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Homology and cohomology groups of commutativeBanach algebras and analytic polydiscs

Published online by Cambridge University Press:  07 August 2001

L. I. Pugach
Affiliation:
Department of Mathematics, The University, Newcastle upon Tyne NE1 7RU, England
M. C. White
Affiliation:
Department of Mathematics, The University, Newcastle upon Tyne NE1 7RU, England
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Abstract

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In this paper we deduce the existence of analytic structure in a neighbourhood of a maximal ideal M in the spectrum of a commutative Banach algebra, A, from homological assumptions. We assume properties of certain of the cohomology groups H^n(A,A/M), rather than the stronger conditions on the homological dimension of the maximal ideal the first author has considered in previous papers. The conclusion is correspondingly weaker: in the previous work one deduces the existence of a Gel'fand neighbourhood with analytic structure, here we deduce only the existence of a metric neighbourhood with analytic structure. The main method is to consider products of certain co-cycles to deduce facts about the symmetric second cohomology, which is known to be related to the deformation theory of algebras.

1991 Mathematics Subject Classification. 46J20, 46M20.

Type
Research Article
Copyright
2000 Glasgow Mathematical Journal Trust
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