Elliptic Theory and Noncommutative Geometry Nonlocal Elliptic Operat
This comprehensive yet concise book deals with nonlocal elliptic differential operators, whose coefficients involve shifts generated by diffeomorophisms of the manifold on which the operators are defined. The main goal of the study is to relate analy
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H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. Olshevsky (Storrs) M. Putinar (Santa Barbara) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) T. Kailath (Stanford) H. Langer (Vienna) P. D. Lax (New York) H. Widom (Santa Cruz)
Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze Universität Potsdam, Germany
Jerome A. Goldstein The University of Memphis, TN, USA
Sergio Albeverio Universität Bonn, Germany
Nobuyuki Tose Keio University, Yokohama, Japan
Michael Demuth Technische Universität Clausthal, Germany
Elliptic Theory and Noncommutative Geometry Nonlocal Elliptic Operators Vladimir E. Nazaikinskii Anton Yu. Savin Boris Yu. Sternin
Birkhäuser Basel · Boston · Berlin
A P D E
Advances in Partial Differential Equations
Authors: Vladimir E. Nazaikinskii Institute for Problems in Mechanics Russian Academy of Sciences Prosp. Vernadskogo 101-1 119526 Moscow Russia e-mail: [email protected]
Anton Yu. Savin Boris Yu. Sternin Independent University of Moscow Bolshoy Vlasyevskiy Pereulok 11 119002 Moscow Russia e-mail: [email protected] [email protected]
2000 Mathematical Subject Classification: 19K, 35J, 39B, 46L, 58J
Library of Congress Control Number: 2008924711
Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
ISBN 978-3-7643-8774-7 Birkhäuser Verlag AG, Basel - Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.
© 2008 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-7643-8774-7
e-ISBN 978-3-7643-8775-4
987654321
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Contents Preface
xi
Introduction
1
I Analysis of Nonlocal Elliptic Operators
3
1 Nonlocal Functions and Bundles 1.1 Group Algebras and Crossed Products . . . . . 1.1.1 Group Algebras . . . . . . . . . . . . . . 1.1.2 C ∗ -Crossed Products . . . . . . . . . . . 1.1.3 Isomorphism Theorem . . . . . . . . . . 1.1.4 Smooth Crossed Products . . . . . . . . Motivation . . . . . . . . . . . . . . . . Sufficient Condition for Locality . . . . Groups of Polynomial Growth . . . . . . Tempered Actions . . . . . . . . . . . . Schweitzer theorem . . . . .
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