Character amenability of Banach algebras

Abstract

AbstractWe introduce the notion of character amenable Banach algebras. We prove that character amenability for either of the group algebra L1(G) or the Fourier algebra A(G) is equivalent to the amenability of the underlying group G. Character amenability of the measure algebra M(G) is shown to be equivalent to G being a discrete amenable group. We also study functorial properties of character amenability. For a commutative character amenable Banach algebra A, we prove all cohomological groups with coefficients in finite-dimensional Banach A-bimodules, vanish. As a corollary we conclude that all finite-dimensional extensions of commutative character amenable Banach algebras split strongly.