No abstract

Generalizing the notion of left amenability for so-called F-algebras [12], we study the concept of ϕ-amenability of a Banach algebra A, where ϕ is a homomorphism from A to C. We establish several characterizations of ϕ-amenability as well as some hereditary properties. In addition, some illuminating examples are given.

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.

harmonic analysis-Research exposition (monographs, survey articles). msc | Abstract harmonic analysis-Abstract harmonic analysis-Measure algebras on groups, semigroups, etc. msc | Abstract harmonic analysis-Abstract harmonic analysis-L 1-algebras on groups, semigroups, etc. msc | Abstract harmonic analysis-Abstract harmonic analysis-Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. msc | Abstract harmonic analysis-Abstract harmonic analysis-Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups. msc | Functional analysis-Research exposition (monographs, survey articles). msc | Topological groups, Lie groups-Research exposition (monographs, survey articles). msc Classification:

Let G be a locally compact abelian group. The notion of Zak transform on L 2 (R d) extends to L 2 (G). Suppose that G is compactly generated and its connected component of the identity is non-compact. Generalizing a classical result for L 2 (R), we then prove that if f ∈ L 2 (G) is such that its Zak transform Zf is continuous on G × G, then Zf has a zero.