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.