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
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