Noncommutative Analysis, Operator Theory and Applications PDF
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Operator Theory Advances and Applications 252 Daniel Alpay Fabio Cipriani Fabrizio Colombo Daniele Guido Irene Sabadini Jean-Luc Sauvageot Editors Noncommutative Analysis, Operator Theory and Applications Operator Theory: Advances and Applications Volume 252 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Wien, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Honorary and Advisory Editorial Board: Vadim Adamyan (Odessa, Ukraine) Lewis A. Coburn (Buffalo, NY, USA) Wolfgang Arendt (Ulm, Germany) Ciprian Foias (College Station, TX, USA) Albrecht Böttcher (Chemnitz, Germany) J.William Helton (San Diego, CA, USA) B. Malcolm Brown (Cardiff, UK) Thomas Kailath (Stanford, CA, USA) Raul Curto (Iowa, IA, USA) Peter Lancaster (Calgary, Canada) Fritz Gesztesy (Columbia, MO, USA) Peter D. Lax (New York, NY, USA) Pavel Kurasov (Stockholm, Sweden) Donald Sarason (Berkeley, CA, USA) Vern Paulsen (Houston, TX, USA) Bernd Silbermann (Chemnitz, Germany) Mihai Putinar (Santa Barbara, CA, USA) Harold Widom (Santa Cruz, CA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA) Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Beer Sheva, Israel) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands) Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) Ingo Witt (Göttingen, Germany) Daniel Alpay • Fabio Cipriani Fabrizio Colombo • Daniele Guido Irene Sabadini • Jean-Luc Sauvageot Editors Noncommutative Analysis, Operator Theory and Applications Editors Daniel Alpay Fabio Cipriani Department of Mathematics Dipartimento di Matematica Ben-Gurion University of the Negev Politecnico di Milano Beer Sheva, Israel Milano, Italy Fabrizio Colombo Daniele Guido Dipartimento di Matematica Dipartimento di Matematica Politecnico di Milano Università di Roma “Tor Vergata” Milano, Italy Roma, Italy Irene Sabadini Jean-Luc Sauvageot Dipartimento di Matematica Institut de Mathématiques Politecnico di Milano Université Pierre et Marie Curie Milano, Italy Paris, France ISSN 0255-0156 ISSN22 96-4878 (electronic) Operator Theory: Advances and Applications ISBN 978-3-319-29114-7 ISBN 978-3-319-29116-1 (eBook) DOI 10.1007/978-3-319-29116-1 Library of Congress Control Number: 2016945037 Mathematics Subject Classification (2010): 46Hxx, 46Jxx, 46Lxx, 47Axx, 47Lxx, 58Axx © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This book is published under the trade name Birkhäuser. The registered company is Springer International Publishing AG (www.birkhauser-science.com) Contents Preface .................................................................. vii F. Arici, F. D’Andrea and G. Landi Pimsner Algebras and Circle Bundles ............................... 1 S. Bernstein A Fractional Dirac Operator ........................................ 27 V. Bolotnikov On the Sylvester Equation over Quaternions ........................ 43 J. Bourgain and D.-V. Voiculescu The Essential Centre of the mod a DiagonalizationIdeal Commutant of an n-tuple of Commuting Hermitian Operators ...... 77 P. Cerejeiras and N. Vieira Clifford–Hermite Polynomials in Fractional Clifford Analysis ........ 81 F. Cipriani and J.-L. Sauvageot Negative Definite Functions on Groups with Polynomial Growth .... 97 F. Colombo, I. Sabadini and D.C. Struppa An Introduction to Superoscillatory Sequences ...................... 105 H.S.V. de Snoo and H. Woracek Restriction and Factorization for Isometric and Symmetric Operators in Almost PontryaginSpaces ............................. 123 G. Dell’Antonio Measurements vs. Interactions: Tracks in a Wilson Cloud Chamber ................................. 171 M. Elin and D. Shoikhet The Radii Problems for Holomorphic Mappings in J∗-algebras ...... 181 vi Contents U. Franz, A. Kula and A. Skalski L´evy Processes on Quantum Permutation Groups ................... 193 D. Guido and T. Isola New Results on Old Spectral Triples for Fractals .................... 261 W.A. Majewski and L.E. Labuschagne Why Are Orlicz Spaces Useful for Statistical Physics? ............... 271 Preface This volume contains papers written by some of the speakers of the Conference NoncommutativeAnalysis,OperatorTheory,andApplicationsheldinMilanofrom 23 to 27 June 2014, and some invited contributions. The Conference has been an occasion for researchers in different areas to meet and to share their knowledge and ideas. The contents of the volume reflect, we hope, the effort to find a place where researchers from the different areas may interact. All contributed papers represent the most recent achievements in the area as well as “state-of-the-art” expositions. TheEditorsaregratefultothecontributorstothisvolumeandtothereferees, for their painstaking and careful work. The Editors thank the Dipartimento di Matematica, Politecnico di Milano for hosting the Conference and for assigningto them the FARB (Fondo di Ateneo per la Ricerca di Base). Moreover they thank the Fondazione Gruppo Credito Valtellinese as well as Birkh¨auser Basel for the financial support. October 2015 The Editors OperatorTheory: Advances andApplications,Vol.252,1–25 (cid:2)c 2016SpringerInternational Publishing Pimsner Algebras and Circle Bundles Francesca Arici, Francesco D’Andrea and Giovanni Landi Abstract. We report on the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. We illustrate severalresultswithexamplesofquantumweightedprojectiveandlensspaces and θ-deformations. MathematicsSubjectClassification(2010).Primary19K35;Secondary55R25, 46L08. Keywords. Pimsner algebras, quantum principal bundles, graded algebras, noncommutativegeometry, quantum projective and lens spaces. Contents 1. Introduction ........................................................ 2 2. Hilbert C∗-modules and Morita equivalence ......................... 3 2.1. Hilbert C∗-modules ............................................... 3 2.2. Morita equivalence ................................................ 5 2.3. Self-Morita equivalence bimodules ................................. 6 2.4. Frames ............................................................ 8 3. Pimsner algebras and generalized crossed products .................. 9 3.1. The Pimsner algebra of a self-Morita equivalence .................. 9 3.2. Generalized crossed products ...................................... 11 3.3. Algebras and circle actions ........................................ 12 3.4. Six-term exact sequences .......................................... 13 4. Principal bundles and graded algebras .............................. 15 4.1. Noncommutative principal circle bundles and line bundles ......... 15 4.2. Line bundles ...................................................... 16 4.3. Strongly graded algebras .......................................... 17 4.4. Pimsner algebras from principal circle bundles ..................... 18 5. Examples ........................................................... 19 5.1. Quantum weighted projective and quantum lens spaces ............ 19 5.2. Twisting of graded algebras ....................................... 20 References .............................................................. 23 2 F. Arici, F. D’Andrea and G. Landi 1. Introduction This paper is devoted to Pimsner (or Cuntz–Krieger–Pimsner)algebras, focusing ontheir connectionswithnoncommutativeprincipal circlebundles aswell aswith (strongly) graded algebras. Pimsner algebras,which were introduced in the seminal work [22], provide a unifying framework for a range of important C∗-algebras including crossed prod- ucts by the integers, Cuntz–Kriegeralgebras [8, 9], and C∗-algebrasassociated to partialautomorphisms[11].Duetotheirflexibilityandwiderangeofapplicability, there has been recently an increasing interest in these algebras (see for instance [13, 24]). A related class of algebras, known as generalized crossed products, was independentlyinventedin[1].Thetwonotionscoincideinmanycases,inparticular inthoseofinterestforthepresentpaper.Herewewilluseamoregeometricalpoint ofview, showinghow certainPimsneralgebras,coming froma self-Morita equiva- lencebimodule overaC∗-algebra,canbethoughtofasalgebrasoffunctionsonthe total space of a noncommutative principal circle bundle, along the lines of [4, 10]. Classically, starting from a principal circle bundle P over a compact topo- logical space X, an important role is in the associated line bundles. Given any of these, the corresponding module of sections is a self-Morita equivalence bimodule for the commutative C∗-algebra C(X) of continuous functions over X. Suitable tensor powers of the (sections of the) bundle are endowed with an algebra struc- ture eventually giving back the C∗-algebra C(P) of continuous functions over P. This is just a Pimsneralgebraconstruction.By analogythen, one thinks ofa self- Morita equivalence bimodule over an arbitrary C∗-algebra as a noncommutative line bundle and of the corresponding Pimsner algebra as the ‘total space’ algebra of a principal circle fibration. TheEulerclassofa(classical)linebundle hasanimportantuseinthe Gysin sequence in complex K-theory, that relates the topology of the base space X to that of the total space P of the bundle. This sequence has natural counterparts in the context of Pimsner algebras, counterparts given by two sequences in KK- theorywithnaturalanaloguesoftheEulerclassandacentralroleplayedbyindex maps of canonical classes. In order to make this review self-contained, we start in §2 from recalling the theory of Hilbert modules and Morita equivalences, focusing on those definitions and results that will be needed in the sequel of the paper. Then §3 is devoted to Pimsner’s construction and to the construction of generalized crossed products. This is followed by the six-term exact sequences in KK-theory. We next move in §4 to noncommutative principal circle bundles and graded algebras and recall howprincipalityofthe actioncanbe translatedintoanalgebraicconditiononthe induced grading. This condition is particularly relevant and it resembles a similar condition appearing in the theory of generalized crossed products. We then show how all these notions are interconnected and can be seen as different aspects of the same phenomenon. Finally, §5 is devotedto examples: we illustrate how theta deformedandquantumweightedprojectiveandlensspacesfitintotheframework.
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