Pure state transformations induced by linear operators

AbstractWe generalise Wigner's theorem to its most general form possible for B(h) in the sense of completely characterising those vector state transformations of B(h) that appear as restrictions of duals of linear operators on B(h). We then use this result to similarly characterise all pure state transformations of general C*-algebras that appear as restrictions of duals of linear operators on the underlying algebras. This result may variously be interpreted as either a non-commutative Banach–Stone theorem, or (in the bijective case) a pure state-based description of Wigner symmetries. These results extend the work of Shultz [Comm. Math. Phys. 82 (1982) 497–509] (who considered only the case of bijections), and also complements and completes the investigation of linear maps with pure state preserving adjoints begun in [Labuschagne and Mascioni, Adv. Math. 138 (1998) 15–45]

Linear Mappings That Preserve The Derivational Structure Of C*-Algebras

Given a C -algebra A and a suitable set of derivations on A, we consider the algebras A n of n-differentiable elements of A as described in [B], before passing to an analysis of important classes of bounded linear maps between two such spaces. We show that even in this general framework, all the main features of the theory for the case C (n) (U) ! C (p) (V ) where U and V are open balls in suitable Banach spaces, are preserved (see for example [A-G-L], [Gu-L], [Ja] and [L]). With this clear analogy as motivation, we conclude by introducing the concept of C (n) -diffeomorphisms on the pure state space of a C -algebra. As a consequence of the theory developed we obtain a non-commutative Singer-Wermer theorem for C -algebras. 1 2 1 1991 Mathematics Subject Classification. Primary 46L57, 46L87; Secondary 46J15, 46L89, 58B30. 2 Keywords and phrases: C -algebra, non-commutative, composition operator, diffeomorphism, derivation. 2 1. Introductory Constructs and Defin..

Dynamics on noncommutative Orlicz spaces

Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [26] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces 〈Lcosh−1, L log(L + 1)〉, since this framework gives a better description of regular observables, and also allows for a well-defined entropy function. In the present paper we “complete” the picture by addressing the issue of the dynamics of such a system, as described by a Markov semigroup corresponding to some Dirichlet form (see [4, 13, 14]). Specifically, we show that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation strategy for extending the action of such maps to the appropriate intermediate spaces of the pair (L∞, L1). As a consequence, we obtain that completely positive quantum Markov dynamics naturally extends to the context proposed in [26

On entropy for general quantum systems

In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac’s formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theor

Multiplication operators on non-commutative spaces

Boundedness and compactness properties of multiplication operators on quantum (non-commutative) function spaces are investigated. For endomorphic multiplication operators these properties can be characterized in the setting of quantum symmetric spaces. For non-endomorphic multiplication operators these properties can be completely characterized in the setting of quantum -spaces and a partial solution obtained in the more general setting of quantum Orlicz space