791 research outputs found
Pure state transformations induced by linear operators
No full textAbstractWe 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]
On entropy for general quantum systems
No full textIn 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
Dynamics on noncommutative Orlicz spaces
No full textQuantum 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 Applications of Orlicz Spaces to Statistical Physics
No full textWe present a new rigorous approach based on Orlicz spaces for
the description of the statistics of large regular statistical systems, both
classical and quantum. The pair of Orlicz spaces we explicitly use are,
respectively, built on the exponential function (for the description of regular
observables) and on an entropic type function (for the corresponding
states). They form a dual pair (both for classical and quantum systems).
This pair has the advantage of being general enough to encompass regular
observables, and specific enough for the latter Orlicz space to select
states with a well-defined entropy function. Moreover for small quantum
systems, this pair is shown to agree with the classical pairing of bounded
linear operators on a Hilbert space, and the trace-class operators.Grant Number N N202 208238; Foundation
for Polish Science TEAM project cofinanced by the EU European Regional
Development Fund for W.A. Majewski; National Research
Foundation for L.E. Labuschagn
Weighted noncommutative Banach function spaces
No full textWe review the concept of a weighted noncommutative Banach function space. This concept constitutes a generalisation of the by now well-known theory of noncommutative Banach function spaces associated with a semifinite von Neumann algebra. In this review we remind the reader of the quantum statistical problem which gave birth to this concept, we investigate the extent to which a weighted theory of measurable operators agrees with the standard theory, we explore competing methods for defining such spaces, before finally describing the monotone interpolation theory of such space
A time-variant norm constrained interpolation problem arising from relaxed commutant lifting
No full textA time-variant analogue of an interpolation problem equivalent to the relaxed commutant lifting problem is introduced and studied. In a somewhat less general form the problem already appears in the analysis of the set of all solutions to the three chain completion problem. The interpolants are upper triangular operator matrices of which the columns induce contractive operators. The set of all solutions of the problem is described explicitly. The results presented are time-variant analogues of the main theorems in [A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to the relaxed commutant lifting problem, Acta Sci. Math. (Szeged) 72 (2006), 299--318]
Representing silk design: Nicholas Joubert de L'Hiberderie and Le dessinateur pour les etoffes d'or, d'argent et de soie (Paris, 1765)
No full textLe Dessinateur pour les étoffes d'or, d'argent et de soie was published in Paris in 1765 depite the reservations of the silk-weaving guild of Lyons, receiving a good response in the Enlightenment press. As the first description on French of the trade of silk designer to appear in the public domain, it has become an important work on which much subsequent history has been based, often rather uncritically. This article delves into the representation of design offered by this text, evaluating it against the personal experience of its author, the literary and manufacturing heritage on which he drew, and the readership for women it was intended. The analysis is based on the form and content of the book, the structure, vocabulary and illustrations. Comparative data are drawn from two other important publications on silk manufacture, the relevant sections of Denis Diderot's Encyclopédie and Jean Paulet's L'Art du fabriquant
Marketing communications plan for L.E. Jones company Menominee, Michigan
No full textThe author researched the feasibility of improving marketing communications for L.E. Jones Company of Menominee, Michigan. Based on career experience and scholarship gained through the Cardinal Stritch Master of Business Administration program, the author was able to identify the problems facing the firm, analyze the marketing communications needs, recommend various alternatives, assist in the selection of alternatives, and create the actual advertising and sales support materials. These improvements in marketing communications will permit the firm to more confidently pursue new business in an increasingly competitive market
Derivations on operator algebras
No full textDissertation (MSc)--University of Pretoria, 2004.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.Mathematics and Applied MathematicsMScUnrestricte
Collectively compact and collectively strictly singular sets of linear operators
No full textDissertation (MSc (Mathematics))--University of Pretoria, 1994.In this thesis the concept of collectively compact sets of operators is studied. As a reason for the study of such operators it is shown how collectively compact sets of operators are applicable to an approximation theory for Fredholm integral equations of the second kind where the kernel is continuous. In this case the integral operator mapping C[a, b] into C[a, b] is compact and the set of numerical-integral operators approximating the integral operator is collectively compact. Convergence theorems and error bounds are given for this type of situation. Once the importance of the concept of collective compactness has been established, properties of such sets of operators are studied. A characterisation of collectively compact sets of operators in terms of countable subsets is given. In addition, a comparison between totally bounded sets and collectively compact sets of compact operators is done since the approximation theory mentioned above is applicable to sets of operators that are collectively compact but not totally bounded. Perturbation theorems involving perturbations of semi-Fredholm operators with collectively compact sets of operators are also studied. The concept of collectively strictly singular sequences of operators is defined and perturbation theorems for perturbations of semi-Fredholm operators with collectively strictly singular sequences of operators are given. It is probable that the concept of collective strict singularity might be applicable in establishing an approximation theory for Fredholm integral equations of the second kind with measurable, discontinuous kernel where the integral operator maps the Lebesgue space £ 1 into £ 1• The concept of collectively strictly cosingular sequences of operators naturally arises and is therefore defined. It is noted that analogous perturbation theorems to the ones proved for collectively strictly singular sequences of operators could easily be proven by suitably dualising the proofs for the above-mentioned theorems.Mathematics and Applied MathematicsMSc (Mathematics
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