On the \(\ast\)- semisimplicity of the \(\ell^1\)-algebra on an abelian \(\ast\)-semigroup

Authors

  • S.J. Bhatt Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar- 388120, Gujarat, India
  • P.A. Dabhi Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar, Gujarat, India, PIN-388120
  • H.V. Dedania Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar- 388120, Gujarat, India

Keywords:

$\ast$-semigroup, Banach $\ast$-algebra, semisimplicity, $\ast$-semisimplicity

Abstract

Towards an involutive analogue of a result on the semisimplicity of $\ell^1(S)$ by Hewitt and Zuckerman, we show that given an abelian $\ast$-semigroup $S$, the commutative convolution Banach $\ast$-algebra $\ell^1(S)$ is $\ast$-semisimple iff hermitian bounded semicharacters on $S$ separate the points of $S$; and we search for an intrinsic separation property on $S$ equivalent to the $\ast$-semisimplicity. Very many natural involutive analogues of separation property by Hewitt and Zuckerman are shown not to work, thereby exhibiting intricacies involved in analysis on $S$. 10.1017/S000497271300004X

Published

2013-09-27

Issue

Vol. 88 No. 3 (2013)

Section

Articles
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