On character amenability of Banach algebras

Abstract

Weakly sequentially complete algebra F -algebra We continue our work [E. Kaniuth, A.T. Lau, J. Pym, On ϕ-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 85-96] in the study of amenability of a Banach algebra A defined with respect to a character ϕ of A. Various necessary and sufficient conditions of a global and a pointwise nature are found for a Banach algebra to possess a ϕ-mean of norm 1. We also completely determine the size of the set of ϕ-means for a separable weakly sequentially complete Banach algebra A with no ϕ-mean in A itself. A number of illustrative examples are discussed.

On $${\phi}$$ -contractibility of the Lebesgue–Fourier algebra of a locally compact group

“…Suppose that LA(G) is a left ideal in its second dual, and let ρ ∈ G. Since G is amenable, it follows that there exists a bounded net (f α ) ⊆ L 1 (G) such that for all f ∈ L 1 (G); see Theorem 1.4 of Kaniuth, Lau, and Pym [11], and Corollary 2.4 of Monfared [12]; see also [10]. Fix h 0 ∈ LA(G) such that φ ρ (h 0 ) = 1, and set…”

A characterization of biflatness of Segal algebras based on a character

“…These notions have been studied for various classes of Banach algebras. For more details, we refer the reader to [8,10,11,14].…”

Generalized notions of character amenability

“…For this reason by relaxing some of the constrains in the definition of amenability new concepts have been introduced. The most notable are the concepts of Connes amenability [11,13], weak amenability [2,5] and character amenability [17,18]. More recently, F. Ghahramani and R. J. Loy have introduced and studied the concepts of approximate amenability (contractibility) and uniform approximate amenability (contractibility) for Banach algebras [9,10].…”