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

Authors: Chi-Wai Leung and Chi-Keung Ng
Journal: Proc. Amer. Math. Soc. 134 (2006), 755-763
MSC (2000): Primary 47A60, 32A65
Published electronically: July 19, 2005
MathSciNet review: 2180894
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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.

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Additional Information

Chi-Wai Leung
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong

Chi-Keung Ng
Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China

Keywords: Banach algebras, functional calculus, $*$-regular
Received by editor(s): June 23, 2004
Received by editor(s) in revised form: August 19, 2004, and October 13, 2004
Published electronically: July 19, 2005
Additional Notes: This work was jointly supported by Hong Kong RGC Direct Grant and the National Natural Science Foundation of China (10371058)
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society

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