# Fredholm theory in ordered Banach algebras

Benjamin, Ronalda Abigail Marsha (2016-03)

Thesis (PhD)--Stellenbosch University, 2016

Thesis

ENGLISH ABSTRACT : Since its inception, Fredholm theory has become an important aspect of spectral theory. Among the spectra arising within Fredholm theory is the Weyl spectrum which has been intensively studied by several authors, both in the operator case and in the general situation of Banach algebras. The Weyl spectrum of a bounded linear operator T on a Banach space X is the set T K2K(X) s(T + K), where s(T) denotes the spectrum of T and K(X) the closed ideal of all compact operators on X. A recent result by E. A. Alekhno shows that, if “Banach space" is replaced by an arbitrary complex Banach lattice E, then the Weyl spectrum of T on E can be made more precise, and takes on the form T 0 K2K(E) s(T + K). By an ordered Banach algebra (OBA) we mean a complex unital Banach algebra A containing an algebra cone; that is, a subset C which contains the unit of A and is closed under addition, multiplication and positive scalar multiplication. As is well-known, the algebra of all bounded linear operators on a complex Banach lattice is an important example of an OBA. If A denotes an arbitrary OBA with algebra cone C, B a Banach algebra and T : A ! B a homomorphism with N(T) = fa 2 A : Ta = 0g indicating the null space of T, then the Weyl spectrum T c2N(T) s(a + c) of a 2 A is in general strictly contained in the set T c2C\N(T) s(a + c) — see Example 4.1.13. As a result of this, we investigate the latter set, which we shall refer to as the upperWeyl spectrum of a 2 A. In this work the concept of the upper Browder spectrum of a will also be introduced and results related to these spectra and the underlying sets of elements on which these spectra are defined will be given. This thesis aims to present initial steps taken in the effort of unifying the theory of positivity in OBAs with the general Fredholm theory in Banach algebras.

AFRIKAANSE OPSOMMING : Sedert die bekendstelling daarvan, het die Fredholmteorie ‘n belangrike aspek van spektraalteorie geword. Onder die spektra wat ontstaan in Fredholmteorie is die Weyl spektrum, wat alreeds in diepte bestudeer is deur verskeie outeurs, beide in die operatorkonteks en in willekeurige Banach algebras. Die Weyl spektrum van ‘n begrensde lineêre operator T op ’n Banach ruimte X is die versameling T K2K(X) s(T + K), waar s(T) die spektrum van T voorstel en K(X) die geslote ideaal van kompakte operatore op X. ‘n Resultaat wat onlangs deur E. A. Alekhno bewys is, toon dat, as “Banach ruimte" vervang word met ‘n willekeurige Banach rooster E, dan kan die voorstelling van dieWeyl spektrum van T op E meer presies gemaak word, en dit word gegee deur T 0 K2K(E) s(T + K). Met ‘n geordende Banach algebra (GBA) bedoel ons ’n komplekse unitale Banach algebra A wat ‘n algebra-keël bevat; dit is, ‘n deelversameling C wat die eenheid van A as element het en wat geslote is onder optelling, vermenigvuldiging en positiewe skalaarvermenigvuldiging. Die versameling van begrensde lineêre operatore op ’n komplekse Banach rooster is ’n belangrike voorbeeld van ’n GBA. As A ‘n willekeurige GBA met algebra-keël C voorstel, B ‘n Banach algebra en T : A ! B ‘n homomorfisme met N(T) = fa 2 A : Ta = 0g die nulruimte van T, dan is dieWeyl spektrum T c2N(T) s(a+c) van a 2 A in die algemeen eg bevat in die versameling T c2C\N(T) s(a + c) — kyk na Voorbeeld 4.1.13. As gevolg hiervan, ondersoek ons die laasgenoemde versameling, wat ons die bo-Weyl spektrum van a 2 A sal noem. In hierdie werk word die konsep van die bo-Browder spektrum van a ook bekend gestel en resultate wat verband hou met hierdie spektra en met die onderliggende versamelings van elemente waarop hierdie spektra gedefineer is sal gegee word. Die doel van hierdie tesis is die bekendstelling van die beginstappe wat geneem is in die poging om die teorie van positiwiteit in GBAs

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