  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).
