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Abstract
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Let A be a Banach algebra. We say that a pair (G, U) is a (topologically
Gelfand theory ) Gelfand theory for A if the following hold: (G1) U is a C*-
algebra and G : A ? U is a homomorphism which induces the (homeomorphism)
bijection p 7? p ? G from Ub onto Ab; (G2) for everymaximal modular
left ideal L, G(A) 6? 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 )
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