Idempotents of norm one and Banach algebra representations of compact groups

Abstract:

Let $G$ be a finite group of order $n$ and let $A$ be a (real or complex) Banach algebra. Rudin and Schneider [3] ask whether a mapping $f:G \to A$ satisfying $||f(x)|| = 1$ and $f(x) = (1/n){\Sigma _{y \in G}}f(x{y^{ - 1}})f(y)$ is necessarily a homomorphism (Question 1, p. 602). They give an affirmative answer if $A$ is either commutative and semisimple or strictly convex. Here, we prove this result for general Banach algebras, and at the same time prove the natural generalization to compact groups. This allows us to characterize norm one idempotents in generalized group algebras.