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Automatic continuity and second order cohomology

Part of: Topological algebras, normed rings and algebras, Banach algebras Methods of category theory in functional analysis

Published online by Cambridge University Press:  09 April 2009

Volker Runde
Volker Runde
Affiliation:
Department of Mathematical Sciences University of Alberta Edmonton, Alberta T6G 2G1 Canada e-mail: runde@math.ualberta.ca
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Abstract

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Many Banach algebras A have the property that, although there are discontinuous homomorphisms from A into other Banach algebras, every homomorphism from A into another Banach algebra is automatically continuous on a dense subspace—preferably, a subalgebra—of A. Examples of such algebras are C*-algebras and the group algebras L1(G), where G is a locally compact, abelian group. In this paper, we prove analogous results for , where E is a Banach space, and . An important rôle is played by the second Hochschild cohomology group of and , respectively, with coefficients in the one-dimensional annihilator module. It vanishes in the first case and has linear dimension one in the second one.

MSC classification

Secondary: 46H40: Automatic continuity 46M20: Methods of algebraic topology (cohomology, sheaf and bundle theory, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 2 , April 2000 , pp. 231 - 243
Copyright
Copyright © Australian Mathematical Society 2000

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