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# Functional Calculus and *-Regularity of a Class of Banach Algebras

Chi-Wai Leung and Chi-Keung Ng
Proceedings of the American Mathematical Society
Vol. 134, No. 3 (Mar., 2006), pp. 755-763
Stable URL: http://www.jstor.org/stable/4098424
Page Count: 9
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## Abstract

Suppose that (A, G, α) 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 π is a "nice representation" of A, we define a new Banach *-algebra $l_{\pi}^{1}(G, A)$ 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 $l_{\pi}^{1}(G, A)$ is *-regular.

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