DOCSLIB.ORG

Noncommutative Calculus and Operads

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Noncommutative Calculus and Operads Topics in Noncommutative Geometry Clay Mathematics Proceedings Volume 16 Topics in Noncommutative Geometry Third Luis Santaló Winter School - CIMPA Research School Topics in Noncommutative Geometry Universidad de Buenos Aires Buenos Aires, Argentina July 26–August 6, 2010 Guillermo Cortiñas Editor American Mathematical Society Clay Mathematics Institute 2010 Mathematics Subject Classification. Primary 14A22, 18D50, 19K35, 19L47, 19L50, 20F10, 46L55, 53D55, 58B34, 81R60. Cover photo of the front of Pabell´on 1 of Ciudad Universitaria courtesy of Adri´an Sarchese and Edgar Bringas. Back cover photo of Professor Luis Santal´ois courtesy of his family. Library of Congress Cataloging-in-Publication Data Luis Santal´o Winter School-CIMPA Research School on Topics in Noncommutative Geometry (2010 : Buenos Aires, Argentina) Topics in noncommutative geometry : Third Luis Santal´o Winter School-CIMPA Research School on Topics in Noncommutative Geometry, July 26–August 6, 2010, Universidad de Buenos Aires, Buenos Aires, Argentina / Guillermo Corti˜nas, editor. page cm. — (Clay mathematics proceedings ; volume 16) Includes bibliographical references. ISBN 978-0-8218-6864-5 (alk. paper) 1. Commutative algebra—Congresses. I. Corti˜nas, Guillermo, editor of compilation. II. Title. QA251.3.L85 2012 512.55—dc23 2012031426 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2012 by the Clay Mathematics Institute. All rights reserved. Published by the American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA. Printed in the United States of America. The Clay Mathematics Institute retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the Clay Mathematics Institute home page at http://www.claymath.org/ 10987654321 171615141312 Contents Preface vii Classifying Morita Equivalent Star Products 1 Henrique Bursztyn and Stefan Waldmann Noncommutative Calculus and Operads 19 Boris Tsygan Some Elementary Operadic Homotopy Equivalences 67 Eduardo Hoefel Actions of Higher Categories on C*-Algebras 75 Ralf Meyer Examples and Applications of Noncommutative Geometry and K-Theory 93 Jonathan Rosenberg Rational Equivariant K-Homology of Low Dimensional Groups 131 J.-F. Lafont, I. J. Ortiz and R. J. Sanchez-Garc´ ´ıa Automata Groups 165 Andrzej Zuk Spectral Triples and KK-Theory: A Survey 197 Bram Mesland Deformations of the Canonical Spectral Triples 213 R. Trinchero Twisted Bundles and Twisted K-Theory 223 Max Karoubi A Guided Tour Through the Garden of Noncommutative Motives 259 Gonc¸alo Tabuada v Preface This volume contains the proceedings of the third Luis Santal´o Winter School, organized by the Mathematics Department and the Santal´o Mathematical Research Institute of the School of Exact and Natural Sciences of the University of Buenos Aires (FCEN). This series of schools is named after the geometer Luis Santal´o. Born in Spain, the celebrated founder of Integral Geometry was a Professor in our department where he carried out most of his distinguished professional career. This edition of the Santal´o School took place in the FCEN from July 26 to August 6 of 2010. On this occasion the school was devoted to Noncommutative Geometry, and was supported by several institutions; the Clay Mathematics Insti- tute was one of its main sponsors. The topics of the school and the contents of this volume concern Noncommu- tative Geometry in a broad sense: it encompasses the various mathematical and physical theories that incorporate geometric ideas to study noncommutative phe- nomena. One of those theories is that of deformation quantization. A main result in this area is Kontsevich’s formality theorem. It implies that a Poisson structure on a manifold can always be formally quantized. More precisely it shows that there is an isomorphism (although not canonical) between the moduli space of formal deformations of Poisson structures on a manifold and the moduli space of star products on the manifold. The question of understanding Morita equivalence of star products under (a specific choice of) this isomorphism was solved by Henrique Bursztyn and Stefan Waldmann; their article in the present volume gives a survey of their work. They start by discussing how deformation quantization arises from the quantization problem in physics. Then they review the basics on star products, the main results on deformation quantization and the notion of Morita equivalence of associative algebras. After this introductory material, they present a description of Morita equivalence for star products as orbits of a suitable group action on the one hand, and the B-field action on (formal) Poisson structures on the other. Finally, they arrive at their main results on the classification of Morita equivalent star products. The next article, by Boris Tsygan, reviews Tamarkin’s proof of Kontsevich’s formality theorem using the theory of operads. It also explains applications of the formality theorem to noncommutative calculus and index theory. Noncommuta- tive calculus defines classical algebraic structures arising from the usual calculus on manifolds in terms of the algebra of functions on this manifold, in a way that is vii viii PREFACE valid for any associative algebra, commutative or not. It turns out that noncom- mutative analogs of the basic spaces arising in calculus are well-known complexes from homological algebra. These complexes turn out to carry a very rich algebraic structure, similar to the one carried by their classical counterparts. It follows from Kontsevich’s formality theorem that when the algebra in question is the algebra of functions, those noncommutative geometry structures are equivalent to the classical ones. Another consequence is the algebraic index theorem for deformation quanti- zations. This is a statement about a trace of a compactly supported difference of projections in the algebra of matrices over a deformed algebra. It turns out that all the data entering into this problem (namely, a deformed algebra, a trace on it, and projections in it) can be classified using formal Poisson structures on the manifold. The algebraic index theorem implies the celebrated index theorem of Atiyah-Singer and its various generalizations. Operad theory, mentioned before as related to Kontsevich’s formality theorem, is the subject of Eduardo Hoefel’s paper. It follows from general theory that the Fulton-MacPherson operad Fn and the little discs operad Dn are equivalent. Hoefel gives an elementary proof of this fact by exhibiting a rather explicit homotopy equivalence between them. Many important examples in noncommutative geometry appear as crossed products. Ralf Meyer’s lecture notes deal with several crossed-products of C∗- algebras, (e.g. crossed-products by actions of locally compact groups, twisted crossed-products, crossed products by C∗correspondences) and discuss notions of equivalence of these constructions. The author shows how these examples lead nat- urally to the concept of a strict 2-category and to the unifying approach of a functor from a group to strict 2-categories whose objects are C∗-algebras. In addition, this approach enables the author to define in all generality the crossed product of a C∗-algebra by a group acting by Morita-Rieffel bimodules. Jonathan Rosenberg’s article is concerned with two related but distinct top- ics: noncommutative tori and Kasparov’s KK-theory. Noncommutative tori are certain crossed products of the algebra of continuous functions on the unit circle by an action of Z. They turn out to be noncommutative deformations of the al- gebra of continuous functions on the 2-torus. The article reviews the classification of noncommutative tori up to Morita equivalence, and of bundles (i.e. projec- tive modules) over them, as well as some applications of noncommutative tori to number theory and physics. Kasparov’s K-theory is one of the main homological invariants in C∗-algebra theory. It can be presented in several equivalent manners; the article reviews Kasparov’s original definition in terms of Kasparov bimodules (or generalized elliptic pseudodifferential operators), Cuntz’s picture in terms of quasi-homomorphisms, and Higson’s description of KK as a universal homology ∗ theory for separable C -algebras. It assigns groups KK∗(A, B)toanytwosepara- ble C∗-algebras A and B; usual operator K-theory and K-homology are recovered by setting A (respectively B) equal to the complex numbers. The article also considers the equivariant version of KK for C∗-algebras equipped with a group action, and considers its applications to the K-theory of crossed products, includ- ing the Pimsner-Voiculescu sequence for crossed products with Z (used for example to compute the K-theory of noncommutative tori), Connes’ Thom isomorphism PREFACE ix for crossed products with R and the Baum-Connes conjecture for crossed products with general locally compact groups. The Baum-Connes conjecture predicts that the K-theory of the reduced C∗- G algebra Cr(G) of a group G is the G-equivariant K-homology KK∗ (EG, C)of the classifying space for proper actions.
Load more

Recommended publications
  • 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.
    [Show full text]
  • New Publications Offered by the AMS
    New Publications Offered by the AMS appropriate generality waited for the seventies. These early Algebra and Algebraic occurrences were in algebraic topology in the study of (iter- ated) loop spaces and their chain algebras. In the nineties, Geometry there was a renaissance and further development of the theory inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, Almost Commuting derived categories, Morse theory, symplectic and contact EMOIRS M of the geometry, combinatorics, knot theory, moduli spaces, cyclic American Mathematical Society Elements in Compact cohomology, and, not least, theoretical physics, especially Volume 157 Number 747 string field theory and deformation quantization. The general- Almost Commuting Lie Groups Elements in ization of quadratic duality (e.g., Lie algebras as dual to Compact Lie Groups Armand Borel, Institute for commutative algebras) together with the property of Koszul- Armand Borel Advanced Study, Princeton, NJ, ness in an essentially operadic context provided an additional Robert Friedman computational tool for studying homotopy properties outside John W. Morgan and Robert Friedman and THEMAT A IC M A L N A S O C I C of the topological setting. R I E E T M Y A FO 8 U 88 John W. Morgan, Columbia NDED 1 University, New York City, NY The book contains a detailed and comprehensive historical American Mathematical Society introduction describing the development of operad theory Contents: Introduction; Almost from the initial period when it was a rather specialized tool in commuting N-tuples; Some characterizations of groups of homotopy theory to the present when operads have a wide type A; c-pairs; Commuting triples; Some results on diagram range of applications in algebra, topology, and mathematical automorphisms and associated root systems; The fixed physics.
    [Show full text]
  • Noncommutative Geometry and the Spectral Model of Space-Time
    S´eminaire Poincar´eX (2007) 179 – 202 S´eminaire Poincar´e Noncommutative geometry and the spectral model of space-time Alain Connes IHES´ 35, route de Chartres 91440 Bures-sur-Yvette - France Abstract. This is a report on our joint work with A. Chamseddine and M. Marcolli. This essay gives a short introduction to a potential application in physics of a new type of geometry based on spectral considerations which is convenient when dealing with noncommutative spaces i.e. spaces in which the simplifying rule of commutativity is no longer applied to the coordinates. Starting from the phenomenological Lagrangian of gravity coupled with matter one infers, using the spectral action principle, that space-time admits a fine structure which is a subtle mixture of the usual 4-dimensional continuum with a finite discrete structure F . Under the (unrealistic) hypothesis that this structure remains valid (i.e. one does not have any “hyperfine” modification) until the unification scale, one obtains a number of predictions whose approximate validity is a basic test of the approach. 1 Background Our knowledge of space-time can be summarized by the transition from the flat Minkowski metric ds2 = − dt2 + dx2 + dy2 + dz2 (1) to the Lorentzian metric 2 µ ν ds = gµν dx dx (2) of curved space-time with gravitational potential gµν . The basic principle is the Einstein-Hilbert action principle Z 1 √ 4 SE[ gµν ] = r g d x (3) G M where r is the scalar curvature of the space-time manifold M. This action principle only accounts for the gravitational forces and a full account of the forces observed so far requires the addition of new fields, and of corresponding new terms SSM in the action, which constitute the Standard Model so that the total action is of the form, S = SE + SSM .
    [Show full text]
  • 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 ).
    [Show full text]
  • Space and Time Dimensions of Algebras With
    Space and time dimensions of algebras with applications to Lorentzian noncommutative geometry and quantum electrodynamics Nadir Bizi, Christian Brouder, Fabien Besnard To cite this version: Nadir Bizi, Christian Brouder, Fabien Besnard. Space and time dimensions of algebras with appli- cations to Lorentzian noncommutative geometry and quantum electrodynamics. Journal of Mathe- matical Physics, American Institute of Physics (AIP), 2018, 59 (6), pp.062303. 10.1063/1.4986228. hal-01398231v2 HAL Id: hal-01398231 https://hal.archives-ouvertes.fr/hal-01398231v2 Submitted on 16 May 2019 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. Space and time dimensions of algebras with application to Lorentzian noncommutative geometry and quantum electrodynamics Nadir Bizi, Christian Brouder, and Fabien Besnard Citation: Journal of Mathematical Physics 59, 062303 (2018); doi: 10.1063/1.5010424 View online: https://doi.org/10.1063/1.5010424 View Table of Contents: http://aip.scitation.org/toc/jmp/59/6 Published by the American Institute of Physics Articles you may
    [Show full text]
  • Noncommutative Geometry and Flavour Mixing
    Noncommutative geometry and flavour mixing prepared by Jose´ M. Gracia-Bond´ıa Department of Theoretical Physics Universidad de Zaragoza, 50009 Zaragoza, Spain October 22, 2013 1 Introduction: the universality problem “The origin of the quark and lepton masses is shrouded in mystery” [1]. Some thirty years ago, attempts to solve the enigma based on textures of the quark mass matrices, purposedly reflecting mass hierarchies and “nearest-neighbour” interactions, were very popular. Now, in the late eighties, Branco, Lavoura and Mota [2] showed that, within the SM, the zero pattern 0a b 01 @c 0 dA; (1) 0 e 0 a central ingredient of Fritzsch’s well-known Ansatz for the mass matrices, is devoid of any particular physical meaning. (The top quark is above on top.) Although perhaps this was not immediately clear at the time, paper [2] marked a water- shed in the theory of flavour mixing. In algebraic terms, it establishes that the linear subspace of matrices of the form (1) is universal for the group action of unitaries effecting chiral basis transformations, that respect the charged-current term of the Lagrangian. That is, any mass matrix can be transformed to that form without modifying the corresponding CKM matrix. To put matters in perspective, consider the unitary group acting by similarity on three-by- three matrices. The classical triangularization theorem by Schur ensures that the zero patterns 0a 0 01 0a b c1 @b c 0A; @0 d eA (2) d e f 0 0 f are universal in this sense. However, proof that the zero pattern 0a b 01 @0 c dA e 0 f 1 is universal was published [3] just three years ago! (Any off-diagonal n(n−1)=2 zero pattern with zeroes at some (i j) and no zeroes at the matching ( ji), is universal in this sense, for complex n × n matrices.) Fast-forwarding to the present time, notwithstanding steady experimental progress [4] and a huge amount of theoretical work by many authors, we cannot be sure of being any closer to solving the “Meroitic” problem [5] of divining the spectrum behind the known data.
    [Show full text]
  • Shtukas for Reductive Groups and Langlands Correspondence for Functions Fields
    SHTUKAS FOR REDUCTIVE GROUPS AND LANGLANDS CORRESPONDENCE FOR FUNCTIONS FIELDS VINCENT LAFFORGUE This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject. We begin with a short historical account (all notions used below are recalled in the text). The Langlands correspondence [49] is a conjecture of utmost impor- tance, concerning global fields, i.e. number fields and function fields. Many excellent surveys are available, for example [39, 14, 13, 79, 31, 5]. The Langlands correspondence belongs to a huge system of conjectures (Langlands functoriality, Grothendieck’s vision of motives, special val- ues of L-functions, Ramanujan-Petersson conjecture, generalized Rie- mann hypothesis). This system has a remarkable deepness and logical coherence and many cases of these conjectures have already been es- tablished. Moreover the Langlands correspondence over function fields admits a geometrization, the “geometric Langlands program”, which is related to conformal field theory in Theoretical Physics. Let G be a connected reductive group over a global field F . For the sake of simplicity we assume G is split. The Langlands correspondence relates two fundamental objects, of very different nature, whose definition will be recalled later, • the automorphic forms for G, • the global Langlands parameters , i.e. the conjugacy classes of morphisms from the Galois group Gal(F =F ) to the Langlands b dual group G(Q`). b For G = GL1 we have G = GL1 and this is class field theory, which describes the abelianization of Gal(F =F ) (one particular case of it for Q is the law of quadratic reciprocity, which dates back to Euler, Legendre and Gauss).
    [Show full text]
  • 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.
    [Show full text]
  • Noncommutative Stacks
    Noncommutative Stacks Introduction One of the purposes of this work is to introduce a noncommutative analogue of Artin’s and Deligne-Mumford algebraic stacks in the most natural and sufficiently general way. We start with quasi-coherent modules on fibered categories, then define stacks and prestacks. We define formally smooth, formally unramified, and formally ´etale cartesian functors. This provides us with enough tools to extend to stacks the glueing formalism we developed in [KR3] for presheaves and sheaves of sets. Quasi-coherent presheaves and sheaves on a fibered category. Quasi-coherent sheaves on geometric (i.e. locally ringed topological) spaces were in- troduced in fifties. The notion of quasi-coherent modules was extended in an obvious way to ringed sites and toposes at the moment the latter appeared (in SGA), but it was not used much in this generality. Recently, the subject was revisited by D. Orlov in his work on quasi-coherent sheaves in commutative an noncommutative geometry [Or] and by G. Laumon an L. Moret-Bailly in their book on algebraic stacks [LM-B]. Slightly generalizing [R4], we associate with any functor F (regarded as a category over a category) the category of ’quasi-coherent presheaves’ on F (otherwise called ’quasi- coherent presheaves of modules’ or simply ’quasi-coherent modules’) and study some basic properties of this correspondence in the case when the functor defines a fibered category. Imitating [Gir], we define the quasi-topology of 1-descent (or simply ’descent’) and the quasi-topology of 2-descent (or ’effective descent’) on the base of a fibered category (i.e.
    [Show full text]
  • Noncommutative Geometry, the Spectral Standpoint Arxiv
    Noncommutative Geometry, the spectral standpoint Alain Connes October 24, 2019 In memory of John Roe, and in recognition of his pioneering achievements in coarse geometry and index theory. Abstract We update our Year 2000 account of Noncommutative Geometry in [68]. There, we de- scribed the following basic features of the subject: I The natural “time evolution" that makes noncommutative spaces dynamical from their measure theory. I The new calculus which is based on operators in Hilbert space, the Dixmier trace and the Wodzicki residue. I The spectral geometric paradigm which extends the Riemannian paradigm to the non- commutative world and gives a new tool to understand the forces of nature as pure gravity on a more involved geometric structure mixture of continuum with the discrete. I The key examples such as duals of discrete groups, leaf spaces of foliations and de- formations of ordinary spaces, which showed, very early on, that the relevance of this new paradigm was going far beyond the framework of Riemannian geometry. Here, we shall report1 on the following highlights from among the many discoveries made since 2000: 1. The interplay of the geometry with the modular theory for noncommutative tori. 2. Great advances on the Baum-Connes conjecture, on coarse geometry and on higher in- dex theory. 3. The geometrization of the pseudo-differential calculi using smooth groupoids. 4. The development of Hopf cyclic cohomology. 5. The increasing role of topological cyclic homology in number theory, and of the lambda arXiv:1910.10407v1 [math.QA] 23 Oct 2019 operations in archimedean cohomology. 6. The understanding of the renormalization group as a motivic Galois group.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]