On some group algebra modules related to Wiener’s algebraM₁

Abstract

TENG-SUN LIU, ARNOUD VAN ROOIJ AND JU-KWEI1* Our notations are basically the same as those used in [3]. We use, however, C to denote the complex number field. Throughout the paper, G is a locally compact group with a left Haar measure λ. Instead of C Q0 (G), L P (G) etc. we write C oo , L P etc. We view L t as a subspace of M. We identify two functions that are equal almost everywhere.For a function f on G define /' bywhere Δ denotes the modular function of G. Then /" = / and (/•*)'= *'•/' for /^ei, If B is a left Banach module over L x (see [3; 32.14]), then J3* becomes a left Banach module by Every jeB induces the multiplier f\-*f*j.The following theorem is essentially due to Rieffel [6] and is proved in A Radon measure on G is a linear functional μ: C oo ~> C such that for every compact set CcG there exists a number c such that(J e C oo , Supp jcC).The Radon measures form a vector space which we denote by R. For μe R and for an f e L λ with compact support we define f*μeR and μ*feR by These formulas reduce to the familiar convolution formulas in case μeL p .Every Radon measure is a linear combination of positive Radon measures [1]. Thus, if μ e R and if X is a relatively compact Borel subset of G, we can in a natural way define μ(X). Further, if μe R and if A is any Borel set there is a unique ξ A μeR such that ξ A μ(X) = μ(X Π A) for all relatively compact Borel sets X. There ON SOME GROUP ALGEBRA MODULES 509 exists a unique |μ\ e R such that ξ A \μ\ = \ξ Λ μ| for every compact set A (see [1; Ch. 13]

A characterization of minimal homogeneous Banach spaces

“…For A = C°(G) (space of continuous functions on G vanishing at infinity) Theorem 5 gives the main result of [6] (cf. [1, Proposition VIII], and [12]). We have called (C^G)),,^ "Wiener's algebra" because it coincides with Wiener's classical algebra for G = R".…”

“…That can also be shown directly, using Theorem 2 (for similar assertions cf. [1], [2], [7] or [12]). Corollary 4.…”

“…Let B satisfy the general hypothesis, and let g G As, g ^ 0 be given such that g(x) = const on some open set U G G. Then Bmin = Bx = {/ G BXoc, \\f\\Bl-fj{lyg)f\\B dy < oo J, (12) and the norms || \\B¡ and || H^ are equivalent on i?^.…”

A characterization of biflatness of Segal algebras based on a character

“…This is a Segal algebra in L 1 (G), called the Lebesgue-Fourier algebra of G. We note that if G is unimodular, then LA(G) is a symmetric abstract Segal algebra in L 1 (G). Moreover, it is worthwhile to mention that the Weiner's algebra M 1 , see [13] and the Feichtinger's Segal algebra S 0 (G), see [4] and [19] are important examples of symmetric Segal algebras. As a consequence, we have the following.…”

A Characterization of Minimal Homogeneous Banach Spaces

“…Paris 290 (1980), 791-794. Generalizations of the spaces considered above and in [1], [2], [6]- [8], [10] and [12] are to be treated in forthcoming papers.…”