INTERMEDIATE BIMODULES FOR CROSSED PRODUCTS OF VON NEUMANN ALGEBRAS

Transcription

1 INTERMEDIATE BIMODULES FOR CROSSED PRODUCTS OF VON NEUMANN ALGEBRAS Roger Smith (joint work with Jan Cameron) COSy Waterloo, June 2015

2 REFERENCE Most of the results in this talk are taken from J. Cameron and R.R. Smith. Bimodules in crossed products of von Neumann algebras. Adv. Math. 274 (2015),

3 CROSSED PRODUCTS M B(H) is a factor. A discrete group G acts on M by automorphisms α g with α g outer for g e (there is no u U(M) so that α g (x) = uxu for x M). Outerness is equivalent to: if t M and xt = tα g (x) for x M, then t = 0. π : M B(H l 2 (G)) is π(x)(ξ δ h ) = α h 1(x)ξ δ h, ξ H, x M, h G. A representation g u g is u g (ξ δ h ) = ξ δ gh, ξ H, g, h G. M α G is generated by {π(x) : x M} {u g : g G} u g π(x)u g = π(α g (x)), x M, g G.

4 We also write this as M α G = {xg : x M, g G} subject to the relations gx = α g (x)g. There is a normal faithful conditional expectation E : M α G M defined on generators by { xg, g = e, E(xg) = 0, g e. Each x M α G has a Fourier series x g G x g g where x g = E(xg 1 ). The series does not converge in the usual von Neumann algebra topologies so we need the Bures topology or B-topology (Mercer).

5 THE B-TOPOLOGY M N with a faithful normal conditional expectation E : N M (think E : M α G M). For each normal state φ on M there is a seminorm x (φ E(x x)) 1/2, x N. The B-topology is defined by these seminorms and x λ x in the B-topology if and only if equivalently lim λ φ E((x λ x) (x λ x)) = 0, φ M, w lim λ E((x λ x) (x λ x)) = 0. (Mercer) For each x M α G, its Fourier series converges in the B-topology to x.

6 EXAMPLE The B-topology is closely connected to the w -topology. Let x λ x in the B-topology for elements in M α G. Write the Fourier series as x λ = g G x λ,g g, x = g G x g g. For a particular g G and a normal state φ on M, φ(x λ,g x g ) = φ E((x λ x)g 1 ) (φ E((x λ x) (x λ x))) 1/2 0 (limit over λ) (C-S) so x g = w lim λ x λ,g.

7 CHODA S THEOREM Theorem (H. Choda) Let M be a factor and let G act on M by outer automorphisms. If N is a von Neumann algebra with M N M α G and there is a faithful normal conditional expectation E : M α G N, then there is a subgroup H G so that N = M α H. For M with separable predual, Izumi-Longo-Popa showed that the existence of such a conditional expectation is always true. Question What can be said about intermediate w -closed M-bimodules? Easier to answer for B-closed M-bimodules, a subclass of the w -closed ones.

8 CARTANS Recall that a masa A in a factor M is Cartan if its unitary normalizers in M generate M, and there is a normal conditional expectation onto A. Put M in standard position on H with a cyclic and separating vector ξ and modular conjugation J. A = (A JAJ), masa in B(H). Each w -closed A-bimodule X M gives an A-invariant subspace Xξ 2 and the projection P X onto it is in A. Theorem (Cameron-Pitts-Zarikian) X P X is a bijection from B-closed A-bimodules to projections in A.

9 MAIN RESULT For x = g G x g g M α G, let supp(x) = {g G : x g 0} G. For an M-bimodule X M α G, let S X = {supp(x) : x X} G and for S G, let X S = {x M α G : supp(x) S} M α G.

10 Theorem (Cameron-S.) S X S = span B {Mg : g S} is a bijection between subsets S G and B-closed M-bimodules in M α G. The inverse map is X S X. In particular the B-closed intermediate von Neumann algebras are M α H for subgroups H G. If G has the approximation property (AP) of Haagerup-Kraus, then the w -closed and B-closed M-bimodules coincide.

11 APPROXIMATION PROPERTY The predual of L(G) is the Fourier algebra A(G) consisting of the functions g l g ξ, η where ξ, η l 2 (G). A completely bounded multiplier is a function f on G such that f A(G) A(G) and the associated multiplication operator M f is completely bounded. The space of completely bounded multipliers is denoted by M 0 (A(G)) and is a Banach space in the cb-norm. It is a dual space with predual denoted by Q(G). 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 functions on G. (Haagerup-Kraus). This is a large class of groups that contains all amenable discrete groups and also the weakly amenable groups of Cowling-Haagerup. The AP class is closed under semidirect products so is strictly larger than the weakly amenable class.

12 KEY LEMMA Lemma Let M be a factor and let X be a w -closed M-bimodule in M α G. If g 0 G is such that x g0 0 for some x = g G x g g X, then g 0 X. In particular {supp(x) : x X} X.

13 SPECIAL CASE Assume M is hyperfinite and take an amenable subgroup Γ U(M) that generates M. Fix an invariant mean and write it as Γ du, u Γ. Take x = x g g X with x g0 0. We want to pick out g 0. Using basic properties of factors we can first make x g0 a nonzero projection and then 1. Consider y = uxα g 1(u ) du X. 0 Since Γ ux g gα g 1 0 (u ) = ux g α gg 1(u )g, 0 the g 0 -coefficient of y is 1 while for g g 0 y g = ux g α gg 1(u ) du. 0 Γ

14 By invariance vy g α gg 1(v ) = y g, v Γ, 0 vy g = y g α gg 1(v), v Γ, 0 xy g = y g α gg 1(x), x M. 0 Thus y g = 0 for g g 0 since α gg 1 0 We get y = g 0 X. is outer. The general case requires a different type of averaging.

15 CHRISTENSEN-SINCLAIR THEOREM Let L cb (M, M) be the cb-maps of M to M, L cb (M, M) M the subspace of right M-module maps, which are x tx for some t M. Take all strings β = (m j ) where j m jm j = 1, m j M. For a cb-map φ : M M, form φ β (x) = j φ(xm j )m j, x M. Theorem (Chris.-Sin.) (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), x M.

16 For an x = g G x g g with x g0 = 1, form j m j xα g 1 0 (m j ) = g G j m j x g α gg 1 0 (mj ) g, and take the limit over (β) to get rid of x g for g g 0, and leaving g 0. This is achieved by applying the C-S theorem to the cb-map φ(x) = α g0 g 1(x)α g 0 g 1(x g), x M.

17 GENERAL CHODA THEOREM If M N M α G, M a factor, N a von Neumann algebra then is a subgroup and Taking w -closures gives H = {g G : g N} M α,r H = span {Mg : g H} N N B = M α H. N = M α H so all intermediate von Neumann algebras are M α H for subgroups H and there is always a faithful normal conditional expectation E : M α G M α H = N without a separable predual assumption.

18 NONFACTOR CASE Let {z g : g G} be a set of central projections in M and let N = span w {Mz g g : g G} M α G. When is N a von Neumann algebra? (i) Unital: so z e = 1. (ii) Self-adjoint: so (Mz g g) = g 1 Mz g = Mα g 1(z g )g 1 = Mz g 1g 1 = α g 1(z g ) = z g 1. (iii) Closed under multiplication: so z g gz h h = z g α g (z h )gh = z g α g (z h ) z gh.

19 Theorem (Cameron-S.) Every intermediate von Neumann algebra is N = span w {Mz g g : g G} where {z g : g G} satisfy (i)-(iii). There is a faithful normal conditional expectation E : M α G N given by E( x g g) = x g z g g.

20 BACK TO FACTORS When M is a factor, z g {0, 1}. Put H = {g G : z g = 1}. (ii) α g 1(z g ) = z g 1 = H is closed under inverses, (iii) z g α g (z h ) z gh = H is closed under multiplication so the intermediate algebras of Choda reappear as M α H.

21 A α G Let A be abelian. It is Cartan in A α G. Assume τ is a trace on A and the α g are trace preserving. Also the action of G is free and ergodic so A α G is a II 1 factor. Let ξ be a tracial vector for A. Then ξ δ e is a cyclic and separating vector for A α G and if N = span w {Az g g : g G} then the projection P N corresponding to N has range span 2 {Az g ξ δ g : g G}.

22 MERCER S EXTENSION THEOREM Recall that a masa A in a factor M is Cartan if its unitary normalizers in M generate M. Theorem (Mercer, Cameron-Pitts-Zarikian) Let A be a Cartan masa in a factor M. Let X be a w -closed A-bimodule that generates M and let θ : X X be a w -continuous isometric surjective isomorphism such that θ A is a -automorphism and θ(a 1 xa 2 ) = θ(a 1 )θ(x)θ(a 2 ), x X, a i A. Then θ extends to a -automorphism θ of M.

23 ANOTHER VERSION Theorem (Cameron-S.) Let M be a factor. Let X be a w -closed M-bimodule that generates M α G and let θ : X X be a w -continuous isometric surjective isomorphism such that θ M is a -automorphism and θ(m 1 xm 2 ) = θ(m 1 )θ(x)θ(m 2 ), x X, m i M. Then θ extends to a -automorphism θ of M α G.

24 SKETCH and Apply θ to S = {g G : g X}, Y = span {Mg : g S} M α,r G, Y X. gx = α g (x)g, x M, g S, This gives θ(g)θ(x) = θ(α g (x))θ(g), x M, g S. θ(g) θ(g), θ(g)θ(g) M (M α G) = C1, so θ(g) is a unitary in M α G that normalizes M. Then θ(g) = uh for some h G and u U(M). We get h S so θ maps Y to Y.

25 C -ENVELOPES Recall that a C -algebra A is said to be the C -envelope of a unital operator space X if there is a completely isometric unital embedding ι : X A so that ι(x) generates A, and if B is another C -algebra with a completely isometric unital embedding 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 unique C -envelope denoted C env(x).

26 Lemma The C -envelope of Y = span {Mg : g S} M α,r G (where S is the set of group elements in X) is M α,r G and θ : Y Y extends to a -automorphism φ : M α,r G M α,r G. Moreover, there is an automorphism π of G such that φ(mg) = Mπ(g), g G.

27 FINAL STEP Lemma Let φ be a -automorphism of M α,r G such that φ(m) = M. Then φ extends to a -automorphism θ of M α G. On Fourier series, θ( x g g) = φ(x g )φ(g). To summarize: the original θ : X X was restricted to Y X, then extended to φ : M α,r G M α,r G, and extended again to θ : M α G M α G.