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Functional calculus and -regularity of a class of Banach algebras
Authors:Chi-Wai Leung   Chi-Keung Ng
Affiliation:Department of Mathematics, The Chinese University of Hong Kong, Hong Kong ; School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People's Republic of China
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.

Keywords:Banach algebras   functional calculus   $*$-regular
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