This work primarily provides some detail of results on domain properties of closed (unbounded) derivations on C*-algebras. The focus is on Section 4: Domain Properties where a combination of topological and algebraic conditions for certain results are illustrated. Various earlier results are incorporated into the proofs of Section 4. Section 1: Basics lists some basic functional analysis results, operator algebra theory (of particular importance is the continuous functional calculus and certain results on the state and pure state space) and a special section on operator closedness. Some Hahn-Banach results are also listed. The results of this section were obtained from various sources (Zhu, K. [24], Kadison, R.V. and Ringrose, J.R. [8], Goldberg, S. [6], Rudin, W. [20], Sakai, S. [22], Labuschagne, L.E. [10] and others). The development of the representation theory presented in Section 1.1.7 was compiled from Bratteli, O. and Robinson, D.W. [3], Section 2.3. Section 2: Derivations provides some background to the roots of derivations in quantum mechanics. The results of Section 2.2 (Commutators) are due to various authors, mainly obtained from Sakai, S. [22]. A detailed proof of Theorem 45 is given. Section 2.3 (Differentiability) contains some Singer-Wermer results mainly obtained from Mathieu, M. and Murphy, G.J. [13] and Theorem 50 is proved in detail. Section 2.4 deals with conditions for bounded derivations (Sakai, S. [22] and (Johnson-Sinclair, cf. (Sakai, S. [22])), and Theorem 51 is proved in detail. Section 2.5 deals with the well published derivation theorem (Sakai, S.[22], Section 2.5 and Bratteli, O. and Robinson, D.W. [3], Corollary 3.2.47) and a slightly weaker version of the W *-algebra derivation theorem as published in Bratteli, O. and Robinson, D.W. [3], Corollary 3.2.47, is proved here. Section 3: Derivations as generators first introduces some basic semi-group theory (obtained from Pazy, A. [16], Section 1.1 and 1.2) after which the well-behavedness property is introduced in Section 3.2. Some general results mainly obtained from Sakai, S. [22], Section 3.2, is detailed. The ;proofs of Theorems 61 and 62 makes use of various previous results and were conducted in detail. Section 3.3 (Well-behavedness and generators) draws a link between the well-behavedness property and conditions for a derivation to be a semi-group generator. The results are obtained from Pazy, A. [16], Section 1.4, and Bratteli, O. and Robinson, D.W. [3], Section 3.2.4 Special care was taken in the outlined proof of Theorem 68. A proof of a domain characterization theorem (due to Bratteli, O. and Robinson, D.W. [3], Proposition 3.2.55) is provided (Theorem 69) and used in the construction of the counter example of Section 4.6. Section 4: Domain properties is occupied with un-bounded derivations on C*-algebras and their domain properties. Some initial complex function theory is developed after which four important domain preserving theorems are proved in full detail: the inverse function (Section 4.2), the exponential function (Section 4.3), Fourier analysis on the domain (Section 4.4) and C2-functions on the domain (Section 4.5). The non domain preserving C1 function counter example is presented in Section 4.6. The results of Section 4 appear in Bratteli, O. and Robinson, D.W. [3], Section 3.2.2, and Sakai, S. [22], Section 3.3, and the counter example is due to McIntosh, A. [11]. All the results in Section 4 are presented in full detail not available in this format from any of the sources used. Some Topelitz operator theory is used with reference to Brown, A. and Halmos, P.R. [4], 94, and the Fourier coefficients of a required function is calculated. Some results on direct sum spaces and the core of a linear operator were used from Kadison, R.V. and Ringrose, J.R. [8], Section 2.6 and page 160, as well as Zhu, K. [24], Section 14.2. / Dissertation (MSc (Mathematics and Applied Mathematics))--University of Pretoria, 2004. / Mathematics and Applied Mathematics / unrestricted
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:up/oai:repository.up.ac.za:2263/30574 |
| Date | 23 February 2005 |
| Creators | Holm, Rudolph |
| Contributors | Labuschagne, Louis, upetd@ais.up.ac.za |
| Publisher | University of Pretoria |
| Source Sets | South African National ETD Portal |
| Detected Language | English |
| Type | Dissertation |
| Rights | © 2004, 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. |
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