Derivations on operator algebras

Abstract

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.