On character amenability of Banach algebras

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