On the C*-Algebras of Foliations in the Plane
The main result of this original research monograph is the classification of C*-algebras of ordinary foliations of the plane in terms of a class of -trees. It reveals a close connection between some most recent developments in modern analysis and low-dime
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1257
Xiaolu Wang
On the C*-Algebras of Foliations in the Plane
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1257
Xiaolu Wang
On the C*-Algebras of Foliations in the Plane
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
XiaoluWANG Department of Mathematics University of Chicago Chicago, Illinois 60637, USA
Mathematics Subject Classification (1980): 34C35, 46L55, 46L80, 57R30
ISBN 3-540-17903-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17903-8 Springer-Verlag New York Berlin Heidelberg
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. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
TO MY FATHER
I would rather schematize the structure of mathematics by a complicated graph, where the vertices are the various parts of mathematics and the edges describe the connections between them.
These
connections sometimes go one way, sometimes both ways, and the vertices can act both as sources and sinks.
The development of the
individual topics is of course the life and blood of mathematics, but, in the same way as a graph is more than the union of its vertices, mathematics is much more than the sum of its parts.
It is the
presence of those numerous, sometimes unexpected edges, which makes mathematics a coherent body of knowledge, and testifies to its fundamental unity. A. Borel
I have the feeling that we don't understand at all the extraordinary interplay of combinatorics and what I would call "conceptual" mathematics.
J. Dieudonne
Contents
o.
Introduction.
l.
Foliations of the pl ane
10
2.
Various trees and graphs.
35
3.
Distinguished trees
4.
The C* -algebras of fo1 iations of the plane.
.
1
.
.
53
. 1-11
Bibliography.
159
Notation and Symbols.
162
Subject Index
..
.
163
§o. 0.1
Introduction In [Cl] A. Connes introduced C*-algebras of foliations as an
important ingredient in his noncommutative integration theory.
Linking
topology, geometry and analy3is, it has become an important area of research; see [C2] for a more detailed treatment.
Many general proper-
ties of C*-algebras of foliations have also been studied by T. Fack, G. 5kandalis, M. Hilsumand others; see [F-5], [F], [H-5]. A very interesting and natural problem is to determine for various manifolds the structure of the C*-algebras of special types of foliations, or at least their K-theory.
This wil
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