Monotone Complete C*-algebras and Generic Dynamics
This monograph is about monotone complete C*-algebras, their properties and the new classification theory. A self-contained introduction to generic dynamics is also included because of its important connections to these algebras. Our knowledge and underst
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Kazuyuki Saitô J. D. Maitland Wright
Monotone Complete C*-algebras and Generic Dynamics
Springer Monographs in Mathematics
More information about this series at http://www.springer.com/series/3733
Kazuyuki Saitô • J.D. Maitland Wright
Monotone Complete C*-algebras and Generic Dynamics
123
Kazuyuki Saitô Sendai, Japan
J.D. Maitland Wright University of Aberdeen Aberdeen, Scotland Christ Church University of Oxford Oxford, UK
ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-1-4471-6773-0 ISBN 978-1-4471-6775-4 (eBook) DOI 10.1007/978-1-4471-6775-4 Library of Congress Control Number: 2015958093 Mathematics Subject Classification: 46L05, 46L35, 37A20, 37A55, 46L36, 06A12, 54G05, 06F05 Springer London Heidelberg New York Dordrecht © Springer-Verlag London 2015 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 Springer-Verlag London Ltd. is part of Springer Science+Business Media (www.springer.com)
Abstract
This book is about monotone complete C -algebras, their properties, and the new classification theory (using spectroid invariants and a classification semigroup). A basic account of generic dynamics is included because of its important connections to these algebras. Each bounded, upward-directed net of real numbers has a limit. Monotone complete algebras of operators have a similar property. In particular, every von Neumann algebra is monotone complete, but the converse is false. The small von Neumann factors can be labelled by the set of real numbers. But there are many more (2R ) small monotone complete C -algebras which are factors. The aim of this book is to give an account of monotone complete C -algebras which includes recent advances but also indicates the many mysteries and open problems which remain.
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Preface
First, we wish to thank Professor Takesaki who encouraged us to write this book. Without him, we would never have started; without the unfailing support of Yuko and Harvinder, w
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