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Fredholm classes in operator algebras
Thesis (PhD)--University of Pretoria, 1995.
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| Format: | Thesis |
| Language: | English |
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University of Pretoria
2022
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| _version_ | 1867613551238053888 |
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| access_status_str | Open Access |
| author2 | Vermaak, Jacobus A. |
| author_browse | Vermaak, Jacobus A. |
| author_facet | Vermaak, Jacobus A. |
| collection | Thesis |
| dc_rights_str_mv | © 2020 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. |
| description | Thesis (PhD)--University of Pretoria, 1995. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/85468 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:37:56.779Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2022 |
| publishDateRange | 2022 |
| publishDateSort | 2022 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/85468 Fredholm classes in operator algebras Vermaak, Jacobus A. UCTD Fredholm classes Algebras Thesis (PhD)--University of Pretoria, 1995. In the first part of the thesis the divisors of zero in the "Calkin" algebra of a von Neumann algebra relative to an arbitrary closed ideal are used to define classes of Fredholm-type operators. The geometrical, algebraic and topological properties of these classes are studied completely and shown to be similar to properties possessed by the Fredholm classes. It is also investigated precisely in which cases these classes coincide with the Fredholm classes, and hence useful characterisations of the semi-Fredholm operators in the von Neumann algebra setting are obtained. In the second part of the thesis lifting theorems for a number of properties of elements in the "Calkin" algebra are proved and numerous unsolved problems are stated. The study is concluded with a Fredholm theory for closed densely defined operators affiliated to a von Neumann algebra. Although unanswered questions ( which are described in the thesis) remam, the results are reasonably complete, especially with respect to certain norm closed ideals which are of principal interest in the theory of operator algebras. Chapter 1 contains a summary of the notation used throughout the thesis as well as some preliminary results. In Chapter 2 the left and right Fredholm operators relative to any closed ideal I in any von Neumann algebra A are characterised in terms of the left and right topological divisors of zero in the quotient algebra A/I. A characterisation of the semi-Fredholm operators in a semifinite von Neumann algebra, relative to the closed ideal generated by the projections with finite trace, is proved and then used to give a description of the essential spectrum in terms of the eigenvalues in the Calkin algebra. Chapter 3 contains a few lifting results on properties of elements in the "Calkin" algebra. In Chapter 4 some of the results contained in Chapter 2 are extended to similar results for T-measurable Fredholm operators relative to the closure of the trace class in the topology of convergence in measure. Mathematics and Applied Mathematics PhD Unrestricted 2022-05-17T11:21:37Z 2022-05-17T11:21:37Z 30/7/2021 1995 Thesis * https://repository.up.ac.za/handle/2263/85468 en © 2020 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. application/pdf University of Pretoria |
| spellingShingle | UCTD Fredholm classes Algebras Fredholm classes in operator algebras |
| title | Fredholm classes in operator algebras |
| title_full | Fredholm classes in operator algebras |
| title_fullStr | Fredholm classes in operator algebras |
| title_full_unstemmed | Fredholm classes in operator algebras |
| title_short | Fredholm classes in operator algebras |
| title_sort | fredholm classes in operator algebras |
| topic | UCTD Fredholm classes Algebras |
| url | https://repository.up.ac.za/handle/2263/85468 |