Noncommutative Geometry & Operator Algebras Seminar
Fall 2004

Organizer: Dietmar Bisch

Tuesdays, 4:00pm-5:00pm 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 K-theory. 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
  • no meeting

• Date: 9/21/04
  • Speaker: Guoliang Yu, Vanderbilt University
  • Title: Group actions on Banach spaces and operator K-theory
  • Abstract: In this talk, I will discuss groups actions on Banach spaces and its applications to K-theory 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 non-amenable 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 non-isomorphic 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 well-known that the Temperley-Lieb 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 Fuss-Catalan 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.

• 10/16-10/17/04 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, 4:00pm-5:00pm in SC 1432
  • Speaker: Jesse Peterson, UCLA
  • Title: A 1-cohomology characterization of property (T) in von Neumann algebras
  • Abstract: We obtain a characterization of property (T) for von Neumann algebras in terms of 1-cohomology, similar to the Delorme-Guichardet Theorem for groups.

• Date: 10/20/04 (Wednesday) Special Talk in the Topology & Group Theory Seminar, 4:10-5:00pm in SC 1424
  • Speaker: Sorin Popa, UCLA
  • Title: Some calculations of 1-cohomology 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 measure-preserving 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 1-cohomology 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 algebraic-combinatorial 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 non-isomorphic subfactors.

• Date: 10/28/04 (Thursday) Mathematics Colloquium, 4:10pm-5:00pm 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 quasi-isometries 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 KK-theory. When the group under consideration is profinite (that is, compact totally disconnected) they then showed that this theory is naturally isomorphic to complexified equivariant KK-theory, 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 one-sided operator space multipliers. Then I will present an extremely useful ``geometric'' characterization of these multipliers, obtained jointly with Blecher and Effros. Applications to non-self-adjoint operator algebras, dual operator spaces, and one-sided M-ideals will be explained.

• Date: 11/19/04 (Friday) Special Seminar Talk, 3:00pm-4:00pm 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 self-adjoint 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 non-commutative geometry and topology. I will present recent results on coarse embeddings into Hilbert and, more generally, L_p-spaces 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, 4:10pm-5:00pm 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 non-smooth 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 non-smooth 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.

Past NCGOA seminars