Abstract. We obtain characterizations of left character amenable Banach algebras in terms of the existence of left φ-approximate diagonals and left φ-virtual diagonals. We introduce the left character amenability constant and find this constant for some Banach algebras. For all locally compact groups G, we show that the Fourier-Stieltjes algebra B(G) is C-character amenable with C < 2 if and only if G is compact. We prove that if A is a character amenable, reflexive, commutative Banach algebra, then A ∼ = C n for some n ∈ N. We show that the left character amenability of the double dual of a Banach algebra A implies the left character amenability of A, but the converse statement is not true in general. In fact, we give characterizations of character amenability of L 1 (G) * * and A(G) * * . We show that a natural uniform algebra on a compact space X is character amenable if and only if X is the Choquet boundary of the algebra. We also introduce and study character contractibility of Banach algebras.

We introduce the notion of character amenable Banach algebras. We prove that character amenability for either of the group algebra L 1 (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.

Abstract. Given Banach algebras A and B with spectrum σ(B) = ∅, and given θ ∈ σ(B), we define a product A × θ B, which is a strongly splitting Banach algebra extension of B by A. We obtain characterizations of bounded approximate identities, spectrum, topological center, minimal idempotents, and study the ideal structure of these products. By assuming B to be a Banach algebra in C 0 (X) whose spectrum can be identified with X, we apply our results to harmonic analysis, and study the question of spectral synthesis, and primary ideals.

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