Abstract
AbstractIn this paper, we characterize amenability of locally compact groups in terms of the properties of Orlicz Figà-Talamanca Herz algebras.
Highlights
Let G be a locally compact group and let Ap(G) the Figa-Talamanca Herz algebra introduced by Herz [10]
The following are equivalent: a) The group G is amenable. b) The Banach algebra Ap(G) possesses a bounded approximate identity. c) Every closed cofinite ideal is of the form I(E), where E is a finite subset of G. d) The Banach algebra Ap(G) factorizes weakly. e) Each homomorphism from Ap(G) with finite dimensional range is continuous
We begin this section with the main result of this paper on the characterization of amenable groups in terms of the existence of bounded approximate identities in AΦ(G)
Summary
Let G be a locally compact group and let Ap(G) the Figa-Talamanca Herz algebra introduced by Herz [10]. The following theorem on the characterization of amenability in terms of the Ap(G) algebras is well-known. The following are equivalent: a) The group G is amenable. C) Every closed cofinite ideal is of the form I(E), where E is a finite subset of G. d) The Banach algebra Ap(G) factorizes weakly. The equivalence of the statements a) and b) was due to Herz [11]. In [16], we have introduced and studied the LΦ-versions of the Figa-Talamanca Herz algebras. The space AΦ(G) is defined as the space of all continuous functions u, where u is of the form. This paper has the modest aim of proving the above said equivalent statements in the context of AΦ(G) algebras. We shall begin with some preliminaries that are needed in the sequel
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