The First Cohomology Group of a Maximal Ideal Space

Abstract

Given Banach alebras $\mathfrak{A}_1$ and $\mathfrak{A}_2$ with maximal ideal spaces $\mathcal{M}_1$ and $\mathcal{M}_2$ , if $\mathfrak{A}_1$ and $\mathfrak{A}_2$ are isomorphic as algebras, then $\mathcal{M}_1$ and $\mathcal{M}_2$ are homeomorphic. It is thus to be expected that the topology of $\mathcal{M}\left( \mathfrak{A} \right)$ is reflected in the algebraic structure of $\mathfrak{A}$ , for an arbitrary Banach algebra $\mathfrak{A}$ .