K-Theory and Operator Algebras: Proceedings of a Conference held at the University of Georgia in Athens, Georgia, April 21–25, 1975

199 Pages·1977·2.937 MB·English

by  M. F. Atiyah (auth.), Bernard B. Morrel, I. M. Singer (eds.)

Lecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
575
K-Theory
and Operator Algebras
Proceedings of a Conference held at
the University of Georgia in Athens,
Georgia, April 21-25, 1975
Edited by
B. B. Morrel and I. M. Singer
Spri nger-Verlag
Berlin· Heidelberg· New York 1977
Editors
Bernard B. Morrel
Department of Mathematics
Swain Hall East, Indiana University
Bloomington, IN 47401/USA
I. M. Singer
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139/USA
Library of Congress Cataloging in Publication Data
ConferenCE; on K-Theory and Operato:::' Al~"etras, "JEi',I
si ty of Georgia, 1975.
K-tteory a:1d operator algebras.
(Lecture notes in mathemat::.cs ; 575)
1. K-theory--Congresses. 2. Operator algebras-
Congresses. I. Morrel, Bernar::1 B. II.
Isadore Manuel, ::"924- III. Title. IV.
Lecture notes in mathematiss (Eerlin) ; 575.
Q,A3.I28 no. 575 [G,A6l2.33J 5l0' .8s 77-1050
[5l4' .23 J
AMSSubjectClassifications(1970): 18F25, 46-02, 46 L05, 46 L 10,47-02,
47G05, 55-02, 55B15, 58-02, 58GlO, 58G15
ISBN 3-540-08133-X Springer-Verlag Berlin' Heidelberg· New York
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Preface
This volume records most of the talks given at the
Conference on K-theory and Operator Algebras held at the
University of Georgia in Athens, Georgia, April 21-
April 25, 1975. The purpose of the conference was to
review the known connections between operator theory
and K-theory and explore possible new ones. Consequently,
some of the papers present historical background, some
develop new ideas, some are expository in an attempt to
acquaint experts in the one field with recent develop
ments in the other, and some pose new problems which
further developments might solve.
We are happy to express our thanks to the National
Science Foundation for sponsoring the meeting, to the
University of Georgia for providing funds for additional
participants, to the Mathematics Department for its gracious
hospitality, to Ms. Teddy Schultz, Ms. Carol Ledbetter and
Ms. Ann Ware for the typing, to the authors for their
manuscripts and their patience, and to Springer-Verlag
for publishing this volume.
Bernard Morrel
I. M. Singer
CONTENTS
1. M. F. ATIYAH
A survey of K-theory
4. LAWRENCE G. BROWN
Characterizing ext (X) . 10
S. RICHARD W. CAREY and JOEL D. PINCUS
Almost commuting algebras . . . 19
6. R. G. DOUGLAS
Extensions of C·-algebras and K-homology 44
7. KARL HEINRICH HOFMA~N
Bundles and sheaves are equivalent in the
category of Banach spaces . . . . . . 53
8. JEROME KAMINKER
Topological obstructions to perturbations
of pairs of operators .. . ..... . 70
9. RONNIE LEE and R. H. SZCZARBA
On algebraic K-theory and the homology of
congruence subgroups . . . . . . 78
10. DUSA MACDUFF
Configuration spaces 88
11. CARL PEARCY and NORBERTO SALINAS
Extensions of C·-algebras and the
reducing essential matricial spectra
of an operator •.. . . . . . . . . 96
v
13. GRAEME SEGAL
K-homology theory and algebraic K-theory 113
14. I. M. SINGER
Some remarks on operator theory and index
theory . . . . . . . . . . . . 12R
15. MASAMICHI TAKESAKI
Factors of type III • • . . . . . • . . . 1. 39.
16. JOSEPH L. TAYLOR
Twisted products of Banach algebras and
third tech cohomolog~ . . . . 157
17. J. B. WAGONER
H-cobordisms, pseudo-isotopies, and analytic
torsion. . . . . . . . . ... . .... 175
Conference Part icipants
Joel Anderson Karl Hofmann
Michael Atiyah John Hollingsworth
Edward Azoff Richard Kadison
Richard Bouldin Jerome Kaminkel'
John Bunce H. W. Kim
Manfred Breuer Dusa MacDuff
Lawrence G. Brown Bernard Morrel
Richard Carey Judith l\lorrel
Kevin Clancey R. D. Moyer
D. N. Clark Paul ~luhly
Lewis Coburn Catherine Olsen
E. H. Connell Carl pearcy
lain Craw William T. Pelletier
James Deddens Joel Pincus
James Deel lain Raeburn
Allen Devinatz William G. Rosen
R. G. Douglas Shoichil'o Sakai
Maurice Dupre Sorberto Sal inas
David A. Edwards David G. Schaeffer
John Ernest Claude Schochet
Nazih Faour Graeme Segal
Peter Fillmore I. ill. Singer
Robert E. Goad .James Simons
E. C. Gootman J. G. Stampfli
Paul Halmos Robert Szczarba
Herbert Halpern ~asamichi Takesaki
Allen Hatcher J. L. Taylor
William Hel ton Javier Thayer
Harold l\"idom
A Sl~V~Y OF K-THEORY
M.F. Atiyah, Oxford
Introduction
In this talk I shall describe the way K-theory enters in various
branches of mathematics. shall follow the historical development
and emphasize those aspects of most relevance to this conference.
shall not therefore dwell too much on the more algebraic parts of the
subject.
Let me first make some very general remarks on the nature of
K-theory. Roughly speaking K-theory may be described as the linear
algebra of large matrices, also called stable linear algebra, and it
deals primarily with such notions as idempotents (projections) and
units (invertible matrices). Its main feature is that it is an
abelian theory, despite the non-abelian character of matrices. This
stems from the fact that although A and B may not commute, A ~ I and
I ~ B do commute. Thus by increasing the size of our matrices we can,
for certain purposes, reduce to an abelian situation.
When we transfer K-theory from one area of mathematics to ano-
ther certain formal similarities remain. However each area has dif-
ferent problems and techniques, and the success of K-theory hinges on
the fact that in many areas it has proved possible to link it up with
natural classical problems.
§l. Algebraic Geo~etry
K-theory was introduced into Algebraic Geometry by Grothendieck,
though the preliminary ground-work had been laid by Serre. For an
affine variety V c en, with coordinate ring A(V), we consider
finitely-generated projective ACV)-modules E, i.e. E is a direct
summand of a free module A(V)N and so is given by an idempotent in the
N x N matrix algebra over A(V). The isomorphism classes of such
modules form an abelian semigroup under ~ and KO(V) is the correspond
ing abelian group. Thus KO(V) is the universal group for studying
2
abelian invariants of projective modules.
For a projective variety one can use graded rings and modules
or, better still, one can use the geometrical language of vector
bundles. Thus we now consider an algebraic vector bundle E over V
or equivalently a locally free sheaf over the sheaf of functions of
V (the sheaf is given by the sections of E). Again we define KO(V)
although now, since short exact sequences do not split, we impose a
stronger equivalence relative so that KO(V) is universal for abelian
invariants aCE) such that aCE) aCE') + aCE") whenever
°
~ E' ~ E ~ E" ~ 0
is an exact sequence of vector bundles.
Note that the language of vector bundles works also in the affine
reEl
case since the global sections of E form a finitely-generated
projection ACV)-module and tile correspondence E ~ rCE) is bijective.
The two baSic examples, which motivated Grothendieck, of abelian
invariants of E are
q
i) 1(CE) ~"0 C-ll dim HqCY,El
q
ii) chCE), the Chern character of E.
Here (i) is the sheaf cohomology Euler characteristic (V assumed pro-
jective): it takes values in Z, while chCE) takes values in the
rational cohomology ring of V. Thus we have homomorphisms
If we use all coherent sheaves instead of locally free ones (in
the affine case this just means dropping the restriction that the
A(1f)-module be projective) we obtain another group denoted by KO(V).
The formal properties of KO(V) and KOCV) can now be summarized as
follows:
a) KO(V) is a ring (under ~) and KO(V) is a ,<O(Vl -module (note that
® does not preserve exact sequences, which is why KO(V) does not have
a ring structure)
b) KOeV)
3
contravariant functor of V while, for proper maps, KO(V) is covar
iant, the map Ka(V) - KO(V') generalizing 'x' (for V' ~ point.
K (V') Z) .
O
c) If V is projective non-singular the natural map KO(V) - KO(V) is
an isomorphism.
In view of these formal properties we see that there is a close
analogy between KO and cohomology, with K playing the role of homol
O
ogy, and c) being Poincare duality. This analogy is in fact quite
deep and has had a Significant influence on the development of K-
theory.
§2. Topology
If X is a compact space we can consider topological vector bun-
dIes (fibre ¢n) over X These correspond to finitely-generated pro-
jective modUles over C(X), the ring of continuo~ complex-valued
functions on X. We then form an abelian group KO(X) as before. Bc-
cause of the analogy with cohomology described in §2 we then define,
for n ~ 1,2, ...
n n
K- (X) = K comp (R x X)
Where, for a locally compact Y we put
Kc omp (Y) = Ker [K(Y U 00) - KCoo) ')
Y OJ ro being the one-point compactification of Y. The famous Batt
periodicity theorem then asserts that
n
and the groups K- (X) (extended to all n E Z by periodici ty) form a
generalized cohomology theory with products. This means they satisfy
all the EilenberG-Steenrod axioms for cohomology except the dimension
axiom (note that Kn(point) ~ Z for 11 even). If X is embeddable in
some RN (e.g. if X is a polyhedron) Alexander duality asserts that
N
Hq(X) ~ aN-q-lCR - X)
where H is reduced cohomology, i.e. modulo a base point. We can
4
therefore define groups
~-q-l(RN _ X), where K is reduced K-theory.
Since RN - X is not compact we must replace it by a compact deforma-
tion retract (imposing some mild restriction on the embedding of X
N
in R ). From this definition it follows that K* is a K*-module and
we even have a Poincare-duality theorem for suitable manifolds (e.g.
complex manifolds) .
Although formally satisfactory it is clear that the above defi-
nition of K* is somewhat artificial. It would be nice to have repre-
sentative objects which were more natural, as we do in algebraic
geometry with coherent sheaves. In [2] I made a first tentative step
in this direction and the work of Brown-Douglas-Fillmore, which we
shall hear about in this conference, provides a very satisfying and
natural analytical solution (see §3). It is also interesting to
compare the algebraic and topological K-theories of §l and §2 when
they both apply, namely when X "V is a proJective algebraic variety.
Clearly we have a homomorphism
~o : Iafl g (X) ~ KOto p eX)
Because K~oP is rather artificial it is not so clear how to define
alg toP
a K° (X) ~ K (X)
O
0
When X is non-singular one can apply "Poincare-duality" on both sides
and the functorial definition of ~ is closely related to the Riemann
Roch theorem (see (3). In the general case Baum and Macpherson [7)
have recently given a definition of Cia' In principle this is tanta
mount to a Riemann-Roch theorem for singular varieties.
§3. Functional Analysis
The K-theory of §2 has a very natural interpretation in terms of
operator theory. I recall that a bounded operator F on a complex
Hilbert space H is called Fredholm if
dim Ker F < '" and dim Coker F < co •
Its index is then defined as