Local Operator Spaces, Unbounded Operators and Multinormed C∗-Algebras
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Differentiation of Operator Functions in Non-Commutative Lp-Spaces B
ARTICLE IN PRESS Journal of Functional Analysis 212 (2004) 28–75 Differentiation of operator functions in non-commutative Lp-spaces B. de Pagtera and F.A. Sukochevb,Ã a Department of Mathematics, Faculty ITS, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands b School of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042 SA, Australia Received 14 May 2003; revised 18 September 2003; accepted 15 October 2003 Communicated by G. Pisier Abstract The principal results in this paper are concerned with the description of differentiable operator functions in the non-commutative Lp-spaces, 1ppoN; associated with semifinite von Neumann algebras. For example, it is established that if f : R-R is a Lipschitz function, then the operator function f is Gaˆ teaux differentiable in L2ðM; tÞ for any semifinite von Neumann algebra M if and only if it has a continuous derivative. Furthermore, if f : R-R has a continuous derivative which is of bounded variation, then the operator function f is Gaˆ teaux differentiable in any LpðM; tÞ; 1opoN: r 2003 Elsevier Inc. All rights reserved. 1. Introduction Given an arbitrary semifinite von Neumann algebra M on the Hilbert space H; we associate with any Borel function f : R-R; via the usual functional calculus, the corresponding operator function a/f ðaÞ having as domain the set of all (possibly unbounded) self-adjoint operators a on H which are affiliated with M: This paper is concerned with the study of the differentiability properties of such operator functions f : Let LpðM; tÞ; with 1pppN; be the non-commutative Lp-space associated with ðM; tÞ; where t is a faithful normal semifinite trace on the von f Neumann algebra M and let M be the space of all t-measurable operators affiliated ÃCorresponding author. -
Introduction to Operator Spaces
Lecture 1: Introduction to Operator Spaces Zhong-Jin Ruan at Leeds, Monday, 17 May , 2010 1 Operator Spaces A Natural Quantization of Banach Spaces 2 Banach Spaces A Banach space is a complete normed space (V/C, k · k). In Banach spaces, we consider Norms and Bounded Linear Maps. Classical Examples: ∗ C0(Ω),M(Ω) = C0(Ω) , `p(I),Lp(X, µ), 1 ≤ p ≤ ∞. 3 Hahn-Banach Theorem: Let V ⊆ W be Banach spaces. We have W ↑ & ϕ˜ ϕ V −−−→ C with kϕ˜k = kϕk. It follows from the Hahn-Banach theorem that for every Banach space (V, k · k) we can obtain an isometric inclusion (V, k · k) ,→ (`∞(I), k · k∞) ∗ ∗ where we may choose I = V1 to be the closed unit ball of V . So we can regard `∞(I) as the home space of Banach spaces. 4 Classical Theory Noncommutative Theory `∞(I) B(H) Banach Spaces Operator Spaces (V, k · k) ,→ `∞(I)(V, ??) ,→ B(H) norm closed subspaces of B(H)? 5 Matrix Norm and Concrete Operator Spaces [Arveson 1969] Let B(H) denote the space of all bounded linear operators on H. For each n ∈ N, n H = H ⊕ · · · ⊕ H = {[ξj]: ξj ∈ H} is again a Hilbert space. We may identify ∼ Mn(B(H)) = B(H ⊕ ... ⊕ H) by letting h i h i X Tij ξj = Ti,jξj , j and thus obtain an operator norm k · kn on Mn(B(H)). A concrete operator space is norm closed subspace V of B(H) together with the canonical operator matrix norm k · kn on each matrix space Mn(V ). -
Quantum Duality, Unbounded Operators, And
JOURNAL OF MATHEMATICAL PHYSICS 51, 063511 ͑2010͒ Quantum duality, unbounded operators, and inductive limits ͒ Anar Dosia Middle East Technical University Northern Cyprus Campus, Guzelyurt, KKTC, Mersin 10, Turkey ͑Received 20 October 2009; accepted 7 April 2010; published online 14 June 2010͒ In this paper, we investigate the inductive limits of quantum normed ͑or operator͒ spaces. This construction allows us to treat the space of all noncommutative con- tinuous functions over a quantum domain as a quantum ͑or local operator͒ space of all matrix continuous linear operators equipped with S-quantum topology. In par- ticular, we classify all quantizations of the polynormed topologies compatible with the given duality proposing a noncommutative Arens–Mackey theorem. Further, the inductive limits of operator spaces are used to introduce locally compact and lo- cally trace class unbounded operators on a quantum domain and prove the dual realization theorem for an abstract quantum space. © 2010 American Institute of Physics. ͓doi:10.1063/1.3419771͔ I. INTRODUCTION The operator analogs of locally convex spaces have been started to develop in Ref. 14 by Effros and Webster. The central goal of this direction is to create a theory of quantum polynormed spaces or quantum spaces, which should reflect the “locally convex space chapters” of quantum functional analysis. This theory has been created as the basic language of quantum physics. To have a comprehensive mathematical model of quantum physics, it is necessary to consider the linear spaces of unbounded Hilbert space operators or “noncommutative variable spaces.”19 This is the mathematical background of Heisenberg’s “matrix mechanics.” The quantizations of a variable space have provided functional analysis with the new constructions, methods, and problems.22 Being a new and modern branch of functional analysis, quantum functional analysis can be divided into the normed13,20 and polynormed ͑or locally convex͒ topics,14,25 as in the classical theory. -
Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems
UNIVERSITY of CALIFORNIA Santa Barbara Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering by Ryan M. Mohr Committee in charge: Professor Igor Mezi´c,Chair Professor Bassam Bamieh Professor Jeffrey Moehlis Professor Mihai Putinar June 2014 The dissertation of Ryan M. Mohr is approved: Bassam Bamieh Jeffrey Moehlis Mihai Putinar Igor Mezi´c,Committee Chair May 2014 Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems Copyright c 2014 by Ryan M. Mohr iii To my family and dear friends iv Acknowledgements Above all, I would like to thank my family, my parents Beth and Jim, and brothers Brennan and Gralan. Without their support, encouragement, confidence and love, I would not be the person I am today. I am also grateful to my advisor, Professor Igor Mezi´c,who introduced and guided me through the field of dynamical systems and encouraged my mathematical pursuits, even when I fell down the (many) proverbial (mathematical) rabbit holes. I learned many fascinating things from these wandering forays on a wide range of topics, both contributing to my dissertation and not. Our many interesting discussions on math- ematics, research, and philosophy are among the highlights of my graduate school career. His confidence in my abilities and research has been a constant source of encouragement, especially when I felt uncertain in them myself. His enthusiasm for science and deep, meaningful work has set the tone for the rest of my career and taught me the level of research problems I should consider and the quality I should strive for. -
Spectral Measures in Locally Convex Algebras by Helmut H
SPECTRAL MEASURES IN LOCALLY CONVEX ALGEBRAS BY HELMUT H. SCHAEFER The University of Michigan, U.S.A.Q) Introduction The central subject of classical spectral theory has been the spectral representation of self-adjoint and normal operators in Hilbert space; after Hilbert had given a complete treatment of the bounded case, the spectra] theory of unbounded self-adjoint and normal operators was developed by yon Neumann and others. (An excellent account can be found in [23].) Abstractions of algebras of Hermitian and normal operators were considered by various authors, notably Stone [22] who characterized such algebras as algebras of conti- nuous functions on a compact (Hausdorff) space. Investigations by Freudenthal [9] and Nakano [15] (especially papers 1 and 2), also leading to spectral theories, went in a different direction. Generalizations of unitary operators to reflexive Banach spaces were considered by Lorch [13]; Lorch also developed an operational calculus for those operators in reflexive Banach spaces that can be represented by a spectral measure [12]. (Taylor [24] developed such a calculus for closed operators on a Banach space whose spectrum does not cover the plane, but necessarily restricted to functions locally holomorphic on the spectrum. There are some recent results in this direction for operators on a locally convex space [26], [16].) The most extensive work on bounded and unbounded operators in a Banach space is due to Dunford, Schwartz, Bade and others (for a detailed bibliography, see [6] and [8]). Dunford considers operators that have a countably additive resolution of the identity, but may differ from a spectral operator (in the sense of Definition 4, Section 4 of this paper) by a quasi-nilpotent. -
The Notion of Observable and the Moment Problem for ∗-Algebras and Their GNS Representations
The notion of observable and the moment problem for ∗-algebras and their GNS representations Nicol`oDragoa, Valter Morettib Department of Mathematics, University of Trento, and INFN-TIFPA via Sommarive 14, I-38123 Povo (Trento), Italy. [email protected], [email protected] February, 17, 2020 Abstract We address some usually overlooked issues concerning the use of ∗-algebras in quantum theory and their physical interpretation. If A is a ∗-algebra describing a quantum system and ! : A ! C a state, we focus in particular on the interpretation of !(a) as expectation value for an algebraic observable a = a∗ 2 A, studying the problem of finding a probability n measure reproducing the moments f!(a )gn2N. This problem enjoys a close relation with the selfadjointness of the (in general only symmetric) operator π!(a) in the GNS representation of ! and thus it has important consequences for the interpretation of a as an observable. We n provide physical examples (also from QFT) where the moment problem for f!(a )gn2N does not admit a unique solution. To reduce this ambiguity, we consider the moment problem n ∗ (a) for the sequences f!b(a )gn2N, being b 2 A and !b(·) := !(b · b). Letting µ!b be a n solution of the moment problem for the sequence f!b(a )gn2N, we introduce a consistency (a) relation on the family fµ!b gb2A. We prove a 1-1 correspondence between consistent families (a) fµ!b gb2A and positive operator-valued measures (POVM) associated with the symmetric (a) operator π!(a). In particular there exists a unique consistent family of fµ!b gb2A if and only if π!(a) is maximally symmetric. -
Operator Spaces
Operator Spaces. Alonso Delf´ın University of Oregon. November 19, 2020 Abstract The field of operator spaces is an important branch of functional analysis, commonly used to generalize techniques from Banach space theory to algebras of operators on Hilbert spaces. A \concrete" operator space E is a closed subspace of L(H) for a Hilbert space H. Almost 35 years ago, Zhong-Jin Ruan gave an abstract characterization of operator spaces, which allows us to forget about the concrete Hilbert space H. Roughly speaking, an \abstract" operator space consists of a normed space E together with a family of matrix norms on Mn(E) satisfying two axioms. Ruan's Theorem states that any abstract operator space is completely isomorphic to a concrete one. On this document I will give a basic introduction to operator spaces, providing many examples. I will discuss completely bounded maps (the morphisms in the category of operator spaces) and try to give an idea of why abstract operator spaces are in fact concrete ones. Time permitting, I will explain how operator spaces are used to define a non-selfadjoint version of Hilbert modules and I'll say how this might be useful for my research. 1 Definitions and Examples Definition 1.1. Let H be a Hilbert space. An operator space E is a closed subspace of L(H). Example 1.2. 1. Any C∗-algebra A is an operator space. 2. Let H1 and H2 be Hilbert spaces. Then L(H1; H2) is regarded as an operator space by identifying L(H1; H2) in L(H1 ⊕ H2) by 0 0 L(H ; H ) 3 a 7! 2 L(H ⊕ H ) 1 2 a 0 1 2 3. -
Riesz-Like Bases in Rigged Hilbert Spaces, in Preparation [14] Bonet, J., Fern´Andez, C., Galbis, A
RIESZ-LIKE BASES IN RIGGED HILBERT SPACES GIORGIA BELLOMONTE AND CAMILLO TRAPANI Abstract. The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space D[t] ⊂H⊂D×[t×]. A Riesz- like basis, in particular, is obtained by considering a sequence {ξn}⊂D which is mapped by a one-to-one continuous operator T : D[t] → H[k · k] into an orthonormal basis of the central Hilbert space H of the triplet. The operator T is, in general, an unbounded operator in H. If T has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces. 1. Introduction Riesz bases (i.e., sequences of elements ξn of a Hilbert space which are trans- formed into orthonormal bases by some bounded{ } operator withH bounded inverse) often appear as eigenvectors of nonself-adjoint operators. The simplest situation is the following one. Let H be a self-adjoint operator with discrete spectrum defined on a subset D(H) of the Hilbert space . Assume, to be more definite, that each H eigenvalue λn is simple. Then the corresponding eigenvectors en constitute an orthonormal basis of . If X is another operator similar to H,{ i.e.,} there exists a bounded operator T withH bounded inverse T −1 which intertwines X and H, in the sense that T : D(H) D(X) and XT ξ = T Hξ, for every ξ D(H), then, as it is → ∈ easily seen, the vectors ϕn with ϕn = Ten are eigenvectors of X and constitute a Riesz basis for . -
Noncommutative Functional Analysis for Undergraduates"
N∞M∞T Lecture Course Canisius College Noncommutative functional analysis David P. Blecher October 22, 2004 2 Chapter 1 Preliminaries: Matrices = operators 1.1 Introduction Functional analysis is one of the big fields in mathematics. It was developed throughout the 20th century, and has several major strands. Some of the biggest are: “Normed vector spaces” “Operator theory” “Operator algebras” We’ll talk about these in more detail later, but let me give a micro-summary. Normed (vector) spaces were developed most notably by the mathematician Banach, who not very subtly called them (B)-spaces. They form a very general framework and tools to attack a wide range of problems: in fact all a normed (vector) space is, is a vector space X on which is defined a measure of the ‘length’ of each ‘vector’ (element of X). They have a huge theory. Operator theory and operator algebras grew partly out of the beginnings of the subject of quantum mechanics. In operator theory, you prove important things about ‘linear functions’ (also known as operators) T : X → X, where X is a normed space (indeed usually a Hilbert space (defined below). Such operators can be thought of as matrices, as we will explain soon. Operator algebras are certain collections of operators, and they can loosely be thought of as ‘noncommutative number fields’. They fall beautifully within the trend in mathematics towards the ‘noncommutative’, linked to discovery in quantum physics that we live in a ‘noncommutative world’. You can study a lot of ‘noncommutative mathematics’ in terms of operator algebras. The three topics above are functional analysis. -
Mathematical Work of Franciszek Hugon Szafraniec and Its Impacts
Tusi Advances in Operator Theory (2020) 5:1297–1313 Mathematical Research https://doi.org/10.1007/s43036-020-00089-z(0123456789().,-volV)(0123456789().,-volV) Group ORIGINAL PAPER Mathematical work of Franciszek Hugon Szafraniec and its impacts 1 2 3 Rau´ l E. Curto • Jean-Pierre Gazeau • Andrzej Horzela • 4 5,6 7 Mohammad Sal Moslehian • Mihai Putinar • Konrad Schmu¨ dgen • 8 9 Henk de Snoo • Jan Stochel Received: 15 May 2020 / Accepted: 19 May 2020 / Published online: 8 June 2020 Ó The Author(s) 2020 Abstract In this essay, we present an overview of some important mathematical works of Professor Franciszek Hugon Szafraniec and a survey of his achievements and influence. Keywords Szafraniec Á Mathematical work Á Biography Mathematics Subject Classification 01A60 Á 01A61 Á 46-03 Á 47-03 1 Biography Professor Franciszek Hugon Szafraniec’s mathematical career began in 1957 when he left his homeland Upper Silesia for Krako´w to enter the Jagiellonian University. At that time he was 17 years old and, surprisingly, mathematics was his last-minute choice. However random this decision may have been, it was a fortunate one: he succeeded in achieving all the academic degrees up to the scientific title of professor in 1980. It turned out his choice to join the university shaped the Krako´w mathematical community. Communicated by Qingxiang Xu. & Jan Stochel [email protected] Extended author information available on the last page of the article 1298 R. E. Curto et al. Professor Franciszek H. Szafraniec Krako´w beyond Warsaw and Lwo´w belonged to the famous Polish School of Mathematics in the prewar period. -
Subspaces and Quotients of Topological and Ordered Vector Spaces
Zoran Kadelburg Stojan Radenovi´c SUBSPACES AND QUOTIENTS OF TOPOLOGICAL AND ORDERED VECTOR SPACES Novi Sad, 1997. CONTENTS INTRODUCTION::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 I: TOPOLOGICAL VECTOR SPACES::::::::::::::::::::::::::::::: 3 1.1. Some properties of subsets of vector spaces ::::::::::::::::::::::: 3 1.2. Topological vector spaces::::::::::::::::::::::::::::::::::::::::: 6 1.3. Locally convex spaces :::::::::::::::::::::::::::::::::::::::::::: 12 1.4. Inductive and projective topologies ::::::::::::::::::::::::::::::: 15 1.5. Topologies of uniform convergence. The Banach-Steinhaus theorem 21 1.6. Duality theory ::::::::::::::::::::::::::::::::::::::::::::::::::: 28 II: SUBSPACES AND QUOTIENTS OF TOPOLOGICAL VECTOR SPACES ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 39 2.1. Subspaces of lcs’s belonging to the basic classes ::::::::::::::::::: 39 2.2. Subspaces of lcs’s from some other classes :::::::::::::::::::::::: 47 2.3. Subspaces of topological vector spaces :::::::::::::::::::::::::::: 56 2.4. Three-space-problem for topological vector spaces::::::::::::::::: 60 2.5. Three-space-problem in Fr´echet spaces:::::::::::::::::::::::::::: 65 III: ORDERED TOPOLOGICAL VECTOR SPACES :::::::::::::::: 72 3.1. Basics of the theory of Riesz spaces::::::::::::::::::::::::::::::: 72 3.2. Topological vector Riesz spaces ::::::::::::::::::::::::::::::::::: 79 3.3. The basic classes of locally convex Riesz spaces ::::::::::::::::::: 82 3.4. l-ideals of topological vector Riesz spaces ::::::::::::::::::::::::: -
The Maximal Operator Space of a Normed Space
Proceedings of the Edinburgh Mathematical Society (1996) 39, 309-323 .<* THE MAXIMAL OPERATOR SPACE OF A NORMED SPACE by VERN I. PAULSEN* (Received 27th June 1994) We obtain some new results about the maximal operator space structure which can be put on a normed space. These results are used to prove some dilation results for contractive linear maps from a normed space into B(H). Finally, we prove CB(MIN(X),MAX(y)) = rj(A', Y) and apply this result to prove some new Grothendieck-type inequalities and some new estimates on spans of "free" unitaries. 1991 Mathematics subject classification: 46L05, 46M05. 1. Introduction Given any (complex) vector space X, let Mn(X) denote the vector space of nxn matrices with entries from X. If B(H) denotes the bounded linear operators on a Hilbert space H then Mn(B(H)) is endowed with a natural norm via the identification, Mn{B(H)) = B(H®•••®H) (n copies). If X is any subspace of B(H) then the inclusion Mn(X)sMn(B(H)) endows Mn(X) with a norm. These are generally called a matrix- norm on X. Subspaces of B(H) are called operator spaces to indicate that they are more than normed spaces, but normed spaces with a distinguished family of norms on the spaces, Mn(X). If X is only a normed space, then any linear isometry <p:X->B(H) for some H endows each Mn(X) with a norm, i.e. makes X an operator space. Generally, for a normed space X the norms on Mn(X) thus obtained are not unique.