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 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.