The relationship between a commutative Banach algebra and its maximal ideal space

Abstract

Our main result is an extension of a theorem due to Novodvorskii and Taylor; we give some special cases. Let A be a commutative Banach algebra with identity, and let Δ be its maximal ideal space. Let B be a Banach algebra with identity; let B −1 denote the invertible group in B and id B denote the set of idempotents in B. Let [ (A \\ ̂ bo B) −1 ] denote the set of path components of (A \\ ̂ bo B) −1 , and [ Δ, B −1] denote the set of homotopy classes of continuous maps of Δ into B −1. We prove that the Gelfand transform on A induces a bijection of [ (A \\ ̂ bo B) −1 ] onto [ Δ, B −1], and extend this result to prove a theorem of Davie. We show that the Gelfand transform induces a bijection of [ id(A \\ ̂ bo B) ] onto [ Δ, id B], and investigate consequences of this result for specific examples of the Banach algebra B.