  We construct a crossed product Banach algebra from a Banach algebra dynamical
system $(A,G,\alpha)$ 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 $L^1(G)$, where more
functional analytic structure is present.
