Generalized notions of character amenability

In this paper, the concepts of approximate character amenability (contractibility), uniform approximate character amenability (contractibility) and w*-approximate character amenability are introduced. We are concerned with the relations among the generalized concepts of character amenability for Banach algebras. We show that approximate character amenability, w*-approximate character amenability and approximate character contractibility are the same properties, as uniform approximate character amenability and character amenability, as uniform approximate character contractibility and character contractibility. The general theory for these concepts is also developed. Moreover, approximate character amenability of several concrete classes of Banach algebras related to locally compact groups and also some discrete semigroups is considered.