Derivations on operator algebras

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 HahnBanach 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, 0. 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, 0. and Robinson, D.W. [3), Corollary 3.2.47) and a slightly weaker version of the W*- algebra derivation theorem as published in Bratteli, 0. 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 (\Vell-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, 0. 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, 0. 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.G.

Grantor dc:publisher

University of Pretoria

Year dc:date.issued

2004

Advisor dc:contributor.advisor

• Labuschagne, L.

• UCTD
• Derivations
• operator algebras

Statement dc:rights

• © 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.

Language dc:language.iso

en

Repository record dc:identifier.uri

https://repository.up.ac.za/handle/2263/85263

OAI identifier oai:identifier

oai:repository.up.ac.za:2263/85263