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Can One Identify Two Unital JB $^* $-Algebras by the Metric Spaces
CAN ONE IDENTIFY TWO UNITAL JB∗-ALGEBRAS BY THE METRIC SPACES DETERMINED BY THEIR SETS OF UNITARIES? MAR´IA CUETO-AVELLANEDA, ANTONIO M. PERALTA Abstract. Let M and N be two unital JB∗-algebras and let U(M) and U(N) denote the sets of all unitaries in M and N, respectively. We prove that the following statements are equivalent: (a) M and N are isometrically isomorphic as (complex) Banach spaces; (b) M and N are isometrically isomorphic as real Banach spaces; (c) There exists a surjective isometry ∆ : U(M) →U(N). We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry ∆ : U(M) → U(N) we can find a surjective real linear isometry Ψ : M → N which coincides with ∆ on the subset eiMsa . If we assume that M and N are JBW∗-algebras, then every surjective isometry ∆ : U(M) → U(N) admits a (unique) extension to a surjective real linear isometry from M onto N. This is an extension of the Hatori–Moln´ar theorem to the setting of JB∗-algebras. 1. Introduction Every surjective isometry between two real normed spaces X and Y is an affine mapping by the Mazur–Ulam theorem. It seems then natural to ask whether the existence of a surjective isometry between two proper subsets of X and Y can be employed to identify metrically both spaces. By a result of P. Mankiewicz (see [34]) every surjective isometry between convex bodies in two arbitrary normed spaces can be uniquely extended to an affine function between the spaces. -
The Index of Normal Fredholm Elements of C* -Algebras
proceedings of the american mathematical society Volume 113, Number 1, September 1991 THE INDEX OF NORMAL FREDHOLM ELEMENTS OF C*-ALGEBRAS J. A. MINGO AND J. S. SPIELBERG (Communicated by Palle E. T. Jorgensen) Abstract. Examples are given of normal elements of C*-algebras that are invertible modulo an ideal and have nonzero index, in contrast to the case of Fredholm operators on Hubert space. It is shown that this phenomenon occurs only along the lines of these examples. Let T be a bounded operator on a Hubert space. If the range of T is closed and both T and T* have a finite dimensional kernel then T is Fredholm, and the index of T is dim(kerT) - dim(kerT*). If T is normal then kerT = ker T*, so a normal Fredholm operator has index 0. Let us consider a generalization of the notion of Fredholm operator intro- duced by Atiyah. Let X be a compact Hausdorff space and consider continuous functions T: X —>B(H), where B(H) is the set of bounded linear operators on a separable infinite dimensional Hubert space with the norm topology. The set of such functions forms a C*- algebra C(X) <g>B(H). A function T is Fredholm if T(x) is Fredholm for each x . Atiyah [1, Appendix] showed how such an element has an index which is an element of K°(X). Suppose that T is Fredholm and T(x) is normal for each x. Is the index of T necessarily 0? There is a generalization of this question that we would like to consider. -
A Short Introduction to the Quantum Formalism[S]
A short introduction to the quantum formalism[s] François David Institut de Physique Théorique CNRS, URA 2306, F-91191 Gif-sur-Yvette, France CEA, IPhT, F-91191 Gif-sur-Yvette, France [email protected] These notes are an elaboration on: (i) a short course that I gave at the IPhT-Saclay in May- June 2012; (ii) a previous letter [Dav11] on reversibility in quantum mechanics. They present an introductory, but hopefully coherent, view of the main formalizations of quantum mechanics, of their interrelations and of their common physical underpinnings: causality, reversibility and locality/separability. The approaches covered are mainly: (ii) the canonical formalism; (ii) the algebraic formalism; (iii) the quantum logic formulation. Other subjects: quantum information approaches, quantum correlations, contextuality and non-locality issues, quantum measurements, interpretations and alternate theories, quantum gravity, are only very briefly and superficially discussed. Most of the material is not new, but is presented in an original, homogeneous and hopefully not technical or abstract way. I try to define simply all the mathematical concepts used and to justify them physically. These notes should be accessible to young physicists (graduate level) with a good knowledge of the standard formalism of quantum mechanics, and some interest for theoretical physics (and mathematics). These notes do not cover the historical and philosophical aspects of quantum physics. arXiv:1211.5627v1 [math-ph] 24 Nov 2012 Preprint IPhT t12/042 ii CONTENTS Contents 1 Introduction 1-1 1.1 Motivation . 1-1 1.2 Organization . 1-2 1.3 What this course is not! . 1-3 1.4 Acknowledgements . 1-3 2 Reminders 2-1 2.1 Classical mechanics . -
Geodesics of Projections in Von Neumann Algebras
Geodesics of projections in von Neumann algebras Esteban Andruchow∗ November 5, 2020 Abstract Let be a von Neumann algebra and A the manifold of projections in . There is a A P A natural linear connection in A, which in the finite dimensional case coincides with the the Levi-Civita connection of theP Grassmann manifold of Cn. In this paper we show that two projections p, q can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of ), if and only if A p q⊥ p⊥ q, ∧ ∼ ∧ where stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal∼ geodesic is unique if and only if p q⊥ = p⊥ q =0. If is a finite factor, any ∧ ∧ A pair of projections in the same connected component of A (i.e., with the same trace) can be joined by a minimal geodesic. P We explore certain relations with Jones’ index theory for subfactors. For instance, it is −1 shown that if are II1 factors with finite index [ : ] = t , then the geodesic N ⊂M M N 1/2 distance d(eN ,eM) between the induced projections eN and eM is d(eN ,eM) = arccos(t ). 2010 MSC: 58B20, 46L10, 53C22 Keywords: Projections, geodesics of projections, von Neumann algebras, index for subfac- tors. arXiv:2011.02013v1 [math.OA] 3 Nov 2020 1 Introduction ∗ If is a C -algebra, let A denote the set of (selfadjoint) projections in . A has a rich A P A P geometric structure, see for instante the papers [12] by H. -
Weak Compactness in the Space of Operator Valued Measures and Optimal Control Nasiruddin Ahmed
Weak Compactness in the Space of Operator Valued Measures and Optimal Control Nasiruddin Ahmed To cite this version: Nasiruddin Ahmed. Weak Compactness in the Space of Operator Valued Measures and Optimal Control. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. pp.49-58, 10.1007/978-3-642-36062-6_5. hal-01347522 HAL Id: hal-01347522 https://hal.inria.fr/hal-01347522 Submitted on 21 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License WEAK COMPACTNESS IN THE SPACE OF OPERATOR VALUED MEASURES AND OPTIMAL CONTROL N.U.Ahmed EECS, University of Ottawa, Ottawa, Canada Abstract. In this paper we present a brief review of some important results on weak compactness in the space of vector valued measures. We also review some recent results of the author on weak compactness of any set of operator valued measures. These results are then applied to optimal structural feedback control for deterministic systems on infinite dimensional spaces. Keywords: Space of Operator valued measures, Countably additive op- erator valued measures, Weak compactness, Semigroups of bounded lin- ear operators, Optimal Structural control. -
Math 259A Lecture 7 Notes
Math 259A Lecture 7 Notes Daniel Raban October 11, 2019 1 WO and SO Continuity of Linear Functionals and The Pre-Dual of B 1.1 Weak operator and strong operator continuity of linear functionals Lemma 1.1. Let X be a vector space with seminorms p1; : : : ; pn. Let ' : X ! C be a Pn linear functional such that j'(x)j ≤ i=1 pi(x) for all x 2 X. Then there exist linear P functionals '1;:::;'n : X ! C such that ' = i 'i with j'i(x)j ≤ pi(x) for all x 2 X and for all i. Proof. Let D = fx~ = (x; : : : ; x): x 2 Xg ⊆ Xn, which is a vector subspace. On Xn, n P take p((xi)i=1) = i pi(xi). We also have a linear map' ~ : D ! C given by' ~(~x) = '(x). This map satisfies j~(~x)j ≤ p(~x). By the Hahn-Banach theorem, there exists an n ∗ extension 2 (X ) of' ~ such that j (x1; : : : ; xn)j ≤ p(x1; : : : ; xn). Now define 'k(x) := (0; : : : ; x; 0;::: ), where the x is in the k-th position. Theorem 1.1. Let ' : B! C be linear. ' is weak operator continuous if and only if it is it is strong operator continuous. Proof. We only need to show that if ' is strong operator continuous, then it is weak Pn operator continuous. So assume there exist ξ1; : : : ; ξn 2 X such that j'(x)j ≤ i=1 kxξik P for all x 2 B. By the lemma, we can split ' = 'k, such that j'k(x)j ≤ kxξkk for all x and k. -
Derivations on Metric Measure Spaces
Derivations on Metric Measure Spaces by Jasun Gong A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2008 Doctoral Committee: Professor Mario Bonk, Chair Professor Alexander I. Barvinok Professor Juha Heinonen (Deceased) Associate Professor James P. Tappenden Assistant Professor Pekka J. Pankka “Or se’ tu quel Virgilio e quella fonte che spandi di parlar si largo fiume?” rispuos’io lui con vergognosa fronte. “O de li altri poeti onore e lume, vagliami ’l lungo studio e ’l grande amore che m’ha fatto cercar lo tuo volume. Tu se’ lo mio maestro e ’l mio autore, tu se’ solo colui da cu’ io tolsi lo bello stilo che m’ha fatto onore.” [“And are you then that Virgil, you the fountain that freely pours so rich a stream of speech?” I answered him with shame upon my brow. “O light and honor of all other poets, may my long study and the intense love that made me search your volume serve me now. You are my master and my author, you– the only one from whom my writing drew the noble style for which I have been honored.”] from the Divine Comedy by Dante Alighieri, as translated by Allen Mandelbaum [Man82]. In memory of Juha Heinonen, my advisor, teacher, and friend. ii ACKNOWLEDGEMENTS This work was inspired and influenced by many people. I first thank my parents, Ping Po Gong and Chau Sim Gong for all their love and support. They are my first teachers, and from them I learned the value of education and hard work. -
PRINCIPLES of APPLIED MATHEMATICS Transformation and Approximation Revised Edition
PRINCIPLES OF To Kristine, Sammy, and Justin, APPLIED Still my best friends after all these years. MATHEMATICS Transformation and Approximation Revised Edition JAMES P. KEENER University of Utah Salt Lake City, Utah ADVANCED BOOK PROGRAM I ewI I I ~ Member of the Perseus Books Group Contents Preface to First Edition xi Preface to Second Edition xvii 1 Finite Dimensional Vector Spaces 1 1.1 Linear Vector Spaces ....... 1 1.2 Spectral Theory for Matrices . 9 1.3 Geometrical Significance of Eigenvalues 17 1.4 Fredholm Alternative Theorem . 24 1.5 Least Squares Solutions-Pseudo Inverses . 25 1.5.1 The Problem of Procrustes .... 41 Many of the designations used by manufacturers and sellers to distinguish their 42 products are claimed as trademarks. Where those designations appear in this 1.6 Applications of Eigenvalues and Eigenfunctions book and Westview Press was aware of a trademark claim, the designations have 1.6.1 Exponentiation of Matrices . 42 been printed in initial capital letters. 1.6.2 The Power Method and Positive Matrices 43 45 Library of Congress Catalog Card Number: 99-067545 1.6.3 Iteration Methods Further Reading . 47 ISBN: 0-7382-0129-4 Problems for Chapter 1 49 Copyright© 2000 by James P. Keener 2 Function Spaces 59 Westview Press books are available at special discounts for bulk purchases in the 2.1 Complete Vector Spaces .... 59 U.S. by corporations, institutions, and other organizations. For more informa 2.1.1 Sobolev Spaces ..... 65 tion, please contact the Special Markets Department at the Perseus Books Group, 2.2 Approximation in Hilbert Spaces 67 11 Cambridge Center, Cambridge, MA 02142, or call (617) 252-5298. -
Course Structure for M.Sc. in Mathematics (Academic Year 2019 − 2020)
Course Structure for M.Sc. in Mathematics (Academic Year 2019 − 2020) School of Physical Sciences, Jawaharlal Nehru University 1 Contents 1 Preamble 3 1.1 Minimum eligibility criteria for admission . .3 1.2 Selection procedure . .3 2 Programme structure 4 2.1 Overview . .4 2.2 Semester wise course distribution . .4 3 Courses: core and elective 5 4 Details of the core courses 6 4.1 Algebra I .........................................6 4.2 Real Analysis .......................................8 4.3 Complex Analysis ....................................9 4.4 Basic Topology ...................................... 10 4.5 Algebra II ......................................... 11 4.6 Measure Theory .................................... 12 4.7 Functional Analysis ................................... 13 4.8 Discrete Mathematics ................................. 14 4.9 Probability and Statistics ............................... 15 4.10 Computational Mathematics ............................. 16 4.11 Ordinary Differential Equations ........................... 18 4.12 Partial Differential Equations ............................. 19 4.13 Project ........................................... 20 5 Details of the elective courses 21 5.1 Number Theory ..................................... 21 5.2 Differential Topology .................................. 23 5.3 Harmonic Analysis ................................... 24 5.4 Analytic Number Theory ............................... 25 5.5 Proofs ........................................... 26 5.6 Advanced Algebra ................................... -
A Framework to Study Commutation Problems Bulletin De La S
BULLETIN DE LA S. M. F. ALFONS VAN DAELE A framework to study commutation problems Bulletin de la S. M. F., tome 106 (1978), p. 289-309 <http://www.numdam.org/item?id=BSMF_1978__106__289_0> © Bulletin de la S. M. F., 1978, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Bull. Soc. math. France, 106, 1978, p. 289-309. A FRAMEWORK TO STUDY COMMUTATION PROBLEMS BY ALFONS VAN DAELE [Kath. Univ. Leuven] RESUME. — Soient A et B deux algebres involutives d'operateurs sur un espace hil- bertien j^, telles que chacune d'elles soit contenue dans Ie commutant de Fautre. On enonce des conditions suffisantes sur A et B, en termes de certaines applications lineaires, ri : A ->• ^ et n' : B -> J^, pour que chacune de ces algebres engendre Ie commutant de Fautre. Cette structure generalise d'une certaine facon celle d'une algebre hilbertienne a gauche; elle permet de traiter Ie cas ou Falgebre de von Neumann et son commutant n'ont pas la meme grandeur. ABSTRACT. — If A and B are commuting *-algebras of operators on a Hilbert space ^, conditions on A and B are formulated in terms of linear maps T| : A -> ^ and n" : B -»• ^ to ensure that A and B generate each others commutants. -
Vector-Valued Holomorphic Functions
Appendix A Vector-valued Holomorphic Functions Let X be a Banach space and let Ω ⊂ C be an open set. A function f :Ω→ X is holomorphic if f(z0 + h) − f(z0) f (z0) := lim (A.1) h→0 h h∈C\{0} exists for all z0 ∈ Ω. If f is holomorphic, then f is continuous and weakly holomorphic (i.e. x∗ ◦ f ∗ ∗ is holomorphic for all x ∈ X ). If Γ := {γ(t):t ∈ [a,b]} is a finite, piecewise smooth contour in Ω, we can form the contour integral f(z) dz. This coincides Γ b with the Bochner integral a f(γ(t))γ (t) dt (see Section 1.1). Similarly we can define integrals over infinite contours when the corresponding Bochner integral is absolutely convergent. Since 1 2 f(z) dz, x∗ = f(z),x∗ dz, Γ Γ many properties of holomorphic functions and contour integrals may be extended from the scalar to the vector-valued case, by applying the Hahn-Banach theorem. For example, Cauchy’s theorem is valid, and also Cauchy’s integral formula: 1 f(z) f(w)= dz (A.2) − 2πi |z−z0|=r z w whenever f is holomorphic in Ω, the closed ball B(z0,r) is contained in Ω and w ∈ B(z0,r). As in the scalar case one deduces Taylor’s theorem from this. Proposition A.1. Let f :Ω→ X be holomorphic, where Ω ⊂ C is open. Let z0 ∈ Ω,r>0 such that B(z0,r) ⊂ Ω.Then W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, 461 Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7, © Springer Basel AG 2011 462 A. -
Notes Which Was Later Published As Springer Lecture Notes No.128: Simplifi- Cation Was Not an Issue, but the Validity of Tomita’S Claim
STRUCTURE OF VON NEUMANN ALGEBRAS OF TYPE III Masamichi Takesaki Abstract. In this lecture, we will show that to every von Neumann algebra M there corresponds unquely a covariant system M, ⌧, R, ✓ on the real line R in such a way that n o f ✓ s M = M , ⌧ ✓s = e− ⌧, M0 M = C, ◦ \ where C is the center off M. In the case that M fis a factor, we have the following commutative square of groups which describes the relation of several important groups such as thef unitary group U(M) of M, the normalizer U(M) of M in M and the cohomology group of the flow of weights: C, R, ✓ : { } e f 1 1 1 ? ? @ ? 1 ? U?(C) ✓ B1( ,?U(C)) 1 −−−−−! yT −−−−−! y −−−−−! ✓ Ry −−−−−! ? ? @ ? 1 U(?M) U(?M) ✓ Z1( ,?U(C)) 1 −−−−−! y −−−−−! y −−−−−! ✓ Ry −−−−−! Ad Ade ? ? ˙ ? @✓ 1 1 Int?(M) Cnft?r(M) H ( ?, U(C)) 1 −−−−−! y −−−−−! y −−−−−! ✓ Ry −−−−−! ? ? ? ?1 ?1 ?1 y y y Contents Lecture 0. History of Structure Analyis of von Neumann Algebras of Type III. Lecture 1. Covariant System and Crossed Product. Lecture 2: Duality for the Crossed Product by Abelian Groups. Lecture 3: Second Cohomology of Locally Compact Abelian Group. Lecture 4. Arveson Spectrum of an Action of a Locally Compact Abelian Group G on a von Neumann algebra M. 1 2 VON NEUMANN ALGEBRAS OF TYPE III Lecture 5: Connes Spectrum. Lecture 6: Examples. Lecture 7: Crossed Product Construction of a Factor. Lecture 8: Structure of a Factor of Type III. Lecture 9: Hilbert Space Bundle. Lecture 10: The Non-Commutative Flow of Weights on a von Neumann algebra M, I.