Almost periodic functionals

Transcription

1 Almost periodic functionals Matthew Daws Leeds Warsaw, July 2013 Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

2 Dual Banach algebras; personal history A Banach algebra A Banach space, algebra, ab a b. which is a dual space A = E, with separate continuity of the product. von Neumann algebras. G a locally compact group, M(G) = C 0 (G) the measure algebra. B(G) = C (G) or B r (G) = C r (G) or M cb A(G) = Q cb (G). E a reflexive Banach space, B(E) = (E E ). Theorem (Daws 07, after Young, Kaiser) Every dual Banach algebra is isometrically a weak -closed subalgebra of B(E) for a reflexive E. Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

3 Weakly almost periodic functionals Let A be a Banach algebra and turn A into an A-bimodule a µ, b = µ, ba, µ a, b = µ, ab (a, b A, µ A ). Then consider the orbit maps L µ : A A ; a a µ, R µ : A A ; a µ a. Definition Say that µ A is weakly almost periodic, µ wap(a) if L µ is a weakly compact operator (i.e. {a µ : a 1} is relatively weakly compact in A ). (Equivalently can use R µ.) Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

4 Turning into an algebra Choose bounded nets (a i ), (b i ) converging to Φ, Ψ A respectively. Then we have two choices for a product: Φ Ψ = lim i lim j a i b j, Φ Ψ = lim j lim i a i b j, the limits in the weak -topology on A. These are the Arens products. For example, given µ A, lim i lim j µ, a i b j = lim i lim j µ a i, b j = lim i Ψ, µ a i = lim i Ψ µ, a i = Φ, Ψ µ. Theorem (Hennefeld, 68) For µ A, we have that Φ Ψ, µ = Φ Ψ, µ for all Φ, Ψ if and only if µ wap(a). Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

5 Universal property; link with dual Banach algebras Theorem (Lau, Loy, Runde,...?) By separate weak -continuity, the product on A extends to wap(a), turning wap(a) into a dual Banach algebra. This is the universal object for dual Banach algebras. That is, if B is a dual Banach algebra and θ : A B a bounded homomorphism, then there is a unique θ : wap(a) B, a weak -continuous homomorphism, with: A wap(a) θ! θ B Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

6 What if I want joint continuity? Definition µ A is almost periodic, µ ap(a), if L µ (equivalently R µ ) is a compact operator. Theorem (Lau) ap(a) is a dual Banach algebra such that the product is jointly weak -continuous, on bounded sets. It s then easy to adapt the argument before, and show that ap(a) is universal for dual Banach algebras where the product is jointly weak -continuous, on bounded sets. Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

7 Example: C -algebras Quigg [1985] studied almost periodic functionals on C -algebras. Let A be a C -algebra, and set M = A a von Neumann algebra. Then µ A is almost periodic if and only if M M = A ; x x µ is compact, say µ ap (M). If N M is a σ-weakly closed ideal, then it has a support projection, and so there is another ideal N with M = N N, also M = N N. If M ap is the largest ideal of M which is a direct sum of matrix algebras, then ap (M) = (M ap ). So ap(a) = M ap. Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

8 For (semi)groups Let S be a discrete semigroup and consider A = l 1 (S): a = s S a s δ s, a = s S a s, δ s δ t = δ st. f l (S) will be almost periodic if and only if the shifts {δ s f : s S} form a relatively compact set (as taking the convex hull doesn t change compactness). Easy to see that ap(a) l (S) will be a unital (commutative) C -algebra. Let S ap be the (compact, Hausdorff) character space, so ap(a) = C(S ap ). Thus M(S ap ) becomes a dual Banach algebra with joint continuity on bounded sets. Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

9 The semigroup S ap ap(l 1 (S)) = C(S ap ), ap(l 1 (S)) = M(S ap ). The map l 1 (S) ap(a) = M(S ap ) determines the product on M(S ap ). Point-evaluation gives a map S S ap. These maps are compatible if we identify S ap with point-masses in M(S ap ). Then S is dense in S ap. So if u S ap there is a net (s i ) S with s i u; similarly let t i v. Then δ u δ v = lim i δ si δ ti = lim i δ si t i. So S ap is a semigroup. Can show that the product is jointly continuous, and that the product on M(S ap ) is convolution. Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

10 Universal property Using some Banach algebra techniques, we started with a semigroup S, and formed a compact (jointly continuous) semigroup S ap. Call such semigroups compact topological semigroups. S ap is universal in the sense that if T is any compact topological semigroup then have: S T! S ap Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

11 For groups If we start with a locally compact group G, form A = L 1 (G), then similarly we find G ap with ap(a) = C(G ap ), and have exactly the same universal property. By joint continuity, as G ap contains a dense subgroup (the image of G) it follows that G ap is a compact group. This is the Bohr compactification of G. As G ap is a compact group, Peter-Weyl tells us that functions of the form G ap C; s (π(s)ξ η), are dense in C(G ap ). Here π : G ap U(n) is a finite-dimensional unitary representation, and ξ, η C n. Such π are in 1-1 correspondence with finite-dimensional unitary representations of G. So such continuous functions are dense in ap(l 1 (G)). Not clear how to see this directly... Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

12 Non-commutative world Let G be a locally compact group, and let π : G U(H) be the universal, strongly continuous, unitary representation of G (direct sum over all such representations). We can integrate this to a map π : L 1 (G) B(H) π(f )ξ = f (s)π(s)ξ ds (f L 1 (G), ξ H). G This is a -homomorphism of L 1 (G); it s the universal one. The closure of π(l 1 (G)) is C (G) the universal group C -algebra. If we replace π by λ the left-regular representation on L 2 (G), we get the reduced group C -algebra C r (G). C (G) C r (G) is an isomorphism precisely when G is amenable. Finally, define VN(G) = C r (G) the group von Neumann algebra. Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

13 Fourier theory If G is an abelian group, then we have the Pontryagin dual Ĝ. The Fourier transform gives a unitary map F : L 2 (G) L 2 (Ĝ). Let C 0 (G) act on L 2 (G) by multiplication, say f M f. The conjugation map M f FM f F 1 gives a -isomorphism C 0 (G) C r (Ĝ). It also gives a normal -isomorphism L (G) VN(Ĝ). So the predual VN(Ĝ) is isomorphic to the algebra L 1 (G). By biduality, VN(G) = L 1 (Ĝ). What happens when G is not abelian? Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

14 Hopf von Neumann algebras There is a normal -homomorphism : VN(G) VN(G) VN(G); λ(s) λ(s) λ(s). That this exists is most easily seen by finding a unitary operator W on L 2 (G G) with (x) = W (1 x)w. is coassociative: ( id) = (id ). Let the predual of VN(G) by A(G); the predual of gives an associative product : A(G) A(G) A(G). This is the Fourier algebra; the map A(G) C 0 (G); ω ( λ(s), ω ) is a contractive algebra homomorphism. So A(G) is a commutative Banach algebra; it is semisimple (and regular, Tauberian) with character space G. Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

15 Almost periodic for the Fourier algebra Question What is ap(a(g))? What relation does it have to a compactification? Let C δ (G) be the C -algebra generated by {λ(s) : s G} inside VN(G). If G is discrete, then C δ (G) = C r (G). If G is discrete and amenable, or G is abelian, then ap(a(g)) = C δ (G). Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

16 Compact quantum groups Unital C -algebra A and coassociative : A A A with cancellation : lin{ (a)(1 b) : a, b A}, lin{ (a)(b 1) : a, b A} are dense in A A. G compact gives C(G) with (f )(s, t) = f (st); G discrete gives C r (G) with as before. Natural morphisms are the Hopf -homomorphisms ; a morphism (A, A ) to (B, B ) is a -homomorphism θ : B A with A θ = (θ θ) B. If φ : G H is a continuous group homomorphism, then may define θ : C(H) C(G) by θ(f ) = f φ. Can extend this to the non-compact world by considering multiplier algebras. Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

17 Quantum Bohr compactification So ltan (2005) considered compactifications in this category. In particular, C δ (G) is the universal object for C r (G). For any compact quantum group (A, A ), have: A MC r (G) G Ĝ! C δ (G)! (Ĝ)ap This gives a justification for looking at C δ (G). Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

18 More on the category LCQG?? Let G be a discrete, non-amenable group; let {e} be the trivial group. The trivial homomorphism G {e} induces a Hopf -homomorphism C = C({e}) C b (G) = MC 0 (G). By duality, there should be a Hopf -homomorphism C r (G) C r ({e}) = C. But such a map existing is equivalent to G being amenable. Work of Ng, Kustermans, and [Meyer, Roy, Woronowicz] resolves this by presenting various different, equivalent notions of a morphism (one being to work with C (G) instead of C r (G)). I checked that So ltan s ideas do give a compactification in this category the resulting compact quantum group is quite mysterious at the C -algebraic level, but the underlying Hopf -algebra is unique. Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

19 Counter-example It s easy to see that always Cδ (G) ap(a(g)). Theorem (Chou ( 90), Rindler ( 92)) There are compact (connected, if you wish) groups G such that ap(a(g)) C δ (G). As G is compact, the constant functions are members of L 2 (G). Let E be the orthogonal projection onto the constants; then E = λ(1 G ) MC r (G). E ap(a(g)) if and only if G is tall. E Cδ (G) if and only if G does not have the weak-mean-zero containment property: there is a net of unit vectors (ξ i ) in ker E with λ(s)ξ i ξ i 2 0 for each s G. [Rindler] Clever choice of G... Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

20 Stronger forms of compact VN(G) is naturally an operator space: we have a family of norms on M n (VN(G)). Then A(G) is also an operator space. The natural morphisms are the completely bounded maps: those whose matrix dilations are uniformly bounded. There are various notions of being completely compact ; they do not interact well with taking adjoints. [Runde, 2011] defined x A(G) to be completely almost periodic if both orbit maps L x and R x are completely compact. If G is amenable, or connected, then cap(a(g)) = {x VN(G) : (x) VN(G) VN(G)}, here is the C -spatial product. So x cap(a(g)) if and only if (x) can be norm approximated by a finite sum n i=1 a i b i. Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

21 Stronger forms of compact cont. Theorem (D.) Let G be discrete. Then (x) VN(G) VN(G) if and only if x C δ (G). Theorem (D.) Let G be a [SIN] group (compact, discrete, abelian... ). Then 2 (x) VN(G) VN(G) VN(G) if and only if x C δ (G). Theorem (Woronowicz, 92) Let G be quantum E(2) (for µ (0, 1)). Then (C 0 (G)) C 0 (G) C 0 (G), and so C 0 (G) (L (G)); but G is not compact! Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22

22 Even stronger forms of compact Definition Say that µ A is periodic if L µ : A A is a finite-rank operator. Say that µ is strongly almost periodic if L µ can be cb-norm approximated by operators of the form L µ with µ periodic. For A(G), equivalently, x is strongly almost periodic if (x x ) can be made arbitrarily small with (x ) finite-rank. Theorem (D.) x VN(G) is strongly almost periodic if and only if x C δ (G). An analogous result holds for all Kac algebras. For a locally compact quantum group, also need to assume things are in D(S) D(S ), which is rather messy... Matthew Daws (Leeds) Almost periodic functionals Warsaw, July / 22