p-operator spaces and harmonic analysis: Feichtinger Figà-Talamanca Herz algebras

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1 p-operator spaces and harmonic analysis: Feichtinger Figà-Talamanca Herz algebras Nico Spronk (Waterloo) Joint work with Serap Öztop (Istanbul) Banach Algebras 2013, August 3, Chalmers U., Göteborg

2 Feichtinger s minimal Segal algebra Definition [Reiter 60s] Let G be a l.c.g. A subspace S 1 (G) L 1 (G) is a Segal algebra if it is dense and has a left translation invariant norm, continuous in G, S 1 L 1 by which it is complete. Theorem [Feichtinger 80] For every l.c. abelian g. G, there exists S 0 (G) L 1 (G) s.t. (a) S 0 (G) is minimal Segal algebra for which χs 0 (G) S 0 (G) (pointwise mult n) with isometric action, for χ in Ĝ; (b) Ŝ0(G) = S 0 (Ĝ) (Fourier transform); (c) H G closed, S 0 (G) H = S 0 (H) and T H (S 0 (G)) = S 0 (G/H) (T H averaging over H); and (d) S 0 (G) γ S 0 (H) = S 0 (G H) (H l.c.a.g., γ proj ve t.p.).

3 Definition [Leinert, Burnham 70s] Let A be a Banach algebra. An abstract Segal algebra is dense ideal S A, which is complete w.r.t. a norm S A by which it is a left Banach A-module. Adding operator space structure... Definition [after Forrest-S.-Wood 07] Let A be completely contractive Banach algebra. An operator Segal algebra is dense ideal S A, which admits a complete operator space structure w.r.t. which S A completely boundedly, and S is a completley bounded left A-module. Operator projective tensor product [Effros-Ruan 90] M, N v.n. alg s, (M N ) = M ˆ N Fourier algebras A(G H) = A(G) ˆ A(H) [Losert 84] A(G H) = A(G) γ A(H) one of G, H a.a.

4 Extending Feichtinger s algebra to non-abelian groups G abelian, A(G) = L 1 (Ĝ) (inv. Fourier trans.); gen. [Eymard 64] Theorem [S. 07] For every l.c.g. G, an operator Segal algebra S 0 (G) A(G) s.t. (a) S 0 (G) is the minimal Segal algebra in A(G) which is closed under left translations, and for which such translations are (complete) isometries and continuous in G; (b) S 0 (G) L 1 (G), and is a(n operator) Segal ideal therein; (c) H G closed, S 0 (G) H = S 0 (H) (c ly) surjectively; (d) N G closed, T N (S 0 (G)) = S 0 (G/N) (c ly) surjectively; and (e) S 0 (G) ˆ S 0 (H) = S 0 (G H) (c ly) isomorphically. Theorem [Öztop-S.] S 0 (G) is the minimal Segal algebra in L 1 (G) for which A(G)S 0 (G) S 0 (G) (pointwise mult n).

5 Figà-Talamanca Herz algebras 1 < p <, 1 p + 1 p = 1, G l.c.g. λ p : G B(L p (G)) p-left reg. rep n, λ p (s)η(t) = η(s 1 t) N p (G) = L p (G) γ L p (G), N p (G) = B(L p (G)) P G : N p (G) C 0 (G), P G ξ η = (s ξ, λ p (s)η ) A p (G) = ranp G (quotient space of N p (G)) PM p (G) = (ker P G ) B(L p (G)), PM p (G) = A p (G) Bipolar theorem PM p (G) = span w λ p (G).

6 Figà-Talamanca Herz algebras Proposition [Daws 10, after Herz 71] PM p (G) PM p (G) = span w λ p (G) λ p (G) = PM p (G G) 2nd idf n spatial: L p (G) p L p (G) = L p (G G). Proposition [Daws 10] ([Stinespring 59], [Herz 70]) W G B(L p (G) L p (G)), W G ξ(s, t) = ξ(s, st) satisfies W G (λ(s) I )W 1 G = λ p(s) λ p (s) pointwise mult n A p (G) γ A p (G) A p (G) defined and cts. Theorem [Herz 73] The Gelfand spectrum of A p (G) is G.

7 p-operator spaces (Pisier, LeMerdy, after Ruan) V vector space, p-op. space structure { n : M n (V) R 0 } s.t. [ (D ) v 0 = max{ v 0 w] n, w m } n+m (M p ) αvβ n α B(l p n ) v n β B(l p n), α, β M n S : V W, S (n) : M n (V) M n (W), S (n) [v ij ] = [Sv ij ] S completely bounded if S pcb = sup n S (n), c ly contractive if S pcb 1, c te quotient if each S (n) quotient Theorem (Pisier, LeMerdy; Ruan if p = 2) V p-operator space, complete isometry π : V B(E), E SQ p Here M n (B(E)) = B(l p (n, E)). Danger. No known Wittstock Haagerup Paulsen extension theorem, i.e. M n = B(l p n) not known to be injective, in any sense.

8 Mapping and dual spaces ([Blecher, Effros Ruan 90s]) M n (CB p (V, W)) = CB p (V, M n (W)) satisfies (D ) and (M p ) Proposition [Daws 10] V = CB p (V, C) isometrically; A p (G) PM p (G) Hence M n (V ) = CB p (V, B(l p n)). Proposition [Daws 10] (a) V B(l p (I )) completely isometrically for some I (b) S : V W complete contraction S : W V complete contraction (c) κ V : V V complete contraction; complete isometry V B(L p (µ)) completley isometrically for some µ V acts on L p if V B(L p (µ)) completely isometrically for some µ

9 Tensor products and weakly completley bounded maps V ˆ p W = V W p p-operator projective tensor product Proposition [Daws 10] ([Blecher Paulsen, Effros Ruan 90s]) (a) (V ˆ p W) = CB p (V, W ) completely isometrically (b) commutative, projective Definition S : V W weakly completely bounded (resp. contractive, quotient) if S : W V is c.b. (resp. c.c., c.i.) Proposition (a) [Daws 10] N p (µ) ˆ p N p (ν) = N p (µ ν), w.c.i ly, µ, ν (b) S : V W w.c.b. (q.) S id : V ˆ p N p (µ) W ˆ p N p (µ) b. (q.) µ (c) S : V W w.c.b. & W acts on L p S c.b.

10 Tensor products of Figà-Talamanca Herz algebras Theorem (after [Daws 10]) (a) u v u v : A p (G) ˆ p A p (H) A p (G H) is w.c.q. (b) A p (G) ˆ p A p (H) = A p (G H) w.c.i ly, if G, H amenable Proof of (b) uses that (A p (G) ˆ p A p (H)) = PMp (G) F PM p (H) CV p (G H). [Cowling 98] [Daws-S. 13] PM p (G) = CV p (G) if G has AP (b), above, holds for G, H with AP. Conjecture: p 2, G, H infinite, A p (G) γ A p (H) A p (G H). Proposition (after [Daws 10]) mult : A p (G) ˆ p A p (G) A p (G) c.c. & w.c.q.; A p (G) c.c.b.a.

11 A special class of ideals K G compact, non-null M p (K) = P G (N p (K)) where N p (K) = L p (K) γ L p (K) N p (G) with quotient o.s. struc. dual o.s.s. from M p (K) M p (K) (Related: [Cowling 98].) Theorem M p (K) operator Segal ideal in A p (G). Proof uses W K B(L p (K G)), W K η(s, t) = η(s, st) M p (K) = 1 K PM p (G)1 K w, then 1 K λ p (s)1 K 1 K λ p (s)1 K I W K (1 K λ p (s)1 K I )W 1 K mult: M p (K) ˆ p A p (G) M p (K) w.c.c. = 1 K λ p (s)1 K λ p (s).

12 Construction of S p 0 (G) l 1 (G) l (G), l (G) B(l p (G)) min. space l 1 (G) max l space acting on Lp ([Öztop-S. 12]); l 1 (G) ˆ p V = l 1 (G) γ V = l 1 (G, V) Q K : l 1 (G) ˆ p M p (K) A p (G), Q(δ s u) = s u S p 0 (G) = ranq K, quotient o.s. struc. dual o.s. struc. Theorem (a) S p 0 (G) operator Segal algebra in A p(g) (b) description independant of compact non-null K (c) can replace M p (K) with other compactly supported op. Segal ideals, say A K p (G) = {u A p (G) : suppu K} (d) minimal Segal algebra in A p (G), closed under bdd. cts. translations Call S p 0 (G) the p-feichtinger Figà-Talamanca Herz algebra.

13 S p 0 (G) as a Segal algebra in L1 (G) weakly compete surjection: S : V W s.t. induced S : V/ ker S W w.c. isomorphism S id : V ˆ N p (N) W ˆ N p (N) surjective Theorem (a) S p 0 (G) pseudo-symmetric operator Segal algebra in L1 (G) (b) S p 0 (G) = ranq K, Q K : L1 (G) ˆ p M p (K) S p 0 (G), Q K (f u) = f u, w.c. surj n (c) minimal Segal algebra in L 1 (G), closed under p twise mult n by A p (G).

14 Functorial property: tensor products Theorem G, H l.c.gs. (a) K G, L H compact non-null; u v u v : M p (K) ˆ p M(L) M p (K L) is w.c.q., injective if A p (G) ˆ p A p (H) = A p (G H) (b) u v u v : S p 0 (G) ˆ p S p 0 (H) Sp 0 (G H) is w.c. surj., injective if A p (G) ˆ p A p (H) = A p (G H) This may be considered main reason for introducing p-op. space: S p 0 (G) γ S p 0 (H) = S p 0 (G H) known only if one G, H discrete.

15 Functorial property: restriction to subgroups H G closed Theorem (after [Derighetti 84]) complete isometry ι : CV p (H) CV p (G) s.t. ι PMp(H) = Res H. Hence Res H : A p (G) A p (H) w.c.q. Theorem (a) Res H (M p (K)) quot.o.s.s. dual o.s.s., Segal ideal in A p (G) (b) Res H : S p 0 (G) Sp 0 (H) c.b. & w.c. surj n Confirmation that algerba (and o.s.s.) agrees with (A p (G), ).

16 Functorial property: averaging over normal subgroup N G closed, T N u(sn) = N u(sn) dn, suitable u A p (G/N) acts as multipliers on A p (G): uv(s) = u(sn)v(s). Theorem (a) M p (K) operator module over A p (G/N) (b) T N : M p (K) A p (G/N) c.b., with q.o.s.s. dual o.s.s., T N (M p (K)) Segal ideal in A p (G/N). (c) T N : S p 0 (G) Sp 0 (G/N) c.b. & w.c.surj n. Confirmation that algebra (and o.s.s.) agrees with (L 1 (G), ).

17 Functorial property: isomorphism Theorem [S. 07] G, H l.c.gs. G = H Φ : S p 0 (G) Sp 0 (H) cts. bij n s.t. Φ(uv) = Φu Φv & Φ(u v) = Φu Φv Just one operation is not sufficent. G discrete, S p 0 (G) = l1 (G) (l 1 (G), ) = (l 1 (H), ) G = H G compact abelian, S p 0 (G) = S p 0 (Ĝ) = l 1 (Ĝ), (S p 0 (G), ) = (l 1 (Ĝ), )

18 Primary reference S. Öztop & N.S. p-operator space structures on Feichtinger Figà-Talamanca Herz algebras, preprint, arxiv:

19 Thank you!