Domination and generalized domination in ordered Banach algebras

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dc.contributor.advisorMouton, Sonjaen_ZA
dc.contributor.advisorBenjamin, Ronaldaen_ZA
dc.contributor.authorRabearivony, Andriamahazosoa Dimbinantenainaen_ZA
dc.contributor.otherStellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.en_ZA
dc.date.accessioned2024-02-28T08:36:42Z
dc.date.accessioned2024-04-26T19:34:55Z
dc.date.available2024-02-28T08:36:42Z
dc.date.available2024-04-26T19:34:55Z
dc.date.issued2024-03
dc.descriptionThesis (PhD)--Stellenbosch University, 2024. en_ZA
dc.description.abstractENGLISH ABSTRACT: Let A be a complex unital Banach algebra. An algebra cone in A is a non-empty subset C of A containing the unit of A and which is closed under addition, multiplication and non-negative scalar multiplication. Any algebra cone C in A corresponds to a partial ordering in A satisfying C = {a ∈ A : 0 ≤ a}. The pair (A, C) is called an ordered Banach algebra (OBA). In this dissertation, we investigate the domination (resp. generalized dom- ination) problems in OBAs, stated as follows: given an OBA (A, C) and two elements a, b ∈ A such that 0 ≤ a ≤ b (resp. ±a ≤ b), under which conditions do we have that a property of b is inherited by a? In 2014, Mouton and Muzundu investigated the domination problem for ergodic elements, where an element a ∈ A is called ergodic if the sequence (Ln 1 aᵏ) is convergent. While their theorem is an outstanding extension k=1 n of an operator theoretic result, the assumption of weak monotonicity of the spectral radius in the quotient algebra is quite strong, limiting its applicability to the regular operators on a Dedekind complete Banach lattice. Our first main result in this dissertation (Theorem 2.3.10) provides a partial solution to this problem in the form of an ergodic domination-type theorem without assuming weak monotonicity of the spectral radius in the quotient algebra. Furthermore, since their result was only for the domination problem, we extend it, as well as most of the other existing domination results, to generalized domination results. (See, in particular, Theorems 3.10.5, 3.12.6 and 3.13.8 regarding ergodicity.) These two contributions provide (partial) answers to two open questions in the survey paper [42] of Mouton and Raubenheimer.en_ZA
dc.description.abstractAFRIKAANSE OPSOMMING: Laat A ’n komplekse unitale Banach-algebra wees. ’n Algebra-kee¨l in A is ’n nie- lee¨ deelversameling C van A wat die eenheidselement van A bevat en wat geslote onder optelling, vermenigvuldiging en nie-negatiewe skalaarvermenigvuldiging is. Enige algebra-kee¨ l C in A stem ooreen met ’n parsie¨ le ordening in A wat aan C = {a ∈ A : 0 ≤ a} voldoen. Die paar (A, C) word ’n geordende Banach- algebra (GBA) genoem. In hierdie proefskrif ondersoek ons die domineringsprobleem (onderskeide- lik, veralgemeende domineringsprobleem) in GBAs: gegee ’n GBA (A, C) en twee elemente a, b ∈ A sodat 0 ≤ a ≤ b (onderskeidelik, ±a ≤ b), onder watter voorwaardes volg dat ’n eienskap van b deur a gee¨ rf word? In 2014 het Mouton en Muzundu die domineringsprobleem vir ergodiese elemente ondersoek, waar ’n element a ∈ A ergodies genoem word as die ry (Ln 1 aᵏ) konvergent is. Hoewel hulle stelling ’n uitstaande uitbreiding van k=1 n ’n operator-teoretiese resultaat is, is die aanname van swak monotoonheid van die spektraalradius in die kwosie¨ nt-algebra besonder sterk, en beperk dit sy toepasbaarheid tot die reguliere operatore op ’n Dedekind-volledige Banach- rooster. Ons eerste hoofresultaat in hierdie proefskrif (Stelling 2.3.10) voorsien ’n gedeeltelike oplossing vir hierdie probleem in die vorm van ’n ergodiese domin- eringstipe stelling sonder die aanname van swak monotoonheid van die spek- traalradius in die kwosie¨ nt-algebra. Verder, omdat hulle resultaat slegs vir die domineringsprobleem was, brei ons dit, sowel as die meeste van die ander bestaande domineringsresultate, uit na veralgemeende domineringsresultate. (Sien, in die besonder, Stellings 3.10.5, 3.12.6 en 3.13.8 aangaande ergodisiteit.) Hierdie twee bydraes voorsien (gedeeltelike) antwoorde op twee oop vrae in die oorsigartikel [42] van Mouton en Raubenheimer.en_ZA
dc.description.versionDoctorateen_ZA
dc.format.extentxv, 95 pagesen_ZA
dc.identifier.urihttps://scholar.sun.ac.za/handle/10019.1/130492
dc.language.isoen_ZAen_ZA
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : Stellenbosch Universityen_ZA
dc.rights.holderStellenbosch Universityen_ZA
dc.subject.lcshBanach algebras -- Mathematical modelsen_ZA
dc.subject.lcshSpectral theory (Mathematics) -- Mathematical modelsen_ZA
dc.subject.lcshDomination problemen_ZA
dc.subject.lcshOperator theory -- Mathematicsen_ZA
dc.subject.lcshGeneralized domination problemen_ZA
dc.subject.nameUCTDen_ZA
dc.titleDomination and generalized domination in ordered Banach algebrasen_ZA
dc.typeThesisen_ZA
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