97i:46094 46Lxx 46-01
Fillmore, Peter A.(3-DLHS)
A user's guide to operator algebras. (English)
Canadian Mathematical Society Series of Monographs and Advanced Texts.
A Wiley-Interscience Publication.
John Wiley & Sons, Inc., New York, 1996. xiv+223 pp. $59.95. ISBN
0-471-31135-9
This "user's guide to operator algebras" is unique among the many
monographs and treatises on operator algebras. Its primary purpose is
to provide the nonspecialist with an accessible and reasonably
complete overview of the subject in a single volume of moderate length
(223 pages). To make his plan work, the author has judiciously
selected material that lies primarily in the mainstream of the
subject. Assuming a knowledge of general functional analysis
(including the Gelfand theory of commutative Banach algebras) and
operator theory through the spectral theorem, he presents elegant and
slick proofs of most results in the early going. However, as he
proceeds through the book fewer and fewer proofs are given.
The reviewer likens the nature of the author's presentation to the way
an expert in a colloquium talk provides an overview of his subject to
nonexperts, making certain to keep the main ideas and notation simple
and clear. Often (but not always) when complicated details show up the
author simply sidesteps them and provides the reader with a suitable
reference. Coupling this with the typography of the book, which is
large with wide margins and considerable open space, the treatment
gives the impression that the subject is somewhat easier and more
accessible than it really is. (One reviewer claims that Fillmore's
book "makes it possible to painlessly glean the outlines of the
subject in a few hours".) However, it is precisely this character of
the book that makes it suitable as a "user's guide", and sets it apart
from other books in the field.
Now, a few words about the contents. There are ten chapters with the
respective titles: Fundamentals; Spectrum and order; Examples and
constructions; von Neumann algebras; Representations; Structure
theory; Tensor products; Crossed products; Factors; and $K$-theory.
The author provides definitions, often in the context of a discussion
(to save space). Some of these are detailed, while others are just
outlined. The writing is clear, so the reader has a good idea of what
is going on. Examples are provided and are treated in the same spirit
as the rest of the material.
The first five chapters constitute an introduction to the basic theory
of $C\sp *$-algebras and von Neumann algebras. The goal in the next
four chapters is to introduce the work of Alain Connes (and others) on
amenable von Neumann algebras and of Vaughn Jones on factors and knot
theory. The final chapter, on $K$-theory, also includes a description
of classifying amenable $C\sp *$-algebras by means of $K$-theoretical
invariants. As was pointed out above, few complete proofs are given in
the last five chapters because of the highly technical details that
are involved.
Although the book contains no formal exercises, there are ample
opportunities for the reader to fill in details of various
constructions and arguments. A bibliography consisting of 150 items
completes the work. This book is likely to be popular, especially
among those who wish to check out the field of operator algebras but
do not have the time or energy to go through the more detailed
treatments that exist. It is a useful and welcome addition to the
literature on operator algebras.
\{See also the following review [MR 97i:46095].\}
Reviewed by Robert S. Doran
_________________________________________________________________
Next Review
95j:46063 46Lxx 19K56 22D25 58B30 58G12 81T13 81V22 81V70
Connes, Alain(F-CDF)
Noncommutative geometry. (English. English summary)
Academic Press, Inc., San Diego, CA, 1994. xiv+661 pp. $59.95. ISBN
0-12-185860-X
FEATURED REVIEW. Alain Connes received the Fields Medal at the Warsaw
Congress in 1983. The citation, which concentrates on his work on von
Neumann algebras, mentions noncommutative geometry in passing, and
states confidently that "the subject will rapidly develop much
further". This large (over 600 pages), dense, fascinating, and
beautiful book is evidence of the accuracy of this prognosis.
To see what is meant by the phrase "noncommutative geometry", consider
ordinary geometry: for example, the geometry of a surface $S$ in
three-dimensional Euclidean space ${\bf R}\sp 3$. Following Descartes,
we study the geometry of $S$ using coordinates. These are just three
functions $x$, $y$, $z$ on ${\bf R}\sp 3$, and their restrictions to
$S$ generate, in an appropriate sense, the algebra $C(S)$ of all
continuous functions on $S$. All the geometry of $S$ is encoded in
this algebra $C(S)$; in fact, the points of $S$ can be recovered
simply as the algebra homomorphisms from $C(S)$ to C. In the language
of physics, one might say that the transition from $S$ to $C(S)$ is a
transition from a "particle picture" to a "field picture" of the same
physical situation.
One notes that the algebra $C(S)$ appearing here is commutative. It is
therefore natural to suggest that it might be valuable to think of
noncommutative algebras in geometric terms, as the algebras of
functions on "noncommutative spaces". Some attempts in this direction
have been made from algebraic geometry, but Connes's program rests on
differential geometry and incorporates deep techniques from functional
analysis. Motivation for this program comes from a number of sources.
One is physics: quantum mechanics asserts that physical observables
such as position and momentum should be modeled by elements of a
noncommutative algebra. Another is a series of examples showing that
objects such as the space of (unitary) representations of a group, or
the space of leaves of a foliation, may be most appropriately modeled
as noncommutative spaces. A third comes from the application of
$K$-theory to operator algebras and the link (in the case of group
$C\sp *$-algebras) to the Novikov higher-signature conjecture and
surgery theory. All these together with other motivations and
applications are discussed in detail in the book.
Connes's book guides the reader through the appropriate noncommutative
versions of four successively more refined kinds of geometric
structure: measure theory, topology (including algebraic topology),
differential topology (manifold theory), and differential (Riemannian)
geometry. To begin with the coarsest of these, noncommutative measure
theory means the theory of von Neumann algebras. Let $H$ be a Hilbert
space, $B(H)$ the set of bounded linear operators on it. A von Neumann
algebra is an involutive subalgebra $A$ of $B(H)$ which is closed in
the weak topology on $B(H)$. The commutative example to keep in mind
is $H=L\sp 2(X,µ)$, where $(X,µ)$ is a measure space, and $A=L\sp
\infty(X,µ)$ acting by multiplication; the weak topology coincides
with the topology of pointwise almost everywhere convergence on
bounded sets. This algebra is in a natural sense a "direct integral"
of copies of ${\bf C}$, which is a factor---that is, a von Neumann
algebra with trivial centre. Already in the original papers of Murray
and von Neumann one finds the classification of factors into three
types: type I, matrix algebras; type II, admitting a "continuous
dimension" function with real values; type III, admitting no dimension
function. Of these, the factors of type III are the most mysterious.
Much of Connes's early work was devoted to elucidating their
structure, and this is described in Chapter V of the book. This
chapter also contains a discussion of the work of Vaughan Jones on
subfactors of factors of type II, which had such remarkable
applications to knot theory.
An example which is significant for many aspects of the book comes
from the theory of foliations. Let $(V,{\scr F})$ be a compact
foliated manifold. We want to study the "space of leaves" $V/{\scr
F}$---initially only as a measure space, but then successively in the
more refined categories mentioned above. If the foliation is
ergodic---think of the irrational-slope flow on a torus---then the
quotient measure space in the usual sense is trivial. But we can form
a noncommutative von Neumann algebra whose elements are measurable
families of operators on the $L\sp 2$ spaces of the leaves of the
foliations, and it turns out that this von Neumann algebra is
equivalent in the non-singular case to the one arising from the usual
measure-theoretic quotient, and that it is interesting even in the
singular case; for instance, the von Neumann algebra of the
irrational-slope flow is of type II, and its "continuous dimension"
function is given by the unique invariant transverse measure on the
foliation. There are remarkable links between this von Neumann algebra
and the more classical invariants of foliations, for example the
result (originally due to Steve Hurder) that if the Godbillon-Vey
class of a foliation is nonzero, then the associated von Neumann
algebra is of type III.
The noncommutative analogue of topology depends on the theory of $C\sp
*$-algebras. These are involutive subalgebras of $B(H)$ which are
closed for the norm topology: the standard (in fact, the only)
commutative example is the algebra of continuous functions vanishing
at infinity on a locally compact Hausdorff space. The noncommutative
quotient construction can be applied in the $C\sp *$-algebraic setting
also. An example of particular interest arises by applying the
quotient construction to the trivial action of a discrete group
$\Gamma$ on a point; one obtains the (reduced) group $C\sp *$-algebra
$C\sp *\sb r(\Gamma)$, which may be regarded as an analytical
completion of the group algebra ${\bf C}\Gamma$. Connections with
geometric topology arise when $\Gamma$ is the fundamental group of a
manifold.
By the theorem of Gelfand and Naimark, the study of commutative $C\sp
*$-algebras is exactly the same as the study of locally compact
Hausdorff spaces. It is natural therefore to ask whether the tools of
algebraic topology extend to the noncommutative $C\sp *$-algebraic
world. It turns out that topological $K$-theory and its $S$-dual
theory, $K$-homology, have natural extensions to the category of $C\sp
*$-algebras. Moreover, there is a natural relationship with index
theory in the case of $C\sp *$-algebras arising from the
noncommutative quotient construction. Recall that Atiyah and Singer
showed that the index of a family of elliptic operators over a base
$B$ should be thought of as an element of the $K$-theory group $K\sp
*(B)$. Analogously, a family of elliptic operators along the leaves of
a foliation, or on the fibres of any other quotient, has an index in
the $K$-theory of the noncommutative quotient $C\sp *$-algebra. A very
general conjecture is then that the whole $C\sp *$-algebra $K$-theory
group is exactly generated by indices of this kind. This is known as
the Baum-Connes conjecture, and it has been verified in a large number
of different situations. It also has noteworthy implications: for
example, the Baum-Connes conjecture for a discrete group $\Gamma$
implies the Novikov conjecture for $\Gamma$, which (via surgery
theory) is crucial to the topology of high-dimensional manifolds
having $\Gamma$ as fundamental group.
Chapter II of the book contains a detailed account of this material.
The general theory is illustrated with many beautiful examples. Connes
gives particular prominence to groupoids arising from geometric
situations, and the $K$-theory maps arising from associated
deformations of algebras. For example, the tangent groupoid is an
algebraic encoding of the idea that the tangent bundle of a manifold
$M$ can be obtained by blowing up the diagonal in $M\times M$; the
associated convolution algebra encapsulates the index-theoretic
aspects of the pseudodifferential calculus.
The next stage in the development is the noncommutative analogue of
differential topology. Granted that $K$-theories seem to be "natural"
(co)homology theories in the noncommutative category, one might still
search for a noncommutative analogue of ordinary (co)homology. The
strategy in the search for the noncommutative analogue of some notion
is first to find a way of formulating the commutative notion in
"function" rather than "point" terms, and then to try replacing the
functions by elements of a noncommutative algebra. Now the version of
cohomology that is most naturally expressed in "function" terms is de
Rham cohomology, but (since it involves differentiation) it must be
formulated in terms of the dense subalgebra of smooth functions inside
the $C\sp *$-algebra of continuous ones. In the absence of a general
smoothing theory for $C\sp *$-algebras (as there is, say, for
high-dimensional topological manifolds) the choice of such a
subalgebra has to be made ad hoc. Often, however, a natural such
subalgebra is available, and one refers to it as a smooth structure on
the $C\sp *$-algebra in question.
Suppose a smooth structure has been chosen. How should we construct
"noncommutative de Rham theory"? Connes's answer is his theory of
cyclic cohomology, expounded in Chapter III of the book. Cyclic
cohomology has received an extensive purely algebraic development in
recent years [see, for example, J.-L. Loday, Cyclic homology,
Springer, Berlin, 1992; MR 94a:19004], but Connes's discussion remains
close to the original geometric motivation. One of the deepest
applications is to the construction of the transverse fundamental
class for a foliation: this is a map from the $K$-theory of the
"transverse space" to the real numbers, and it would have an easy
construction if there were an invariant transverse Riemannian metric.
In general no such metric can be found, but Connes noticed that this
problem was analogous to one that he had already surmounted in his
work on factors of type III. Following this insight, and combining
ideas from cyclic theory (here the choice of an appropriate dense
subalgebra is a very delicate matter) and sophisticated versions of
the Thom isomorphism in $K$-theory, Connes was able to construct the
desired map. Geometric results about foliations, including the theorem
of Hurder mentioned above, are immediate corollaries.
A second motivation for cyclic cohomology, equally significant, is the
idea of "quantized calculus", expounded in Connes's Chapter IV. If $A$
is an algebra of operators on some Hilbert space $H$, and $F$ an
operator on $H$ with $F\sp 2=1$, then the definition $da = [F,a] = Fa
- aF$ obeys the basic rule $d\sp 2=0$ of de Rham theory. If in
addition $F$ and $A$ are "almost commuting" enough that any product of
sufficiently many commutators $[F,a]$ is of trace class, then the
trace can be used to define an "integral" of these "differential
forms" and we have the ingredients for a "noncommutative de Rham
current", i.e. a cyclic cocycle. A helpful example is $A=C\sp
\infty(M)$, $M$ a compact manifold of dimension $n$, and $F$ some
(zeroth order) pseudodifferential involution. The commutators are then
pseudodifferential operators of order $-1$, and any product of more
than $n$ of them is therefore trace-class. In this example we see that
the notion of dimension in quantized calculus is related to the
"degree of traceability" of certain commutators.
Not all cyclic classes arise in this way. In fact, such a cyclic class
belongs to the image of a "Chern character" map from $K$-homology. It
follows, therefore, that if such a cyclic class is paired with an
element of $K$-theory the result must be an integer, because the
evaluation of pairing between $K$-theory and $K$-homology ultimately
reduces to the computation of a Fredholm index. In this way one
obtains integrality results somewhat analogous to those provided by
the Atiyah-Singer index theorem. An application discussed in detail by
Connes is the work of J. Bellissard on the existence of integer
plateaux of conductivity in the quantum Hall effect. This concerns the
flow of electricity in a crystal subjected to electric and magnetic
fields. In general, the "natural" observables in this problem generate
an algebra isomorphic to that associated to the irrational-slope
foliation on a torus, and the integrality arises from an associated
cyclic 2-cocycle.
The last chapter in the book is entitled "The metric aspect of
noncommutative geometry", and begins an investigation of the analogue
of Riemannian geometry and Yang-Mills theory in the noncommutative
context. Many readers will focus with particular interest on the
interpretation of the Standard Model of particle physics in terms of a
noncommutative space which is the product of an ordinary manifold with
a (very small) discrete space. Connes writes: "Our contribution should
be regarded as an interpretation, of a geometric nature, of all the
intricacies of the most accurate phenomenological model of high-energy
physics.$\,\ldots$ It does undoubtedly confirm that high-energy
physics is in fact unveiling the fine structure of space-time.
Finally, it gives a status to the Higgs boson as just another gauge
field, but one corresponding to a finite difference rather than a
differential."
The book is magnificently produced, and contains so much new material
that it completely supersedes the French edition [Geometrie non
commutative, InterEditions, Paris, 1990; MR 92e:58016]. The depth and
variety of Connes' thought will lead the serious reader to explore the
background and proofs of the results in more detail, and he or she
will find a comprehensive bibliography for this purpose.
On the back cover, the publishers describe the book as "the definitive
treatment", but the subject is still developing so rapidly that the
sense of finality conveyed by this phrase hardly seems appropriate.
Vaughan Jones' comment, also on the cover, is more apt: "A milestone
for mathematics".
Reviewed by John Roe
Cited in: 98d:58007 98d:00001 97k:19002 97j:19005 97i:46112 97h:81248
97f:46112 97a:46106 97a:46024 96j:46074 96c:58016
_________________________________________________________________
92d:46141 46Lxx 01A60 46-03
Kadison, Richard V.(1-PA)
Operator algebras---an overview. (English) The legacy of John von
Neumann (Hempstead, NY, 1988), 61--89,
Proc. Sympos. Pure Math., 50,
Amer. Math. Soc., Providence, RI, 1990.
This expository paper is an expanded version of a survey talk given by
the author at the 1988 conference at Hofstra University honoring the
late John von Neumann. The subject of operator algebras (algebras of
operators on a Hilbert space) began a little over sixty years ago with
a paper by von Neumann [Math. Ann. 102 (1929), 370--427; Jbuch 55,
825]. It has since grown into an enormous branch of functional
analysis that has important connections with harmonic analysis, group
representations, ergodic theory, algebraic topology, differential
geometry, $K$-theory, knot theory and quantum physics, to name some of
the principal areas.
The author, who has been one of the central figures in the development
of the subject, begins with a brief description of von Neumann's
original motivation for developing the theory, his (von Neumann's)
collaboration with F. J. Murray, the characterization of commutative
$C\sp *$-algebras by Gelfand and Naimark (M. H. Stone's influence is
mentioned), and the connections of measure theory with the theory of
abelian von Neumann algebras.
He then discusses factors and their classification up to isomorphism,
states, weights, and traces, the GNS-construction, the relation of von
Neumann algebras to their commutants (Tomita-Takesaki theory), and A.
Connes' work on classifying factors of type III. He also treats
crossed products, tensor products and matricial (UHF) algebras. The
principal contributors to each of the above areas and the roles they
played are briefly described together with a little bit of history
thrown in here and there. The bibliography contains six of the main
books on the subject and fifty-one of the most significant papers.
Details of much of what is described can be found in the author's
two-volume work with J. R. Ringrose [Fundamentals of the theory of
operator algebras, Vol. I, Academic Press, New York, 1983; MR
85j:46099; Vol. II, 1986; MR 88d:46106]. The reviewer believes that
this paper is an excellent source for young mathematicians wishing to
get started in the field.
{For the entire collection see MR 91e:00024}.
Reviewed by Robert S. Doran
_________________________________________________________________
85h:55004 55M20 57S17
Fadell, Edward(1-WI); Husseini, Sufian(1-WI)
The Nielsen number on surfaces. (English) Topological methods in
nonlinear functional analysis (Toronto, Ont., 1982), 59--98,
Contemp. Math., 21,
Amer. Math. Soc., Providence, R.I., 1983.
_________________________________________________________________
with CnoFile728139Remarks This paper develops useful tools for
studying the minimum number of fixed points for a homotopy class of
self-maps of a compact surface. The authors present a new procedure
for computing the Nielsen number in this setting, using the "Fox
calculus" of knot theory to compute Husseini's generalized Lefschetz
number [Trans. Amer. Math. Soc. 272 (1982), no. 1, 247--274; MR
83m:55004]. Next they study the Nielsen number on surfaces from the
obstruction theory point of view that they had used, in earlier
papers, for manifolds of dimension at least three, and they show that
there are significant difficulties that arise only in the
two-dimensional setting. When they focus their attention on the
problem of deforming a map, already fixed point free on the
one-skeleton, to a map which is fixed point free on the entire
surface, they obtain as a necessary and sufficient condition the
triviality of a certain subset of the braid group, on two strands, of
the surface. Consequently, much of the paper is taken up with an
investigation of the relevant braid theory.
An important motivation for the study of fixed points on surfaces has
been the search for a counterexample to the Nielsen conjecture [see
the preceding review]. In the present paper, which was written some
time before Jiang discovered his example, the authors define a map $f$
on the closed orientable surface $M$ of genus 3 such that $f$ has
Nielsen number zero. A very extensive analysis of the braid theory
related to this map $f$ fails to determine whether or not every map
homotopic to $f$ has a fixed point. If it does, then $f$ provides a
counterexample to the Nielsen conjecture for closed surfaces, which
remains open because Jiang's example is on the disc-with-two-holes.
The braid theory calculations do, however, furnish a counterexample to
a "local" version of the Nielsen conjecture. The authors exhibit an
open subset $U$ of $M$ and a map $f\colon U\rightarrow M$ which has
local Nielsen number [the authors, Illinois J. Math. 25 (1981), no. 4,
673--699; MR 83i:55002] equal to zero yet if $g\colon U\rightarrow M$
is any map homotopic to $f$ by a compactly-fixed homotopy, then $g$
has a fixed point.
\{For the entire collection see MR 84m:47003\}.
Reviewed by Robert Freeman Brown
Cited in: 96m:55002 90k:55004
© Copyright American Mathematical Society 1998