Crossed products of Banach algebras. I
Abstract
We construct a crossed product Banach algebra from a Banach algebra dynamical system (A,G,α) and a given uniformly bounded class R of continuous covariant Banach space representations of that system. If A has a bounded left approximate identity, and R consists of non-degenerate continuous covariant representations only, then the non-degenerate bounded representations of the crossed product are in bijection with the non-degenerate R-continuous covariant representations of the system. This bijection, which is the main result of the paper, is also established for involutive Banach algebra dynamical systems and then yields the well-known representation theoretical correspondence for the crossed product C*-algebra as commonly associated with a C*-algebra dynamical system as a special case. Taking the algebra A to be the base field, the crossed product construction provides, for a given non-empty class of Banach spaces, a Banach algebra with a relatively simple structure and with the property that its non-degenerate contractive representations in the spaces from that class are in bijection with the isometric strongly continuous representations of G in those spaces. This generalizes the notion of a group C*-algebra, and may likewise be used to translate issues concerning group representations in a class of Banach spaces to the context of a Banach algebra, simpler than L1(G), where more functional analytic structure is present.