Abel Symposia 12 Toke M. Carlsen Nadia S. Larsen Sergey Neshveyev Christian Skau Editors Operator Algebras and Applications The Abel Symposium 2015 ABEL SYMPOSIA Edited by the Norwegian Mathematical Society More information about this series at http://www.springer.com/series/7462 Participants at the Abel Symposium 2015. Photo taken by Andrew Toms Toke M. Carlsen • Nadia S. Larsen • SergeyNeshveyev•ChristianSkau Editors Operator Algebras and Applications The Abel Symposium 2015 123 Editors Toke M. Carlsen Nadia S. Larsen Department of Science and Technology Department of Mathematics University of the Faroe Islands University of Oslo Tórshavn, Faroe Islands Oslo, Norway Sergey Neshveyev Christian Skau Department of Mathematics Department of Mathematical Sciences University of Oslo Norwegian University of Science Oslo, Norway and Technology Trondheim, Norway ISSN 2193-2808 ISSN 2197-8549 (electronic) Abel Symposia ISBN 978-3-319-39284-4 ISBN 978-3-319-39286-8 (eBook) DOI 10.1007/978-3-319-39286-8 Library of Congress Control Number: 2016945020 Mathematics Subject Classification (2010): 46Lxx, 37Bxx, 19Kxx © 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 Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Foreword The Norwegian government established the Abel Prize in mathematics in 2002, and the first prize was awarded in 2003. In addition to honoring the great Norwegian mathematician Niels Henrik Abel by awarding an international prize for outstanding scientific work in the field of mathematics, the prize shall contribute toward raising the status of mathematics in society and stimulate the interest for science among school children and students. In keeping with this objective, the Niels Henrik Abel Board has decided to finance annual Abel Symposia. The topic of the symposia may be selected broadly in the area of pure and applied mathematics. The symposia should be at the highest international level and serve to build bridges between the national and international research communities. The Norwegian Mathematical Society is responsible for the events. It has also been decided that the contributions from these symposia should be presented in a series of proceedings, and Springer Verlag has enthusiastically agreed to publish the series. The Niels Henrik Abel Board is confident that the series will be a valuable contribution to the mathematical literature. Chair of the Niels Henrik Abel Board Helge Holden v Preface Målet for vår vitenskap er på den ene side å oppnå nye resultater, og på den annen side å sammenfatte og belyse tidligere resultater sett fra et høyere ståsted. Sophus Lie1 The Abel Symposium 2015 focused on operator algebras and the wide ramifications the field has spawned. Operator algebras form a branch of mathematics that dates back to the work of John von Neumann in the 1930s. Operator algebras were proposed as a framework for quantum mechanics, with the observables replaced by self-adjoint operators on Hilbert spaces and classical algebras of functions replaced by algebras of operators. Spectacular breakthroughs by the Fields medalists Alain Connes and Vaughan Jones marked the beginning of an impressive development, in the course of which operator algebras established important ties with other areas of mathematics, such as geometry, K-theory, number theory, quantum field theory, dynamical systems, and ergodic theory. The first Abel Symposium, held in 2004, also focused on operator algebras. It is interesting to see the development and the remarkable advances that have been made in this field in the years since, which strikingly illustrate the vitality of the field. The Abel Symposium 2015 took place on the ship Finnmarken, part of the Coastal Express line (the Norwegian Hurtigruten), which offered a spectacular venue. The ship left Bergen on August 7 and arrived at its final destination, Harstad in the Lofoten Islands, on August 11. The scenery the participants saw on the way north was marvelous; for example, the ship sailed into both the Geirangerfjord and Trollfjord. There were altogether 26 talks given at the symposium. In keeping with the organizers’ goals, there was no single main theme for the symposium, but rather a variety of themes, all highlighting the richness of the subject. It is perhaps appropriate to draw attention to one of the themes of the talks, which is the classification program for nuclear C-algebras. In fact, a truly major breakthrough 1“The goal of our science is on the one hand to obtain new results, and on the other hand to summarize and illuminate earlier results as seen from a higher vantage point.” Sophus Lie vii viii Preface in this area occurred just a few weeks before the Abel Symposium 2015—amazing timing! Some of the protagonists in this effort—one that has stretched over more than 25 years and has involved many researchers—gave talks on this very topic at the symposium. The survey article by Wilhelm Winter in this proceedings volume offers a panoramic view of the developments in the classification program leading up to the breakthrough mentioned above. Alain Connes and Vaughan Jones were also among the participants, and they gave talks on topics ranging, respectively, from gravity and the standard model in physics to subfactors, knot theory, and the Thompson group, thus illustrating the broad ramifications of operator algebras in modern mathematics. Ola Bratteli and Uffe Haagerup, two main contributors to the theory of operator algebras, tragically passed away in the months before the symposium. Their legacy was commemorated and honored in a talk by Erling Størmer. One of the articles in this volume is by Uffe Haagerup, and its publication was made possible with the help of three of Haagerup’s colleagues from the University of Copenhagen, to whom he had privately communicated the results shortly before his untimely passing. The articles in this volume are organized alphabetically rather than thematically. Some are research articles that present new results, others are surveys that cover the development of a specific line of research, and yet others offer a combination of survey and research. These contributions offer a multifaceted portrait of beautiful mathematics that both newcomers to the field of operator algebras and seasoned researchers alike will appreciate. Tórshavn, Faroe Islands Toke M. Carlsen Oslo, Norway Nadia S. Larsen Oslo, Norway Sergey Neshveyev Trondheim, Norway Christian Skau April 2016 Contents C-Tensor Categories and Subfactors for Totally Disconnected Groups............................................................................. 1 Yuki Arano and Stefaan Vaes Decomposable Approximations Revisited ..................................... 45 Nathanial P. Brown, José R. Carrión, and Stuart White Exotic Crossed Products......................................................... 61 Alcides Buss, Siegfried Echterhoff, and Rufus Willett On Hong and Szymanski’s´ Description of the Primitive-Ideal Space of a Graph Algebra ............................. 109 Toke M. Carlsen and Aidan Sims Commutator Inequalities via Schur Products ................................ 127 Erik Christensen C-Algebras Associated with Algebraic Actions.............................. 145 Joachim Cuntz A New Look at C-Simplicity and the Unique Trace Property of a Group......................................................................... 161 Uffe Haagerup Equilibrium States on Graph Algebras........................................ 171 Astrid an Huef and Iain Raeburn Semigroup C-Algebras ......................................................... 185 Xin Li Topological Full Groups of Étale Groupoids .................................. 197 Hiroki Matui Towards a Classification of Compact Quantum Groups of Lie Type ...... 225 Sergey Neshveyev and Makoto Yamashita ix x Contents A Homology Theory for Smale Spaces: A Summary......................... 259 Ian F. Putnam On the Positive Eigenvalues and Eigenvectors of a Non-negative Matrix ............................................................................. 271 Klaus Thomsen Classification of Graph Algebras: A Selective Survey ....................... 297 Mark Tomforde QDQ vs. UCT ..................................................................... 321 Wilhelm Winter C-Tensor Categories and Subfactors for Totally Disconnected Groups Yuki Arano and Stefaan Vaes Abstract We