Gelfand theory for non-commutative Banach algebras

Abstract

Let $A$ be a Banach algebra. We call a pair $(G, B)$ a Gelfand theory for $A$ if the following axioms are satisfied: (G 1) $B$ is a $C^\ast$-algebra, and $G : A \to B$ is a homomorphism; (G 2) the assignment $L \mapsto G^{-1}(L)$ is a bijection between the sets of maximal modularleft ideals of $B$ and $A$, respectively; (G 3) for each maximal modular left ideal $L$ of $B$, the linear map $G_L : A / G^{-1}(L) \to B /L $ induced by $B$ has dense range. The Gelfand theory of a commutative Banach algebra is easily seen to be characterized by these axioms. Gelfand theories of arbitrary Banach algebras enjoy many of the properties of commutative Gelfand theory. We show that unital, homogeneous Banach algebras always have a Gelfand theory. For liminal $C^\ast$-algebras with discrete spectrum, we show that the identity is the only Gelfand theory (up to an appropriate notion of equivalence).