Topological invariants of the maximal ideal space of a Banach algebra

Abstract

The theme of this paper is as old as the concept of Banach algebra. Given the functor that assigns to a commutative Banach algebra A its maximal ideal space d, , the urge to interpret topological data concerning AA in terms of the algebraic structure of A is irresistable. The first result in this direction is due to Shilov [29]; it says that each opencompact subset of A, is the support of the Gelfand transform of a unique idempotent in A. A corollary is that A, itself is compact if and only if A has an identity. The Shilov idempotent theorem can be viewed as a characterization of the zero-dimensional Cech cohomology group H”(AA , Z). In fact, it implies immediately that N”(AA , Z) is isomorphic to the additive subgroup of A generated by the idempotents in A. An early result of Brushlinsky [9] points out that if X is compact, then the first Cech group W(X, Z) can be identified with C(X)-l,‘exp(C(X)), where for any commutative Banach algebra A with identity, A-l denotes the group of invertible elements of A and exp(A) denotes the subgroup consisting of elements of the form ea for a E A. Arens [l] and Royden [26] proved that the analogous result holds in general. That is, there is a natural isomorphism