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ON THE $\ast $-SEMISIMPLICITY OF THE ${\ell }^{1} $-ALGEBRA ON AN ABELIAN $\ast $-SEMIGROUP

Part of: Semigroups Commutative Banach algebras and commutative topological algebras Topological (rings and) algebras with an involution

Published online by Cambridge University Press:  15 February 2013

S. J. BHATT ,
P. A. DABHI  and
H. V. DEDANIA
S. J. BHATT
Affiliation:
Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388120, Gujarat, India email [email protected]@yahoo.com
P. A. DABHI*
Affiliation:
Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388120, Gujarat, India email [email protected]@yahoo.com
H. V. DEDANIA
Affiliation:
Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388120, Gujarat, India email [email protected]@yahoo.com
*
✉[email protected]
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Abstract

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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 if and only if Hermitian bounded semicharacters on $S$ separate the points of $S$; and we search for an intrinsic separation property on $S$ equivalent to $\ast $-semisimplicity. Very many natural involutive analogues of Hewitt and Zuckerman’s separation property are shown not to work, thereby exhibiting intricacies involved in analysis on $S$.

Keywords

∗-semigroupBanach ∗-algebrasemisimplicity∗-semisimplicity

MSC classification

Secondary: 46K05: General theory of topological algebras with involution 20M14: Commutative semigroups 46J05: General theory of commutative topological algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 492 - 498
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

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