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On continuity of derivations and epimorphisms on some vector-valued group algebras

Published online by Cambridge University Press:  17 April 2009

Ramesh V. Garimella
Ramesh V. Garimella
Affiliation:
Department of Mathematics, Tennessee Technological University, Cookeville TN 38505, United States of America, e-mail: RVG0037@tntech.edu
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Abstract

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For a locally compact Abelian group G and a commutative Banach algebra B, let L1(G, B) be the Banach algebra of all Bochner integrable functions. We show that if G is compact and B is a nonunital Banach algebra without nontrivial zero divisors, then (i) all derivations on L1(G, B) are continuous if and only if all derivations on B are continuous, and (ii) each epimorphism from a Banach algebra X onto L1(G, B) is continuous provided every epimorphism from X onto B is continuous. If G is noncompact then every derivation on L1(G, B) and every epimorphism from a commutative Banach algebra onto L1(G, B) are continuous. Our results extend the results of Neumann and Velasco for nonunital Banach algebras.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 2 , October 1997 , pp. 209 - 215
Copyright
Copyright © Australian Mathematical Society 1997

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