# Spectrum preserving linear mappings between Banach algebras

 dc.contributor.advisor Mouton, S. dc.contributor.author Weigt, Martin dc.contributor.other Stellenbosch University. Faculty of Science. Dept. of mathematical Sciences (applied, computer, mathematics). en_ZA dc.date.accessioned 2012-08-27T11:35:33Z dc.date.available 2012-08-27T11:35:33Z dc.date.issued 2003-04 dc.identifier.uri http://hdl.handle.net/10019.1/53597 dc.description Thesis (MSc)--University of Stellenbosch, 2003. en_ZA dc.description.abstract ENGLISH ABSTRACT: Let A and B be unital complex Banach algebras with identities 1 and I' en_ZA respectively. A linear map T : A -+ B is invertibility preserving if Tx is invertible in B for every invertible x E A. We say that T is unital if Tl = I'. IfTx2 = (TX)2 for all x E A, we call T a Jordan homomorphism. We examine an unsolved problem posed by 1. Kaplansky: Let A and B be unital complex Banach algebras and T : A -+ B a unital invertibility preserving linear map. What conditions on A, Band T imply that T is a Jordan homomorphism? Partial motivation for this problem are the Gleason-Kahane-Zelazko Theorem (1968) and a result of Marcus and Purves (1959), these also being special instances of the problem. We will also look at other special cases answering Kaplansky's problem, the most important being the result stating that if A is a von Neumann algebra, B a semi-simple Banach algebra and T : A -+ B a unital bijective invertibility preserving linear map, then T is a Jordan homomorphism (B. Aupetit, 2000). For a unital complex Banach algebra A, we denote the spectrum of x E A by Sp (x, A). Let a(x, A) denote the union of Sp (x, A) and the bounded components of
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