# Spectrum preserving linear mappings between Banach algebras

Weigt, Martin (2003-04)

Thesis (MSc)--University of Stellenbosch, 2003.

Thesis

ENGLISH ABSTRACT: Let A and B be unital complex Banach algebras with identities 1 and I' 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 <C \ Sp (x, A). We denote the spectral radius of x E A by p(x, A). A unital linear map T between unital complex Banach algebras A and B is invertibility preserving if and only if Sp (Tx, B) C Sp (x, A) for all x E A. This leads one to consider the problems that arise when, in turn, we replace the invertibility preservation property of T in Kaplansky's problem with Sp (Tx, B) = Sp (x, A) for all x E A, a(Tx, B) = a(x, A) for all x E A, and p(Tx, B) = p(x, A) for all x E A. We will also investigate some special cases that are solutions to these problems. The most important of these special cases says that if A is a semi-simple Banach algebra, B a primitive Banach algebra with minimal ideals and T : A -+ B a surjective linear map satisfying a(Tx, B) = a(x, A) for all x E A, then T is a Jordan homomorphism (B. Aupetit and H. du T. Mouton, 1994).

AFRIKAANSE OPSOMMING: Gestel A en B is unitale komplekse Banach algebras met identiteite 1 en I' onderskeidelik. 'n Lineêre afbeelding T : A -+ B is omkeerbaar-behoudend as Tx omkeerbaar in B is vir elke omkeerbare element x E A. Ons sê dat T unitaal is as Tl = I'. As Tx2 = (TX)2 vir alle x E A, dan noem ons T 'n Jordan homomorfisme. Ons ondersoek 'n onopgeloste probleem wat deur I. Kaplansky voorgestel is: Gestel A en B is unitale komplekse Banach algebras en T : A -+ B is 'n unitale omkeerbaar-behoudende lineêre afbeelding. Watter voorwaardes op A, B en T impliseer dat T 'n Jordan homomorfisme is? Gedeeltelike motivering vir hierdie probleem is die Gleason-Kahane-Zelazko Stelling (1968) en 'n resultaat van Marcus en Purves (1959), wat terselfdertyd ook spesiale gevalle van die probleem is. Ons salook na ander spesiale gevalle kyk wat antwoorde lewer op Kaplansky se probleem. Die belangrikste van hierdie resultate sê dat as A 'n von Neumann algebra is, B 'n semi-eenvoudige Banach algebra is en T : A -+ B 'n unitale omkeerbaar-behoudende bijektiewe lineêre afbeelding is, dan is T 'n Jordan homomorfisme (B. Aupetit, 2000). Vir 'n unitale komplekse Banach algebra A, dui ons die spektrum van x E A aan met Sp (x, A). Laat cr(x, A) die vereniging van Sp (x, A) en die begrensde komponente van <C \ Sp (x, A) wees. Ons dui die spektraalradius van x E A aan met p(x, A). 'n Unitale lineêre afbeelding T tussen unit ale komplekse Banach algebras A en B is omkeerbaar-behoudend as en slegs as Sp (Tx, B) c Sp (x, A) vir alle x E A. Dit lei ons om die probleme te beskou wat ontstaan as ons die omkeerbaar-behoudende eienskap van T in Kaplansky se probleem vervang met Sp (Tx, B) = Sp (x, A) vir alle x E A, O"(Tx, B) = O"(x, A) vir alle x E A en p(Tx, B) = p(x, A) vir alle x E A, onderskeidelik. Ons salook 'n paar spesiale gevalle van hierdie probleme ondersoek. Die belangrikste van hierdie spesiale gevalle sê dat as A 'n semi-eenvoudige Banach algebra is, B 'n primitiewe Banach algebra met minimale ideale is, en T : A -+ B 'n surjektiewe lineêre afbeelding is sodanig dat O"(Tx, B) = O"(x, A) vir alle x E A, dan is T 'n Jordan homomorfisme (B. Aupetit en H. du T. Mouton, 1994).

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