Full Text Available
Note: Clicking the button above will open the full text document at the original institutional repository in a new window.
Domination and generalized domination in ordered Banach algebras
Thesis (PhD)--Stellenbosch University, 2024.
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | en_ZA en_ZA |
| Published: |
Stellenbosch : Stellenbosch University
2024
|
| Subjects: | |
| Tags: |
No Tags, Be the first to tag this record!
|
| _version_ | 1867613818816823296 |
|---|---|
| access_status_str | Open Access |
| author | Rabearivony, Andriamahazosoa Dimbinantenaina |
| author2 | Mouton, Sonja |
| author_browse | Mouton, Sonja Rabearivony, Andriamahazosoa Dimbinantenaina |
| author_facet | Mouton, Sonja Rabearivony, Andriamahazosoa Dimbinantenaina |
| author_sort | Rabearivony, Andriamahazosoa Dimbinantenaina |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2024. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/130492 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA en_ZA |
| last_indexed | 2026-06-10T12:42:11.774Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2024 |
| publishDateRange | 2024 |
| publishDateSort | 2024 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/130492 Domination and generalized domination in ordered Banach algebras Rabearivony, Andriamahazosoa Dimbinantenaina Mouton, Sonja Benjamin, Ronalda Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Banach algebras -- Mathematical models Spectral theory (Mathematics) -- Mathematical models Domination problem Operator theory -- Mathematics Generalized domination problem UCTD Thesis (PhD)--Stellenbosch University, 2024. ENGLISH 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. AFRIKAANSE 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. Doctorate 2024-02-28T08:36:42Z 2024-04-26T19:34:55Z 2024-02-28T08:36:42Z 2024-04-26T19:34:55Z 2024-03 Thesis https://scholar.sun.ac.za/handle/10019.1/130492 en_ZA en_ZA Stellenbosch University xv, 95 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Banach algebras -- Mathematical models Spectral theory (Mathematics) -- Mathematical models Domination problem Operator theory -- Mathematics Generalized domination problem UCTD Rabearivony, Andriamahazosoa Dimbinantenaina Domination and generalized domination in ordered Banach algebras |
| title | Domination and generalized domination in ordered Banach algebras |
| title_full | Domination and generalized domination in ordered Banach algebras |
| title_fullStr | Domination and generalized domination in ordered Banach algebras |
| title_full_unstemmed | Domination and generalized domination in ordered Banach algebras |
| title_short | Domination and generalized domination in ordered Banach algebras |
| title_sort | domination and generalized domination in ordered banach algebras |
| topic | Banach algebras -- Mathematical models Spectral theory (Mathematics) -- Mathematical models Domination problem Operator theory -- Mathematics Generalized domination problem UCTD |
| url | https://scholar.sun.ac.za/handle/10019.1/130492 |
| work_keys_str_mv | AT rabearivonyandriamahazosoadimbinantenaina dominationandgeneralizeddominationinorderedbanachalgebras |