Homology and cohomology groups of commutative Banach algebras and analytic polydiscs

Abstract

Abstract. 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 AY AaM, rather than the stronger conditions on the homological dimension of the maximal ideal the ®rst 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 Classi®cation. 46J20, 46M201. Introduction. This work can be considered as one in the series of papers by the ®rst author which study homological properties of non-idempotent maximal ideals in commutative Banach algebras ([8], [9]). From another point of view it continues the long series of paper by many authors which studied conditions on the vanishing of cohomology groups of Banach algebras.Throughout this paper A denotes a commutative Banach algebra with unit 1, M & A is a ®xed maximal ideal. We denote by w A the space of maximal ideals of A and consider two topologies on w A ± the Gelfand topology and norm (metric) topology, induced by the norm from A Ã . The letter 3 stands for the point in w A , that corresponds to M and for the multiplicative functional annihilating M.The symbol denotes the projective tensor product of Banach spaces [4], and % X M M 3 M is the operator of multiplication. We denote by M 2 the algebraic square of M (that is the linear space generated by ab, (aY b P M)). We also consider the topological square M 2 , which is the closure of M 2 . The ideal M is called idempotent if M M 2 and non-idempotent otherwise. Note that the quotient space MaM 2 is the predual of the space of bounded point derivations at the point 3.