Thesis (MSc)--University of Stellenbosch, 2003. / 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).
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/53597 |
| Date | 04 1900 |
| Creators | Weigt, Martin |
| Contributors | Mouton, S., Stellenbosch University. Faculty of Science. Dept. of mathematical Sciences (applied, computer, mathematics). |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
| Format | 88 p. : ill. |
| Rights | Stellenbosch University |
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