On cohomology groups of Banach algebras

Abstract

In this note we will study the cohomology groups of some special classes of Banach algebras, and show that there are some relationships between the cohomology groups of a Banach algebra (for definition see [2 ]) and the space of maximal ideals of that Banach algebra, We assume that the Banach algebras considered here are the Banach algebras over the field of complex numbers C. Let A = { 1; A } be a Banach algebra which is generated by one element A. Suppose there is a continuous algebra homomorphism X: A-->C with x(A) =r; then C can be regarded as a two sided Banach A-module with Xcc=x(X)c, for cEC and XEA. Now we want to calculate the one dimensional cohomology group of the Banach algebras, which have one generator, with coefficients in C. Let A = { 1, A } and x: A--C be the Banach algebra and continuous algebra homomorphism defined above and suppose that X(A) =-r. Then MA =X-'(0) is a maximal ideal of A and AIZA-C. Let Z' denote the set of all 1-cocycles, then fEZ' if and only if f: A--C is a bounded linear function and satisfying the identity