Bivariant K-theory of groupoids and the noncommutative geometry of limit sets [Elektronische Ressource] / vorgelegt von Bram Mesland

Bram Mesland

Sujets

• Mathematics

Informations

• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2009
• Nombre de lectures: 23
• Langue: English

Extrait

BivariantK-theory of groupoids and the
noncommutative geometry of limit sets
Dissertation
zur
Erlangung des Doktorgrades
der
Mathematisch-Naturwissenschaftlichen Fakult at
der
Rheinischen Friedrich-Wilhelms-Universit at Bonn
Vorgelegt von
Bram Mesland
aus
Amstelveen, Niederlande
Bonn 2009Diese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn unter
http :==hss:ulb:uni-bonn:de=diss online
elektronisch publiziert.Angefertigt mit Genehmigung der
Mathematisch-Naturwissenschaftlichen Fakult at der
Rheinischen Friedrich-Wilhelms-Universit at Bonn.
1.Gutachter: Prof. Dr. Matilde Marcolli
2.Gutachter: Prof.Dr. Matthias Lesch
Tag der Promotion: Freitag, 17.Juli 2009
iiiSummary
We present a categorical setting for noncommutative geometry in the sense of Connes.
This is done by introducing a notion of morphism for spectral triples. Spectral triples are
the unbounded cycles for K-homology ([11]), and their bivariant generalization are the
cycles for Kasparov’sKK-theory ([32]). The central feature ofKK-theory is the Kasparov
product
KK (A;B)
KK (B;C)!KK (A;C):i j i+j

Here A;B and C are C -algebras, and the product allows one to view KK as a category.
The unbounded picture of this theory was introduced by Baaj and Julg ([4]). In this
picture the external product
0 0 0 0
KK (A;B)
KK (A;B )!KK (A
B;A
B );i j i+j
is given by an algebraic formula, as opposed to Kasparov’s original approach, which is
more analytic in nature, and highly technical.
In order to describe the internal Kasparov product of unbounded KK-cycles, we in-
troduce a notion of connection for unbounded cycles (E;D). This is a universal connection
1~r : E! E

(B);B
in the sense of Cuntz and Quillen ([20]), such that [r;D] extends to a completely bounded
operator. The topological tensor product used here is the Haagerup tensor product for

operator spaces. Blecher ([7]) showed this tensor product coincides with the C -module

tensor product, in case both operator spaces areC -modules. His work plays a crucial role
in our construction. The product of two cycles with connection is given by an algebraic
formula and the product of connections can also be de ned. Thus, cycles with connection
form a category, and the bounded transform
12
2(E;D;r)7! (E;D(1 +D ) );
de nes a functor from this category to the category KK.
We also describe a general construction for obtaining KK-cycles from real-valued
groupoid cocycles. IfG is a locally compact Hausdor groupoid with Haar system and
c :G!R a continuous closed cocycle, we show that pointwise multiplication by c in the
convolution algebra C (G), extends to an unbounded regular operator on the completionc

ofC (G) as a C -module over C (H), whereH is the kernel of c. It gives a KK-cycle forc

(C (G);C (H)). In case the groupoidH is unimodular with respect to a quasi-invariant

measure, or more general, if C (H) carries a trace, this KK-cycle gives rise to an index

map K (C (G))!C.1
This result is general enough to be applied in a wide variety of examples. We use it
to obtain the noncommutative torus as a smooth quotient (in the above categorical sense)
of the irrational rotation action on the circle. In the last chapter we sketch the promising
range of applications the above categorical setting and cocycle construction may have in
the noncommutative geometry of limit sets.Acknowledgements
My appreciation goes out to Matilde Marcolli, for guiding me through this
project, suggesting the subject to me, and stimulating me to develop and work
through the technicalities that were involved. On a human level, this thesis would
not have come into existence without her kindness and understanding of the fact
that time is needed to overcome di culties in personal life.
I thank Alain Connes and Nigel Higson for valuable email correspondence dur-
ing crucial phases of the project. Georges Skandalis, Jean Renault and Klaas Lands-
man I thank for guiding me through some of the literature on groupoids.
My Ph.D. studies were supported by an IMPRS-grant from the Max Planck
Gesellschaft, for which I am grateful. The Max Planck Institut fur Mathematik is
a very inspiring environment for doing mathematics, and I have been lucky to have
been able to spend such a long time here. I thank the sta of the MPI for their
kindness and for providing these excellent conditions.
Important parts of the work in this thesis were done during my visits to Florida
State University and the California Institute of Technology. I thank them for their
hospitality and support.
I thank Branimir Cacic, Ioanna Dimitriou, Ivan Dynov, Eugenia Ellis, Tobias
Fritz, Nikolay Ivankov, Javier Lopez, Elke Markert, Nicola Mazzari, Rafael Torres
and Dapeng Zhang for the many inspiring and useful conversations we had, and for
providing some relaxation and fun now and then.
I thank all my friends for the good times we had, for being there, and for keep-
ing their faith in me along the way.
Bob, Annemieke, Lot en Joost, zonder jullie was ik hier nooit gekomen.
viiContents
iii
Summary v
Acknowledgements vii
Introduction 1
Chapter 1. Unbounded bivariant K-theory 5
1. C -modules 5
2. KK-theory 12
3. Operator modules 17
4. Smoothness 22
5. Universal connections 28
6. Correspondences 37
Chapter 2. Groupoids 45
1. A category of groupoids 45
2. C -algebras and -modules 51
3. Cocycles and K-theory 55
4. Crossed products 62
5. Semidirect products 65
Chapter 3. Limit sets 71
1. Limit sets and Patterson-Sullivan measures 71
2. Hyperbolic manifolds 75
3. Fuchsian groups 81
Bibliography 85
ix