Noncommutative Geometry & Operator Algebras Seminar
Fall 2004
Organizer: Dietmar Bisch
Tuesdays, 4:00pm5:00pm in SC 1432
 Date: 8/31/04
 Speaker: Chris Phillips, University of Oregon (visiting Vanderbilt)
 Title: The classification of noncommutative tori
 Abstract:
The irrational rotation algebras, which can be thought of as
noncommutative versions of the ordinary two dimensional torus, have been
one of the central examples in the theory of C*algebras, particularly
the parts of the theory motivated by thinking of it as "noncommutative
topology". Smooth versions of them have played a similar role in Connes'
noncommutative differential geometry.
The simple higher dimensional noncommutative tori are one of the
most natural generalizations. Their structure and properties have been
rather more difficult to determine than in the two dimensional case. For
example, it has been known since about 1980 that two irrational rotation
algebras are isomorphic if and only if they have the same ordered
Ktheory. For the higher dimensional case, this result could be
completely proved only very recently. Furthermore, unlike for the two
dimensional case, isomorphism of the C*algebras does not imply
isomorphism of the smooth algebras.
In these two talks, I will begin by describing the irrational
rotation algebras and their higher dimensional analogs, as well as the
smooth versions. Then I will explain what is known about the isomorphism
classification in both the smooth and C* cases, and I will explain the
connection of the C* classification with the Elliott classification
program. Finally, I will describe some of the steps in the proof of the
isomorphism classification in the C*algebra case. Key roles are played
by Lin's classification result for simple C*algebras with tracial rank
zero, and by versions of the Rokhlin property for group actions on
simple C*algebras.
 Date: 9/7/04
 Speaker: Chris Phillips, University of Oregon (visiting Vanderbilt)
 Title: The classification of noncommutative tori, continued
 Date: 9/14/04
 Date: 9/21/04
 Speaker: Guoliang Yu, Vanderbilt University
 Title: Group actions on Banach spaces and operator Ktheory
 Abstract:
In this talk, I will discuss groups actions on Banach spaces
and its applications to Ktheory of group C*algebras (joint work with
Gennadi Kasparov).
 Date: 9/28/04
 Speaker: Remus Nicoara, Vanderbilt University
 Title: A rigidity result for irrational rotation HT factors
 Abstract:
We present a result on the irrational rotation HT factors
M_\alpha (\Gamma)=L_\alpha Z^2) \rtimes \Gamma, where \Gamma
are arbitrary nonamenable subgroups of SL(2,Z) and
\alpha=e^{2\pi it}, t irrational, showing that for each fixed
\Gamma there exists no separable II_1 factor that contains M_\alpha
(\Gamma) for uncountably many \alpha's. In particular,
\{ M_\alpha (\Gamma)\}_\alpha are nonisomorphic modulo countable sets.
(Joint work with Sorin Popa, UCLA and Roman Sasyk, Purdue
University).
 Date: 10/5/04
 Speaker: Chris Phillips, University of Oregon (visiting Vanderbilt)
 Title: Classifiability of crossed products by minimal homeomorphisms
 Abstract:
This is joint work with Huaxin Lin.
Let $X$ be an infinite compact metric space with finite covering
dimension and let $h \colon X \to X$ be a minimal homeomorphism.
We show that the associated crossed product C*algebra
$A = C^* ({\mathbb{Z}}, X, h)$ has tracial rank zero whenever
the image of $K_0 (A)$ in ${\mathrm{Aff}} (T (A))$ is dense.
No assumption about either smoothness or the number of extreme
tracial states is necessary.
As a consequence, these crossed product C*algebras are
classifiable, and are isomorphic to
simple AH~algebras with real rank zero.
 Date: 10/12/04
 Speaker: Dietmar Bisch, Vanderbilt University
 Title: Intermediate subfactors
 Abstract: It is wellknown that the TemperleyLieb algebras
are the basic algebras of symmetries associated to a subfactor.
If an intermediate subfactor is present Jones and I showed a few
years ago that another tower of algebras  what we called the
FussCatalan algebras  capture the additional symmetry given
by the intermediate subfactor. What happens if two or more intermediate
subfactors occur? The story becomes a lot more complicated and
fascinating. We will discuss some recent work on this question.

10/1610/17/04 Southeastern Section AMS Meeting at Vanderbilt University
Schedule of talks, Special Session "Index theory and the topology of
manifolds"
Schedule of talks, Special Session "Operator theory on function spaces"
Schedule of talks, Special Session "Von Neumann algebras and noncommutative
ergodic theory"
Click here for the meeting program
 Date:
10/19/04 (Tuesday) Special Talk during the Fall Break,
4:00pm5:00pm in SC 1432
 Speaker: Jesse Peterson, UCLA
 Title: A 1cohomology characterization of property (T) in von Neumann
algebras
 Abstract:
We obtain a characterization of property (T) for von Neumann
algebras in terms of 1cohomology, similar to the DelormeGuichardet
Theorem for groups.
 Date:
10/20/04 (Wednesday) Special Talk in the Topology & Group Theory Seminar, 4:105:00pm in SC 1424
 Speaker: Sorin Popa, UCLA
 Title: Some calculations of 1cohomology for actions of groups
 Abstract:
For each group $G$ having an infinite normal subgroup with relative
property (T) and each countable abelian group $\Gamma$ we
construct measurepreserving actions $\sigma_\Gamma$ of $G$
on the probability space such that the 1'st cohomology group
of $\sigma_\Gamma$, $H^1(\sigma_\Gamma,G)$, is
equal to Char$(G) \times \Gamma$. We deduce that $G$ has uncountably
many non stably orbit equivalent actions. We also calculate 1cohomology
groups and show existence of ``many'' non stably orbit equivalent
actions for free products of groups as above.
 Date: 10/26/04
 Speaker: Remus Nicoara, Vanderbilt University
 Title: Some finiteness results for commuting squares
 Abstract:
Commuting squares are squares of inclusions of matrix algebras,
which arise naturally as invariants in Jones' theory of subfactors.
Commuting squares can also be used as construction data for subfactors, and
many explicit examples of subfactors have been obtained this way. By
using a combination of algebraiccombinatorial and analytic methods, we
prove some finiteness results for commuting squares and find a good notion
of primeness (in the sense of isolation) for these objects. As a
consequence, elementary proofs are obtained for theorems such as D. Stefan's
theorem on the finiteness of the number of Hopf structures on a finite
dimensional matrix algebra. The isolation results obtained also suggest
methods of constructing new families of nonisomorphic subfactors.
 Date:
10/28/04 (Thursday) Mathematics Colloquium,
4:10pm5:00pm in SC 1206
 Speaker: Chris Phillips, University of Oregon (visiting Vanderbilt)
 Title: Rational cohomology for Banach algebras
 Abstract:
Let A be a commutative unital Banach algebra. Its maximal ideal space
Max (A) is a compact Hausdorff space which plays a role somewhat similar
to that of the space Spec (R) in algebraic geometry. In particular, A can
be represented, not necessarily faithfully, as an algebra of continuous
functions on Max (A). The Taylor problem asks for a construction of the
(Cech) cohomology of Max (A) "directly from A". For example, H^1 (Max (A); Z)
is the quotient of the invertible group of A by the image of the exponential
map. This, and related descriptions of H^0 (Max (A); Z) and H^2 (Max (A); Z),
have been known since the 1970s, but the program seemed to stop there.
In this talk, after giving some background, we describe a solution for
the _rational_ (Cech) cohomology H^s (Max (A); Q) for arbitrary s, in terms
of the rational homotopy groups of the spaces of last columns of invertible
n by n matrices over A for suitable n. The approach has the promise of
giving something interesting for noncommutative Banach algebras as well.
This is joint work with Greg Lupton, Claude Schochet, and Samuel Smith.
 Date: 11/2/04
 Speaker: Alex Furman, University of Illinois at Chicago
 Title: Ergodic equivalence relations, measure equivalence and higher
rank lattices
 Abstract:
We relate the notion of Orbit Equivalence between group actions in
ergodic theory,
the notion of Measure Equivalence  a measure theoretical analogue of
quasiisometries between groups, and discuss aspects of rigidity for
actions of higher rank lattices.
 Date: 11/9/04
 Speaker: Jeff Raven, Penn State University
 Title: An equivariant bivariant Chern character
 Abstract:
Using notions from homological algebra and sheaf theory Baum and Schneider
defined a bivariant equivariant cohomology theory for spaces which shares many
of the properties of equivariant KKtheory. When the group under consideration
is profinite (that is, compact totally disconnected) they then showed that
this theory is naturally isomorphic to complexified equivariant KKtheory,
and conjectured that the same should be true for all totally disconnected
groups. In my talk I'll survey some of the background material and
discuss how to construct an isomorphism when the group is discrete.
 Date: 11/16/04
 Speaker: Vrej Zarikian, University of Cincinnati
 Title: Operator spaces, multipliers, and ideals
 Abstract:
After reviewing the main ideas of operator space
theory, I will discuss Blecher's theory of
onesided operator space multipliers. Then I will
present an extremely useful ``geometric'' characterization
of these multipliers, obtained jointly with
Blecher and Effros. Applications to nonselfadjoint
operator algebras, dual operator spaces, and onesided
Mideals will be explained.
 Date:
11/19/04 (Friday) Special Seminar Talk,
3:00pm4:00pm in SC 1308
 Speaker: Jon Bannon, University of New Hampshire
 Title: Transitive families of projections in factors of
type II_1
 Abstract:
In this talk we will explore a bit of operator theory relative to
a type II_1 factor. We introduce a notion of transitive family of subspaces
relative to a type $II_1$ factor, and hence a notion of transitive family of
projections in such a factor. We show that whenever M is a factor of type
$II_1$ and $M$ is generated by two selfadjoint elements, then
$M\otimesM_2(\mathbb{C})$ contains a transitive family of 5 projections.
Finally, we exhibit a free transitive family of 12 projections that generate
a type $II_1$ factor.
 Date: 11/23/04
 no meeting, Thanksgiving break
 Date: 12/7/04
 Speaker: Piotr Nowak, Vanderbilt University
 Title: Coarse embeddings and uniform homeomorphisms in the geometry of
Banach spaces
 Abstract:
Coarse embeddings were defined by Gromov in the study of the
large scale geometry of groups and turned out to have important
applications to some major conjectures in noncommutative geometry
and topology. I will present recent results on coarse embeddings
into Hilbert and, more generally, L_pspaces and explain the close
connection with uniformly homoemorphic embeddings studied in the 1980's by
Enflo, Aharoni, Lindenstrauss and others in the context of Banach spaces.
 Date:
12/9/04 (Thursday) Mathematics Colloquium,
4:10pm5:00pm in SC 1206
 Speaker: Victor Nistor, Penn State University
 Title: Analysis on polyhedral domains
 Abstract:
The analysis of elliptic partial differential operators on smooth, bounded domains is well understood and has numerous applications, many outside mathematics. By contrast, the behavior of these operators on nonsmooth domains can be quite different from the one on smooth domains and is much less understood. In my talk, I will first review some known results on boundary value problems on nonsmooth domains. Then I will present an approach to analysis on polyhedral domains that is based on a modification of the usual Sobolev spaces, yielding the so called "Sobolev spaces with weights." One can, for instance, obtain a regularity theorem within these Sobolev spaces with weights that is completely analogous to the usual elliptic regularity on smooth domains. This result, joint work with C. Bacuta and L. Zikatanov, has potential applications to numerical methods. I will also briefly discuss at the end some connections with Operator Algebras, more precisely, with groupoid C^*algebras. The talks is meant to be accessible to a mathematical literate audience, including graduate students.
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