An approach to Gelfand theory for arbitrary Banach algebras

Abstract

Let $A$ be a Banach algebra. We say that a pair $(\mathcal{G},\mathcal{U})$ is a (topologically Gelfand theory) Gelfand theory for $A$ if the following hold: (G1) $\mathcal{U}$ is a C*-algebra and $\mathcal{G}:A\to \mathcal{U}$ is a homomorphism which induces the (homeomorphism) bijection $\pi\mapsto \pi\circ \mathcal{G}$ from $\widehat{\mathcal{U}}$ onto $\widehat{\mathcal{A}}$; (G2) for every maximal modular left ideal $L$, $\mathcal{G}(A)\not\subseteq L$. We show that this definition is equivalent to the usual definition of gelfand theory in the commutative case. We prove that many properties of Gelfand theory of commutative Banach algebras remain true for Gelfand theories of arbitrary Banach algebras. We show that unital homogeneous Banach algebras and postliminal C*-algebras have unique Gelfand theories (up to an appropriate notion of uniqueness ).