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

1991 Mathematics Subject Classification. 46J20, 46M20.