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.

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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.

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