M.Sc.

M.Sc.

No description available.

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).

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: See full text. / AFRIKAANSE OPSOMMING: Sien volteks.

M.Sc. (Mathematics) / Please refer to full text to view abstract

Thesis (MSc)--Stellenbosch University, 2013. / Please refer to full text to view abstract.

O objetivo da dissertação é provar que toda álgebra, sobre o corpo dos números reais, algébrica e absolutamente valuada é de dimensão nita, e portanto isótopa a D . Observamos que H é a álgebra real dos Quatérnios e D R , C , H ou a álgebra real dos Octônios. A demonstração do resultado é feita gradualmente, considerando inicialmente álgebras reais absolutamente valuadas algébrica com unidade, a seguir com unidade e nalmente, algébrica. Na demonstração do teorema será necessário combinar resultados não triviais de álgebras não associativas, análise funcional, álgebras de Banach e técnicas de ultraprodutos de espaços normados. As álgebra absolutamente valuadas não são necessariamente associativas. Abraham Adrian 1947 mostrou que R , C , H e D são as únicas álgebras reais absolutamente valuadas dimensão nita e com unidade; o mesmo Albert dois anos depois, em 1949 , caracterizou Albert em de essas mesmas álgebras como as únicas que são absolutamente valuadas algébricas e com unidade sobre os reais. Em 1960 Fred B. Wright e Kazimierz Urbanik provaram que R , C , D são as únicas álgebra reais absolutamente valuadas e com unidade. Recentemente, em 1997 , Kaidi El-Amin, Maria Isabel Ramírez e Ángel Rodríguez Palacios mostraram que H e toda álgebra real absolutamente valuadas e algébrica é isótopa a uma de estas quatro. Nosso objetivo é desenvolver e unicar os resultados obtidos nestes 4 trabalhos. / Our goal here is to study the absolute valued algebraic real algebras. In order to reach our intention, we regard an absolute valued real algebra and on which one we impose: First, such one is nite-dimensional algebra; second; such one is algebraic algebra; third, such one is with unity; and in the end such one is algebraic algebra. In the latter case, our aim, it needs of certain classic results of functional analysis and others one of Banach algebras; then, we reach that such one real algebra is isotope to one of the classical absolute valued real algebras algebra and D R , C , H or D . Where H is the Quaternions real is the Octonions real algebra. The absolute valued algebras are not necessarily associative. Abraham Adrian Albert was the rst mathematician considering absolute valued algebras in a context not necessarily associative. In 1947 , he proved that any nite-dimensional absolute valued real algebra with unit element is isomorphic to either real eld H or the Octonions algebra D . Two years R , the complex eld C , the Quaternions algebra later, he demonstrated that R , C , H and D are the unique absolute valued algebraic real algebras with unit element. Recently, in 1997 , Kaidi El-Amin, Maria Isabel Ramírez and Ángel Rodríguez Palacios proved that any absolute valued algebraic real algebra is nite-dimensional.

No description available.

The concept of hyperreflexivity has previously been defined for subspaces of $B(X,Y)$, where $X$ and $Y$ are Banach spaces. We extend this concept to the subspaces of $B^n(X,Y)$, the space of bounded $n$-linear maps from $X\times\cdots\times X=X^{(n)}$ into $Y$, for any $n\in \mathbb{N}$. If $A$ is a Banach algebra and $X$ a Banach $A$-bimodule, we obtain sufficient conditions under which $\Zc^n(A,X)$, the space of all bounded $n$-cocycles from $A$ into $X$, is hyperreflexive. To do so, we define two notions related to a Banach algebra: The strong property $(\B)$ and bounded local units (b.l.u). We show that there are sufficiently many Banach algebras which have both properties. We will prove that all C$^*$-algebras and group algebras have the strong property $(\B).$ We also prove that finite CSL algebras and finite nest algebras have this property. We further show that for an arbitrary Banach algebra $A$ and each $n\geq 2$, $M_n(A)$ has the strong property $(\B)$ whenever it is equipped with a Banach algebra norm. In particular, this implies that all Banach algebras are embedded into a Banach algebra with the strong property $(\B)$. With regard to bounded local units, we show that all $C^*$-algebras and many group algebras have b.l.u. We investigate the hereditary properties of both notions to construct more example of Banach algebras with these properties. We apply our approach and show that the bounded $n$-cocycle spaces related to Banach algebras with the strong property $(\B)$ and b.l.u. are hyperreflexive provided that the space of the corresponding $n+1$-coboundaries are closed. This includes nuclear C$^*$-algebras, many group algebras, matrix spaces of certain Banach algebras and finite CSL and nest algebras. We finish the thesis with introducing {\it the hyperreflexivity constant}. We make our results more precise with finding an upper bound for the hyperreflexivity constant of the bounded $n$-cocycle spaces.