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.