Topics in noncommutative geometry : Third Luis Santalo Winter School-CIMPA Research School on Topics in Noncommutative Geometry, July 26-August 6, 2010, Universidad de Buenos Aires, Buenos Aires, Argentina

289 Pages·2012·2.06 MB·English

by  Guillermo Cortinas (eds.)

Topics in Noncommutative
Geometry

Clay Mathematics Proceedings
Volume 16
Topics in Noncommutative
Geometry
Third Luis Santaló Winter School -
CIMPA Research School
Topics in Noncommutative Geometry
Universidad de Buenos Aires
Buenos Aires, Argentina
July 26–August 6, 2010

Guillermo Cortiñas
Editor
American Mathematical Society
Clay Mathematics Institute
2010 Mathematics Subject Classification. Primary 14A22, 18D50, 19K35, 19L47, 19L50,
20F10, 46L55, 53D55, 58B34, 81R60.
Cover photo of the front of Pabello´n 1 of Ciudad Universitaria courtesy of Adria´n
Sarchese and Edgar Bringas.
Back cover photo of ProfessorLuis Santalo´ is courtesy of his family.
Library of Congress Cataloging-in-Publication Data
Luis Santal´o Winter School-CIMPA Research School on Topics in Noncommutative Geometry
(2010: BuenosAires,Argentina)
Topics in noncommutative geometry : Third Luis Santalo´ Winter School-CIMPA Research
SchoolonTopicsinNoncommutativeGeometry,July26–August6,2010,UniversidaddeBuenos
Aires,BuenosAires,Argentina/GuillermoCortin˜as,editor.
pagecm. —(Claymathematicsproceedings;volume16)
Includesbibliographicalreferences.
ISBN978-0-8218-6864-5(alk.paper)
1. Commutative algebra—Congresses. I. Cortin˜as, Guillermo, editor of compilation.
II.Title.
QA251.3.L85 2012
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2012031426
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Contents
Preface vii
Classifying Morita Equivalent Star Products 1
Henrique Bursztyn and Stefan Waldmann
Noncommutative Calculus and Operads 19
Boris Tsygan
Some Elementary Operadic Homotopy Equivalences 67
Eduardo Hoefel
Actions of Higher Categories on C*-Algebras 75
Ralf Meyer
Examples and Applications of Noncommutative Geometry and K-Theory 93
Jonathan Rosenberg
Rational Equivariant K-Homology of Low Dimensional Groups 131
J.-F. Lafont, I. J. Ortiz and R. J. Sa´nchez-Garc´ıa
Automata Groups 165
Andrzej Zuk
Spectral Triples and KK-Theory: A Survey 197
Bram Mesland
Deformations of the Canonical Spectral Triples 213
R. Trinchero
Twisted Bundles and Twisted K-Theory 223
Max Karoubi
A Guided Tour Through the Garden of Noncommutative Motives 259
Gonc¸alo Tabuada
v

Preface
This volume contains the proceedings of the third Luis Santal´o Winter School,
organizedbytheMathematicsDepartmentandtheSantalo´MathematicalResearch
Institute of the School of Exact and Natural Sciences of the University of Buenos
Aires (FCEN). This series of schools is named after the geometer Luis Santalo´.
Born in Spain, the celebrated founder of Integral Geometry was a Professor in our
department where he carried out most of his distinguished professional career.
This edition of the Santal´o School took place in the FCEN from July 26 to
August 6 of 2010. On this occasion the school was devoted to Noncommutative
Geometry, and was supported by several institutions; the Clay Mathematics Insti-
tute was one of its main sponsors.
The topics of the school and the contents of this volume concern Noncommu-
tative Geometry in a broad sense: it encompasses the various mathematical and
physical theories that incorporate geometric ideas to study noncommutative phe-
nomena.
One of those theories is that of deformation quantization. A main result in
this area is Kontsevich’s formality theorem. It implies that a Poisson structure on
a manifold can always be formally quantized. More precisely it shows that there
is an isomorphism (although not canonical) between the moduli space of formal
deformations of Poisson structures on a manifold and the moduli space of star
products on the manifold. The question of understanding Morita equivalence of
star products under (aspecific choice of) this isomorphism was solved by Henrique
Bursztyn and Stefan Waldmann; their article in the present volume gives a survey
of their work. They start by discussing how deformation quantization arises from
thequantizationprobleminphysics. Thentheyreviewthebasicsonstarproducts,
the main results on deformationquantization and the notion of Moritaequivalence
ofassociativealgebras. Afterthisintroductorymaterial, theypresent adescription
of Morita equivalence for star products as orbits of a suitable group action on
the one hand, and the B-field action on (formal) Poisson structures on the other.
Finally, they arrive at their main results on the classification of Morita equivalent
star products.
The next article, by Boris Tsygan, reviews Tamarkin’s proof of Kontsevich’s
formality theorem using the theory of operads. It also explains applications of the
formality theorem to noncommutative calculus and index theory. Noncommuta-
tivecalculusdefinesclassicalalgebraicstructuresarisingfromtheusualcalculuson
manifolds in terms of the algebra of functions on this manifold, in a way that is
vii
viii PREFACE
valid for any associative algebra, commutative or not. It turns out that noncom-
mutative analogs of the basic spaces arising in calculus are well-known complexes
from homological algebra. These complexes turn out to carry a very rich algebraic
structure, similar to the one carried by their classical counterparts. It follows from
Kontsevich’s formality theorem that when the algebra in question is the algebra of
functions,thosenoncommutativegeometrystructuresareequivalenttotheclassical
ones. Another consequence is the algebraic index theorem for deformation quanti-
zations. This is a statement about a trace of a compactly supported difference of
projectionsinthealgebraofmatricesoveradeformedalgebra. Itturnsoutthatall
the dataentering into this problem (namely, a deformed algebra, atrace on it, and
projections in it) can be classified using formal Poisson structures on the manifold.
ThealgebraicindextheoremimpliesthecelebratedindextheoremofAtiyah-Singer
and its various generalizations.
Operadtheory, mentionedbeforeasrelatedtoKontsevich’sformalitytheorem,
is the subject of Eduardo Hoefel’s paper. It follows from general theory that the
Fulton-MacPhersonoperadF andthelittlediscsoperadD areequivalent. Hoefel
n n
gives an elementary proof of this fact by exhibiting a rather explicit homotopy
equivalence between them.
Many important examples in noncommutative geometry appear as crossed
products. Ralf Meyer’s lecture notes deal with several crossed-products of C∗-
algebras, (e.g. crossed-products by actions of locally compact groups, twisted
crossed-products, crossed products by C∗correspondences) and discuss notions of
equivalenceoftheseconstructions. Theauthorshowshowtheseexamplesleadnat-
urallytotheconceptofastrict2-categoryandtotheunifyingapproachofafunctor
froma group tostrict 2-categories whose objectsare C∗-algebras. In addition, this
approach enables the author to define in all generality the crossed product of a
C∗-algebra by a group acting by Morita-Rieffel bimodules.
Jonathan Rosenberg’s article is concerned with two related but distinct top-
ics: noncommutative tori and Kasparov’s KK-theory. Noncommutative tori are
certain crossed products of the algebra of continuous functions on the unit circle
by an action of Z. They turn out to be noncommutative deformations of the al-
gebra of continuous functions on the 2-torus. The article reviews the classification
of noncommutative tori up to Morita equivalence, and of bundles (i.e. projec-
tive modules) over them, as well as some applications of noncommutative tori to
number theory and physics. Kasparov’s K-theory is one of the main homological
invariants in C∗-algebra theory. It can be presented in several equivalent manners;
the article reviews Kasparov’s original definition in terms of Kasparov bimodules
(or generalized elliptic pseudodifferential operators), Cuntz’s picture in terms of
quasi-homomorphisms, and Higson’s description of KK as a universal homology
theory for separable C∗-algebras. It assigns groups KK∗(A,B) to any two separa-
ble C∗-algebras A and B; usual operator K-theory and K-homology are recovered
by setting A (respectively B) equal to the complex numbers. The article also
considers the equivariant version of KK for C∗-algebras equipped with a group
action, and considers its applications to the K-theory of crossed products, includ-
ingthePimsner-VoiculescusequenceforcrossedproductswithZ(usedforexample
to compute the K-theory of noncommutative tori), Connes’ Thom isomorphism
PREFACE ix
for crossed products with R and the Baum-Connes conjecture for crossed products
with general locally compact groups.
The Baum-Connes conjecture predicts that the K-theory of the reduced C∗-
algebra C (G) of a group G is the G-equivariant K-homology KKG(EG,C) of
r ∗
the classifying space for proper actions. The conjecture is known to hold in many
cases. The point of the conjecture is that the K-homology is in principle easier to
compute than the K-theory of C (G), but concrete computations are often very
r
hard, especially if the group contains torsion. The article by Jean-Fran¸cois Lafont,
Ivonne Ortiz and Rub´en Sa´nchez-Garc´ıa is concerned with computing the groups
KKG(EG,C) ⊗Q in the case where G admits a 3-dimensional manifold model
∗
M3 for EG. In this case the authors provide an explicit formula in terms of the
combinatorics of the model M3.
As noted above, groups play a key role in noncommutative geometry; group
theoryisthereforeanimportant tool inthearea. Andrzej Zuk’s articlepresents an
introduction to the theory of groups generated by finite automata. This class con-
tainsseveralremarkablecountablegroups,whichprovidesolutionstolongstanding
questionsinthefield. Forinstance,Aleshin’sautomatagivesagroupwhichisinfin-
tely generated yet torsion; this answers affirmatively a question raised by Burnside
in 1902 of whether such groups could exist. Aleshin’s group is also an example
of a group which is not of polynomial growth yet it is of subexponential growth.
Other remarkable examples of automata groups are also presented, including a
group without uniform exponential growth, and exotic amenable groups.
In Connes’ theory of noncommutative geometry, spectral triples play the role
of noncommutative Riemannian manifolds; they are also the source of elements in
Kasparov’s KK-theory. Bram Mesland’s article deals with the construction of a
category of spectral triples that is compatible with the Kasparov product in KK-
theory. The theory described shows that by introducing a notion of smoothness on
unboundedKK-cycles,theKasparovproductofsuchcyclescanbedefineddirectly,
by an algebraic formula. This allows one to view such cycles as morphisms in a
category whose objects are spectral triples.
ThearticlebyRobertoTrincheroisalsoconcernedwithspectraltriples. Itpro-
vides a link betweenConnes’ noncommutative geometry and quantum field theory.
It displays, in detail, a toy model that aims to shed some light in the dimensional
renormalization framework of Quantum Field Theory, by comparison with the re-
sults obtained under the paradigm of such theory, where a Grasmannian algebra is
the starting point.
MaxKaroubi’sarticleconcernstwistedK-theory,atheoryofveryactivecurrent
research, which was originally defined by Karoubi and Donovan in the late 1960s.
Theapproachpresentedhereisbasedonthenotionoftwisted vector bundles. Such
bundles may be interpreted as modules over suitable algebra bundles. Roughly
speaking, twisted K-theory appears as the Grothendieck group of the category of
twisted vector bundles. Thus a geometric description of the theory is obtained.
The usual operations on vector bundles are extended to twisted vector bundles.
The article also contains a section on cup-products, where it is shown that the