Operator space structure and amenability for Figà-Talamanca–Herz algebras

Lambert, Anselm; Neufang, Matthias; Runde, Volker

doi:10.1016/j.jfa.2003.08.009

Cited by 29 publications

(36 citation statements)

References 21 publications

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“…Problem 37 A 2-convex 2-multi-norm is just an operator sequence space in the sense of [12]. Do any results of [12] generalize to suitable p-multi-norms?…”

Section: Problem 35mentioning

confidence: 99%

Ordered Banach algebras and multi-norms: some open problems

Wickstead¹

2017

Positivity

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This is a list of open problems posed at the workshop on Ordered BanachAlgebras held at the Lorentz Center, Leiden, during the week 21-25 July, 2014.Keywords Banach lattice algebras · Ordered Banach algebras · Multi-normsWe call A a Banach lattice algebra if it is both a Banach lattice and a Banach algebra and furthermore the product of two positive elements is positive.Problem 1 Characterize those Banach lattice algebras A for which the left regular representation a → L a , where L a (x) = ax, into the space of regular operators L r (A) preserves finite suprema and infima, as well as being an algebra homomorphism. Even an answer for Dedekind complete Banach lattice algebras would be of interest, when we are asking about it being a lattice homomorphism. In addition, consider when such representations are either faithful or isometric. The natural norm to take on L r (A) would be the regular norm, but it is conceivable that the question might be of interest for the operator norm as well.This list of problems is based on three preliminary reports produced by Gerard Buskes, Garth Dales and Sonja Mouton. The author would like to extend thanks to them for their work in preparing the original list and to them, Marcel de Jeu and Vladimir Troitsky for further input into this final list.

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“…Problem 37 A 2-convex 2-multi-norm is just an operator sequence space in the sense of [12]. Do any results of [12] generalize to suitable p-multi-norms?…”

Section: Problem 35mentioning

confidence: 99%

Ordered Banach algebras and multi-norms: some open problems

Wickstead¹

2017

Positivity

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“…3.5] or [19] there are certain row and column operator space structures ROW(L p (G)) and COL(L p (G)) which are also L ∞ -homogeneous operator space structures. For 0 < q < ∞ the Figà-Talamanca-Herz algebra A q (G) has been shown in [20] to admit an operator space structure under which it is a completely bounded Banach algebra. If 1 < p < ∞ we can define the Segal p, q-Figà-Talamanca-Herz algebra by…”

Section: 4mentioning

confidence: 99%

Operator Segal algebras in Fourier algebras

2007

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Abstract. Let G be a locally compact group, A(G) its Fourier algebra and L 1 (G) the space of Haar integrable functions on G. We study the Segal algebra S 1 A(G) = A(G) ∩ L 1 (G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of S 1 A(G). We use it to show that the restriction operator u → u| H : S 1 A(G) → A(H), for some non-open closed subgroups H, is a surjective complete quotient map. We also show that if N is a non-compact closed subgroup, then the averaging operator,is a surjective complete quotient map. This puts an operator space perspective on the philosophy that S 1 A(G) is "locally A(G) while globally L 1 ". Also, using the operator space structure we can show that S 1 A(G) is operator amenable exactly when when G is compact; and we can show that it is always operator weakly amenable. To obtain the latter fact, we use E. Samei's theory of hyper-Tauberian Banach algebras.1. Operator Segal algebras 1.1. Notation. For any Banach space X we let B(X ) denote the Banach algebra of bounded linear operators from X to itself, and b 1 (X ) the set of all vectors of norm not exceeding 1.For details on classical harmonic analysis, we use [15,23]. We will always let G denote a locally compact group with a fixed left invariant Haar measure m.is the usual L p -space with respect to m. If f, g are Borel measurable functions and s ∈ G, then for almost every t in G we denote bythe left group action, convolution (when the integrand makes sense) and inversion. We note that L 1 (G) is a Banach algebra with respect to convolution. We let A(G) and B(G) denote the Fourier and Fourier-Stieltjes algebras of G, which are Banach algebras of continuous functions on G and were introduced in [7]. We recall, from that article, that A(G) consists exactly of functions on G of the form u(s) = λ(s)f | g = g * f (s), where λ : G → B(L 2 (G)) is the left regular representation given by λ(s)f = s * f . The dual of A(G) is the von Neumann algebra VN(G), which is generated by λ(G) in B(L 2 (G)).Our standard references for operator spaces are [6,22]. An operator space is a complex Banach space V equipped with an operator space structure: for each space of matrices M n (V) with entries in V, n ∈ N, we have a norm · M n (V) , and the norms satisfy Ruan's axioms in addition to that · M 1 (V) is the norm on V = M 1 (V). A map T from V to another operator space W is said to be completely bounded if the family of linear operatorsis uniformly bounded over n. If A is an algebra and an operator space for which V is a left module over A, we say V is a completely bounded A-module if there is C > 0 so that for eachWe say V is a completely contractive A-module if we can set C = 1. This is the same as asserting that the module multiplication extends to a map on the operator projective tensor product A ⊗ V → V and is bounded at all matrix levels by C. We say A is a completely bounded (contractive) Banach algebra if it itself is a completely bounded (contractive) A-module. We n...

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“…Moreover, it is possible to show (in using a suitable variant of [15,Theorem 5.6.1]) that OA p (G) is the natural predual of the operator space of the completely bounded Fourier IEOT multipliers. We refer to [5,6,14] and [25] for other operator space analogues of A p (G).…”

Section: Introductionmentioning

confidence: 99%

Noncommutative Figà-Talamanca–Herz Algebras for Schur Multipliers

Arhancet

2011

Integr. Equ. Oper. Theory

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In this work, we introduce a noncommutative analogue of the Figà-Talamanca-Herz algebra Ap(G) on the natural predual of the operator space M p,cb of completely bounded Schur multipliers on the Schatten space Sp. We determine the isometric Schur multipliers and prove that the space Mp of bounded Schur multipliers on the Schatten space Sp is the closure in the weak operator topology of the span of isometric multipliers. Mathematics Subject Classification (2000). Primary 46L51; Secondary 46M35, 46L07.

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Operator space structure and amenability for Figà-Talamanca–Herz algebras

Cited by 29 publications

References 21 publications

Ordered Banach algebras and multi-norms: some open problems

Ordered Banach algebras and multi-norms: some open problems

Operator Segal algebras in Fourier algebras

Noncommutative Figà-Talamanca–Herz Algebras for Schur Multipliers

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