Functional calculus and $*$-regularity of a class of Banach algebras

Abstract:

Suppose that $(A,G,\alpha )$ is a $C^*$-dynamical system such that $G$ is of polynomial growth. If $A$ is finite dimensional, we show that any element in $K(G;A)$ has slow growth and that $L^1(G, A)$ is $*$-regular. Furthermore, if $G$ is discrete and $\pi$ is a “nice representation” of $A$, we define a new Banach $*$-algebra $l^1_{\pi }(G, A)$ which coincides with $l^1(G;A)$ when $A$ is finite dimensional. We also show that any element in $K(G;A)$ has slow growth and $l^1_{\pi }(G, A)$ is $*$-regular.