ( α − g ( u ∗ )) (7.6)for all u ∈ U ( Q ) . By proper outerness of the action of G , we must have h g = 0 unless g = g , in which case h g = x g = z. Then zu g ∈ X, so z ∈ J g as required. Lemma 7.5.
With notation as in Lemma 7.4, let Y be the linear span of the spaces { J g u g : g ∈ G } .If X generates N as a von Neumann algebra, then so does Y .Proof. First let X and X be subspaces of N , and suppose that there exist subspaces Y and Y of N so that X i is contained in the k·k − norm closure in N of Y i , i = 1 , . Weclaim that X X is contained in the closure of Y Y in this topology. Let x , x ∈ X and,given ε >
0, pick y ∈ Y so that k x − y k < ε/ (2 k x k ) . Then choose y ∈ Y so that k x − y k < ε/ (2 k y k ) . We then have k x x − y y k ≤ k x x − y x k + k y x − y y k < ε. (7.7)Extending to finite sums of such products, we see that X X ⊆ Y Y k·k ∩ N. This argumentextends by induction to finite products of subspaces X i ⊆ Y i k·k ∩ N, ≤ i ≤ k, so that X X · · · X k ⊆ Y Y · · · Y k k·k ∩ N. We apply the above when each X i is X or X ∗ and each Y i is Y or Y ∗ . If x ∈ X has Fourierseries x = P g ∈ G x g u g , then by Lemma 7.4, we have x g u g ∈ Y for each g ∈ G . Since thisseries converges in k·k -norm, we see that x ∈ Y k·k , so X ⊆ Y k·k . By the above argument,it follows that Alg(
X, X ∗ ) is contained in Alg( Y, Y ∗ ) k·k ∩ N . Since X generates N as a vonNeumann algebra, this says that Alg( Y, Y ∗ ) is k·k -dense in N , and so Y generates N . Lemma 7.6.
With the notation of Lemma 7.5, the C ∗ -envelope of Y is C ∗ ( Y ) .Proof. Let J be a nonzero ideal in the C ∗ -algebra C ∗ ( Y ). We may apply Lemma 5.3 with A = C ∗ ( Y ) to conclude that J ∩ Q = { } . Now let i : Y → C ∗ env ( Y ) be a completely isometric unital embedding and let i ′ : Y → C ∗ ( Y ) be the identity embedding. Then there is a surjective ∗ -homomorphism π : C ∗ ( Y ) → C ∗ env ( Y ) such that 23 ◦ i ′ ( y ) = i ( y ) , y ∈ Y. (7.8)Let J = ker π . If J is a nonzero ideal then it has a nonzero intersection with Q , so choose q ∈ Q with q = 0 and π ( q ) = 0 . Then also i ( q ) = 0 , a contradiction.We now turn to the main question of this section, in which we consider w ∗ -closed bimod-ules Q ⊆ X ⊆ N (in the notation above), and isometric w ∗ -continuous maps θ : X → X which restrict to ∗ -automorphisms of Q that fix the subfactor M . We aim to show that anysuch map extends to a ∗ -automorphism of the containing II factor N . Notice that this is ageneralization of our work above on crossed products, in the special case that the containingfactor is of type II . Lemma 7.7.
Let X be a w ∗ -closed Q -bimodule with Q ⊆ X ⊆ N , and let Y be as in Lemma7.5. Let θ : X → X be a w ∗ -continuous, surjective isometry whose restriction to Q is a ∗− automorphism such that θ ( M ) = M and θ ( q xq ) = θ ( q ) θ ( x ) θ ( q ) , q , q ∈ Q, x ∈ X. (7.9) Then θ maps Y onto Y . Moreover, θ is completely isometric.Proof. For each g ∈ G , J g is a w ∗ -closed ideal in Q , so is Qp for some central projection p ∈ Z ( Q ). Since θ ( ju g ) = θ ( j ) θ ( pu g ) for j ∈ J g , it suffices to prove that θ ( zu g ) ∈ Y whenever z ∈ Z ( Q ) and zu g ∈ X. Applying θ to the equation qzu g = zqu g = zu g α − g ( q ) , q ∈ Q, (7.10)we obtain θ ( q ) θ ( zu g ) = θ ( zu g ) θ [ α − g ( q )] , q ∈ Q. (7.11)Let P g ∈ G y g u g be the Fourier series of θ ( zu g ). Then, computing coefficients, we have θ ( q ) y g = y g α g [ θ ( α − g ( q ))] , q ∈ Q, g ∈ G. (7.12)Replacing q by θ − ( q ) , we get qy g = y g ( α g ◦ θ ◦ α − g ◦ θ − )( q ) q ∈ Q. (7.13)We claim that θ ( zu g ) has at most one nonzero Fourier coefficient y g . Suppose thatthere are distinct group elements g and g for which y g and y g are nonzero. Then theautomorphisms ψ = α g ◦ θ ◦ α − g ◦ θ − and ψ = α g ◦ θ ◦ α − g ◦ θ − (7.14)of Q both fail to be properly outer on Q from (7.13). As Q is generated by M and its relativecommutant, and by our hypotheses both M and M ′ ∩ N are invariant under θ , by Lemma7.1 this implies that both automorphisms are inner on M . Thus, the composition ψ ◦ ψ − = α g ◦ α − g = Ad ω ( g , g − ) ◦ α g g − (7.15)24s an inner automorphism of M . This is a contradiction. Thus, the Fourier series of θ ( zu g )has only one nonzero coefficient, so θ ( zu g ) lies in Y . To see that θ maps Y onto itself,replace q i by θ − ( q i ) , i = 1 ,
2, in the equation θ ( q xq ) = θ ( q ) θ ( x ) θ ( q ) , q , q ∈ Q, x ∈ X (7.16)and apply θ − to see that θ − ( q xq ) = θ − ( q ) θ − ( x ) θ − ( q ) , q , q ∈ Q, x ∈ X. (7.17)Thus, θ − satisfies the same properties that θ was assumed to have, so we may repeat theargument above to see that θ − also maps Y into itself. This completes the proof that θ maps Y onto itself.To prove that θ is a complete isometry, note that θ is a surjective isometry from X ontoitself. The subalgebra Q of X is norming [5], so we may use the Q -bimodularity properties of X and θ , and follow the proof of [31, Theorem 1.4] to show that θ is a complete contraction.The same argument can be used for θ − : X → X , so θ is a complete isometry from X ontoitself.Since θ : Y → Y is a surjective, complete isometry, by the universal property of C ∗ -envelopes and Lemma 7.6, θ extends to a ∗ -automorphism φ of C ∗ ( Y ). Our next objectiveis to show that this map extends further to a ∗ -automorphism of N , for which we will needthe following lemma, analogous to Lemma 7.7, and for which we require some notation. For g ∈ G , let K g ⊆ Q be the set of elements j so that ju g ∈ Alg(
Y, Y ∗ ). Then K g is an algebraicideal in Q and, by examining finite products from Y and Y ∗ , we see that K g is also the set ofelements which occur as the u g -coefficients in Fourier series of elements of Alg( Y, Y ∗ ). Sincethis algebra is w ∗ -dense in N , we also see that K g is w ∗ -dense in Q for each g ∈ G . Nowlet I g denote the norm closure of K g , which is an ideal in Q . Approximating elements ofC ∗ ( Y ) by elements from Alg( Y, Y ∗ ) shows that I g is also the set of elements which occur asthe u g -coefficients in the Fourier series of elements of C ∗ ( Y ) and ju g ∈ C ∗ ( Y ) for all j ∈ I g .The conclusion of the next result is similar to that of the previous one, but requires differentmethods since the ideals I g may not be w ∗ -closed in contrast to the ideals J g . Lemma 7.8.
With the above notation, for each g ∈ G there exists a unitary w g ∈ Q suchthat φ ( ju g ) = θ ( j ) w g u σ ( g ) , j ∈ I g , (7.18) where σ is an automorphism of G .Proof. We first establish that all elements φ ( ju g ), g ∈ G , j ∈ I g , have Fourier series of lengthat most one, so suppose that this fails for particular choices k ∈ G and j ∈ I k . For each j ∈ I k write the Fourier series of φ ( ju k ) as φ ( ju k ) = X g ∈ G x j,g u g . (7.19)By assumption there are two distinct elements g , g ∈ G such that x j ,g i = 0 for i = 1 , j , we may assume that both have norm at least one.25ow, for j ∈ I k , ju k xu ∗ k j ∗ = jα k ( x ) j ∗ , x ∈ Q, (7.20)so multiplication on the right by ju k leads to ju k xα − k ( j ∗ j ) = jα k ( x ) j ∗ ju k , x ∈ Q. (7.21)Apply φ in (7.21) and use (7.19) to reach X g ∈ G x j,g u g φ ( x ) φ ( α − k ( j ∗ j )) = X g ∈ G φ ( j ) φ ( α k ( x )) φ ( j ∗ ) x j,g u g , x ∈ Q, j ∈ I k . (7.22)Comparison of coefficients in (7.22) gives x j,g α g ( φ ( x )) α g ( φ ( α − k ( j ∗ j ))) = φ ( j ) φ ( α k ( x )) φ ( j ∗ ) x j,g , x ∈ Q, j ∈ I k , g ∈ G. (7.23)If we define ∗ -automorphisms of Q by β g = α g ◦ φ ◦ α − k ◦ φ − , g ∈ G , and make the substitution y = φ ( α k ( x )), then (7.23) becomes φ ( j ) yφ ( j ∗ ) x j,g = x j,g β g ( y ) α g ( φ ( α − k ( j ∗ j ))) , y ∈ Q, j ∈ I k , g ∈ G. (7.24)Choose an increasing contractive positive approximate identity { e λ : λ ∈ Λ } for I k . Since I k is w ∗ -dense in Q , we have the additional property thatSOT − lim λ e λ = 1 . (7.25)Consider now r, j ∈ I k . Then X g ∈ G x rj,g u g = φ ( rju k ) = φ ( r ) φ ( ju k ) = X g ∈ G φ ( r ) x j,g u g , (7.26)so comparison of coefficients in (7.26) leads to φ ( r ) x j,g = x rj,g , r, j ∈ I k , g ∈ G. (7.27)Similarly, for j, s ∈ I k , X g ∈ G x js,g u g = φ ( jsu k ) = φ ( ju k α − k ( s ))= X g ∈ G x j,g u g φ ( α − k ( s )) = X g ∈ G x j,g α g ( φ ( α − k ( s ))) u g , (7.28)so x js,g = x j,g α g ( φ ( α − k ( s ))) , j, s ∈ I k , g ∈ G. (7.29)Replacing r by e λ in (7.27) gives φ ( e λ ) x j,g = φ ( e λ ) x e λ j,g = φ ( e λ ) x e λ ,g α g ( φ ( α − k ( j ))) , λ ∈ Λ , g ∈ G, j ∈ I k , (7.30)where the second equality follows from (7.29) with e λ and j in place of j and s respectively.26rom (7.25) we may choose λ so large that k φ ( e λ ) x j ,g i k ≥ / , λ ≥ λ , i = 1 , . (7.31)It follows from (7.30) that k φ ( e λ ) x e λ ,g i k ≥ / , λ ≥ λ , i = 1 , , (7.32)and in particular that all of these elements are nonzero. Returning to (7.24) and replacing j by e λ , g by g i , and y by x , we see that φ ( e λ ) xφ ( e λ ) x e λ ,g i = x e λ ,g i β g i ( x ) α g i ( φ ( α − k ( e λ ))) , x ∈ Q, λ ≥ λ , i = 1 , . (7.33)Then (7.33) has the form of (5.3) with a = φ ( e λ ) and b = φ ( e λ ) x e λ ,g i and it follows from(7.32) that the hypotheses of Lemma 5.1 are met with x = 1. By that lemma, we concludethat β g and β g are not properly outer on Q and so their restrictions to M are both innerby Lemma 7.1. Thus α g ◦ α − g = β g ◦ β − g is inner on M , in contradiction to Lemma 7.2since g = g . This proves that φ ( ju g ) has a Fourier series of length at most one whenever ju g ∈ C ∗ ( Y ).For a particular g ∈ G , if there exist nonzero elements j , j ∈ I g so that φ ( j i u g ) = k i u h i for i = 1 , h = h , then the Fourier series of φ (( j + j ) u g ) has length two, contradictingwhat we have already proved. Thus there is a map σ : G → G and linear contractions µ g : I g → Q , g ∈ G , so that φ ( ju g ) = µ g ( j ) u σ ( g ) , g ∈ G, j ∈ I g . (7.34)Fix an arbitrary g ∈ G and let { f λ : λ ∈ Λ } be a contractive positive approximate identityfor I g with the property of (7.25). Since { µ g ( f λ ) : λ ∈ Λ } is a set of contractions, we maydrop to a subnet and assume that there is an element w g ∈ Q such that w ∗ − lim λ µ g ( f λ ) = w g .Since lim λ k jf λ − j k = 0 for each j ∈ I g , (7.34) gives φ ( ju g ) = lim λ φ ( jf λ u g ) = lim λ φ ( j ) µ g ( f λ ) u σ ( g ) = φ ( j ) w g u σ ( g ) , j ∈ I g . (7.35)It remains to show that σ is an automorphism of G and that each w g is a unitary. Take g, h ∈ G and choose increasing positive contractive approximate identities { e λ,g : λ ∈ Λ g } and { e ν,h : ν ∈ Λ h } for I g and I h respectively, both tending strongly to 1. Then the equation φ ( e λ,g α g ( e ν,h ) u g u h ) = φ ( e λ,g u g e ν,h u h ) = φ ( e λ,g u g ) φ ( e ν,h u h ) (7.36)shows that φ ( e λ,g α g ( e ν,h ) u g u h ) lies simultaneously in Qu σ ( g ) σ ( h ) and Qu σ ( gh ) . Sufficientlylarge choices of λ and ν ensure that e λ,g α g ( e ν,h ) = 0, and thus σ ( gh ) = σ ( g ) σ ( h ), provingthat σ is a homomorphism.Since w g is a contraction, it suffices to show that w g w ∗ g = 1; the finiteness of Q will thenestablish that w ∗ g w g = 1. If w g w ∗ g = 1, then there exist ε > p for w g w ∗ g such that k pw g k ≤ − ε . Then e λ,g φ − ( p ) ∈ I g for λ ∈ Λ g , so φ ( e λ,g φ − ( p ) u g ) = φ ( e λ,g ) pw g u σ ( g ) . (7.37)27hus k φ ( e λ,g φ − ( p ) u g ) k ≤ k pw g k ≤ − ε, λ ∈ Λ g . (7.38)On the left hand side, the limit over λ is 1 since e λ,g → w g is a unitary.The above arguments apply equally to φ − and so there exist unitaries v g ∈ Q for g ∈ G and a homomorphism ρ : G → G such that φ − ( ju g ) = φ − ( j ) v g u ρ ( g ) , g ∈ G, j ∈ I g . (7.39)Computing ju g as both φ − ( φ ( ju g )) and φ ( φ − ( ju g )) leads easily to the conclusion that ρσ and σρ are both the identity map on G . This shows that σ is an automorphism andcompletes the proof. Lemma 7.9.
The ∗ -automorphism φ of C ∗ ( Y ) extends to a ∗ -automorphism of N .Proof. Lemma 7.8 establishes that φ ( ju g ) = φ ( j ) w g u σ ( g ) (7.40)for j ∈ I g , g ∈ G , an automorphism σ of G and unitaries w g ∈ Q . Based on this, theremainder of the proof follows that of Lemma 6.2 with suitable changes of notation. Weomit the details.Denote the extension of the ∗ -automorphism φ to all of N by θ . We know that this mapcoincides with the original isometric bimodule map θ on Y , and we will have proved ourmain result – that θ is the unique extension of θ to a ∗ -automorphism of N – if we can showthat the two maps coincide on the ultraweakly closed bimodule X . To do this, we first provethe following lemma. Lemma 7.10. If x ∈ X has Fourier series x = P g ∈ G x g u g , then θ ( x ) has Fourier series P g ∈ G θ ( x g u g ) .Proof. If x ∈ Q and g ∈ G are such that xu g ∈ Y , then x lies in the ideal J g = Xu − g ∩ Q of Lemma 7.4, and thus in the larger ideal I g of Lemma 7.8. By this last lemma, there isan automorphism σ of G such that θ maps J g u g into I σ ( g ) u σ ( g ) , and since θ maps Y to Y wesee that θ takes J g u g into J σ ( g ) u σ ( g ) . The same observations applied to θ − also show that θ − ( J σ ( g ) u σ ( g ) ) ⊆ J g u g for g ∈ G. This establishes that θ ( J g u g ) = J σ ( g ) u σ ( g ) . To prove the result, proceed by contradiction and suppose that we have an element x = P g ∈ G x g u g ∈ X for which θ ( x ) = P g ∈ G y g u g ∈ X with y σ ( g ) u σ ( g ) = θ ( x g u g ) for some g ∈ G. Subtracting y σ ( g ) u σ ( g ) from both sides of the equation θ ( x ) = X g ∈ G y g u g , (7.41)and noting that y σ ( g ) u σ ( g ) is the image under θ of some element of J g u g distinct from x g u g , we obtain an element x = P g ∈ G x g u g ∈ X with x g = 0, for which θ ( x ) = P g ∈ G y g u g has y σ ( g ) = 0. 28y further reductions similar to those in Lemma 7.7, we may also assume that x g isa nonzero central projection in Q . Note that these operations do not change the fact that y σ ( g ) = 0 . We now use familiar averaging techniques to pick out individual terms in therespective Fourier series.For any unitary w ∈ Q, we have θ ( w ) θ X g ∈ G x g u g ! θ ( α − g ( w ∗ )) = θ X g ∈ G wx g α g α − g ( w ∗ ) u g ! , (7.42)by the Q -bimodularity property of θ. On the other hand, θ ( w ) X g ∈ G y g u g θ ( α − g ( w ∗ )) = X g ∈ G θ ( w ) y g α g ◦ θ ◦ α − g ( w ∗ ) u g . (7.43)This says that for any w ∈ U ( Q ) , the operator θ (cid:16)P g ∈ G wx g α g ◦ α − g ( w ∗ ) u g (cid:17) lies in the weakclosure of the convex set K = conv (X g ∈ G vy g α g ◦ θ ◦ α − g ◦ θ − ( v ∗ ) u g : v ∈ U ( Q ) ) . Then since θ is w ∗ -continuous, applying θ to the element of minimal k·k -norm in the weakclosure of the convex set K ′ = conv (X g ∈ G wx g α g α − g ( w ∗ ) u g : w ∈ U ( Q ) ) also gives an element of the weak closure of K . By the methods used in Lemma 7.4, we knowthat the element h of minimal k·k -norm in the weak closure of K ′ has the form h = x g u g , so this says that θ ( x g u g ) ∈ K, and has the form q σ ( g ) u σ ( g ) for some q σ ( g ) ∈ Q. But since y σ ( g ) = 0 , every element of K has zero σ ( g ) − coefficient, so this says that θ ( x g u g ) = 0 , acontradiction to the fact that θ is isometric on X . This proves the desired result.We come now to the main result of this section, our version of Mercer’s theorem forregular inclusions of subfactors. Theorem 7.11.
Let M ⊆ N be a regular inclusion of II factors. Denote by Q the vonNeumann algebra generated by M and M ′ ∩ N, and let X be a w ∗ -closed Q -bimodule with Q ⊆ X ⊆ N , which generates N as a von Neumann algebra. Suppose that θ : X → X is a w ∗ -continuous, surjective isometry whose restriction to Q is a ∗ -automorphisms with θ ( M ) = M and which satisfies θ ( q xq ) = θ ( q ) θ ( x ) θ ( q ) , for all x ∈ X and q , q ∈ Q. Then θ has a unique extension to a ∗ -automorphism θ : N → N. roof. Let θ be the extension to N of the ∗ -automorphism φ : C ∗ ( Y ) → C ∗ ( Y ) obtained inLemma 7.8. We know that θ coincides with θ on the subspace Y ⊆ N , and so we are doneif we can show that these two maps coincide on X . By Lemma 7.10, we have that for all x = P g ∈ G x g u g ∈ X, θ ( x ) = X g ∈ G θ ( x g u g ) , (7.44)and since θ and θ coincide on Y , the same argument also shows that θ ( x ) = X g ∈ G θ ( x g u g ) (7.45)for all such x. Thus, since θ ( x g u g ) = θ ( x g u g ) for each g , we see that θ ( x ) = θ ( x ). A longstanding problem is the question of whether each separable von Neumann algebrais singly generated. There has been considerable work on this problem [12, 13, 19, 30, 39,41] with the result that the only remaining open case is for finite von Neumann algebras.Recently Shen [34] studied this question and introduced a numerical quantity G ( M ) whichis related to the number of generators of M . He showed that if G ( M ) < /
4, then M isgenerated by a projection p and a hermitian element h (and thus singly generated by p + ih )and noted that the same proof shows single generation when G ( M ) < / G ( M ) = 0 for various II factors,hyperfinite, Cartan or nonprime, thus providing many examples of singly generated factors.Indeed, the question of single generation is equivalent to whether G ( M ) = 0 for all M [13],since any example where G ( M ) > G ( · ) under the formation of crossed products,and this will give new classes of singly generated factors.For a II factor M with normalized trace τ , one way to define the Shen invariant G ( M )is as follows. Consider a fixed element x ∈ M and a fixed set P = { p , . . . , p k } of orthogonalprojections summing to I and with equal traces τ ( p i ) = 1 /k . Relative to these projections, x can be written as a k × k matrix ( x ij ), and I ( x, P ) is defined to be the number of nonzeromatrix entries divided by k . For a finite subset X = { x , . . . , x n } ⊆ M , we define I ( X, P ) = n X i =1 I ( x i , P ) . (8.1)If M is not finitely generated then we set G ( M ) = ∞ ; otherwise we define G ( M ) = inf I ( X, P ) (8.2)taken over all finite sets X of generators and all finite sets of projections P as above. If P = { p , . . . , p k } is such a set of projections, then we may form a new set Q = { q , . . . , q k } bysplitting each p i as an orthogonal sum of two projections of trace 1 / (2 k ). It is straightforward30o verify that I ( x, Q ) ≤ I ( x, P ), and by applying this halving argument an arbitrary numberof times, we see that the sets of projections in (8.2) can be restricted to have sizes that exceedany given integer. Lemma 8.1.
Let G be a countable discrete group that acts on a II factor M by outerautomorphisms, and let p ∈ M be a nonzero projection. Then there exists an element a ∈ p ( M ⋊ α G ) p so that every coefficient in the Fourier series of a = P g ∈ G a g g is nonzero.Proof. Let { g , g , . . . } be an enumeration of the group elements. Since M is a factor, thereexist elements x n ∈ M , n ≥
1, so that px n α g n ( p ) = 0, and by scaling we may assume that k x n k ≤ − n . Then x = P n ≥ x n g n defines an element of M ⋊ α G , and pxp = ∞ X n =1 px n g n p = ∞ X n =1 px n α g n ( p ) g n , (8.3)so all the Fourier coefficients of pxp are nonzero. Then a = pxp ∈ p ( M ⋊ α G ) p is the desiredelement. Theorem 8.2.
Let M be a II factor and let G be a countable discrete group acting on M by outer automorphisms. Then G ( M ⋊ α G ) ≤ G ( M ) . (8.4) Proof.
The result is clear if G ( M ) = ∞ , so we may assume that this quantity is finite. Given ε >
0, choose an integer K so that K > p /ε . By the remarks preceding Lemma 8.1, thereis a set of projections P = { p , . . . , p k } with k ≥ K and a set of generators X = { x , . . . , x n } for M so that I ( X, P ) < G ( M ) + ε/ . (8.5)By Lemma 8.1, there exists an element x n +1 ∈ p ( M ⋊ α G ) p whose Fourier coefficients areall nonzero. By Remark 4.6, Y = { x , . . . , x n +1 } is a generating set for M ⋊ α G , and I ( Y, P ) = I ( X, P ) + I ( x n +1 , P ) = I ( X, P ) + 1 /k < G ( M ) + ε/ /K < G ( M ) + ε. (8.6)Thus G ( M ⋊ α G ) < G ( M ) + ε , and the result follows since ε > Remark . If G ( M ) = 0, then G ( M ⋊ α G ) = 0 from [34], so Theorem 8.2 gives nothing newin this case. However, there are finitely generated factors M for which G ( M ) is currentlyunknown. Such examples include free group factors where G ( L ( F n ) ≤ ( n − / n ≥ M whose Shen invariants are yet to be determined but which lie below 1/4. Anycrossed product of such an algebra is singly generated by Theorem 8.2. (cid:3) eferences [1] H. Aoi, A construction of equivalence subrelations for intermediate subalgebras. J. Math.Soc. Japan , (2003), 713–725.[2] W.B. Arveson, Subalgebras of C ∗ -algebras. Acta Math., (1969), 141–224.[3] W.B. Arveson, Subalgebras of C ∗ -algebras. II. Acta Math., (1972), 271–308.[4] D. Bures,
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