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The relationship between a commutative Banach algebra and its maximal ideal space
Authors:Iain Raeburn
Institution:Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 U.S.A.
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.
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