Spring 2009

- Date:
**1/9/09****Cyril Houdayer, UC Los Angeles**- Title:
**An example of strongly solid group von Neumann algebra** - Abstract: I will give an example of non-amenable ICC group \Gamma for which I'll show that the group von Neumann algebra L(\Gamma) is strongly solid, i.e. for any diffuse amenable subalgebra P \subset L(\Gamma), the normalizer of P generates an amenable von Neumann subalgebra of L(\Gamma). Moreover, L(\Gamma) is not isomorphic to a free group factor.

- Date:
**1/16/09****Emily Peters, UC Berkeley**- Title:
**Generators and relations for subfactor planar algebras** - Abstract: A generators-and-relations presentation for a planar algebra can be a nice, short way to summarize the large amount of data of a planar algebra. We'll talk about what sort of generators and relations one can expect for subfactor planar algebras, and discuss some one- and two-generator examples, such as D_2n, E_6, Haagerup, and D_2n^(1).

- Date:
**Wednesday, 1/21/09, 4:10-5:30pm****Adrian Ioana, Clay Research Fellow**- Title:
**Relative property (T) for the subequivalence relations associated with the action of SL(2,Z) on T^2.** - Abstract: Let S be the equivalence relation induced by the action of SL(2,Z) on T^2. Then any ergodic subequivalence relation R of S is either hyperfinite or rigid (or has relative property (T)), in the sense of Popa.

- Date:
**1/30/09****Stuart White, University of Glasgow**- Title:
**Perturbations of nuclear C*-algebras.**. - Abstract: In the early 70's Kadison and Kastler equipped the set of all operator algebras on B(H) with a metric by comparing the distance between the unit balls of two operator algebras in the Hausdorff metric. They conjectured that sufficiently close operator algebras must be unitarily conjugate. This was the subject of much research in the 70's and 80's and their conjecture was verified by Christensen when both algebras are injective von Neumann algebras and subequently when one algebra is an injective von Neumann algebra, using an extra ingredient of Raeburn and Taylor. In this case the distance from the unitarily implementing an isomorphism to the identity can be controlled by the distance between the two algebras involved. In this talk we will explain recent progress on this conjecture for separable nuclear C*-algebras and show that sufficiently close separable nuclear C*-algebras must be isomorphic. We'll also discuss one-sided versions of these concepts and how they give rise to characterisations of direct limits. This is joint work with Erik Christensen, Allan Sinclair, Roger Smith and Wilhelm Winter.

- Date:
**2/6/09****Jesse Peterson, Vanderbilt University**- Title:
**Derivations on group-measure space constructions** - Abstract: In this talk we will investigate the structure of a class of closable derivations on von Neumann algebras coming from group-measure space constructions. We will then show how to apply these results to obtain new examples of von Neumann algebras which do not arise as group-measure space constructions, for example the von Neumann algebra L( SL(3, Z) * G ) where G is any non-trivial group.

- Date:
**2/13/09****Paramita Das, Vanderbilt University**- Title:
**Diagonal planar algebra** - Abstract: To every group with a finite set of generators and a scalar 3-cocycle, we associate a planar algebra (which we call diagonal planar algebra) in a recent work (arXiv:0811.1084); we also proved that the planar algebra associated to a diagonal subfactor is a diagonal planar algebra. In this talk I will prove the converse, namely, every subfactor whose planar algebra is isomorphic to a diagonal planar algebra, must necessarily be a diagonal subfactor. This is joint work with Dietmar Bisch and Shamindra Ghosh.

- Date:
**2/20/09****Mrinal Raghupathi, Vanderbilt University**- Title:
**Nevanlinna-Pick interpolation and Fuchsian groups** - Abstract:
In this talk I will describe the Nevanlinna-Pick problem from
classical function theory.

Let $R$ be a region in $\mathbb{C}^d$, let $z_1,\ldots,z_n$ be $n$ distinct points in $R$, and let $w_1,\ldots,w_n$ be $n$ complex numbers in the unit disk $\mathbb{D}$. A typical Nevanlinna-Pick interpolation problem is concerned with finding necessary and sufficient conditions for the existence of a holomorphic map $f:R\to \mathbb{D}$ such that $f(z_j)=w_j$ for $j=1,\ldots,n$.

We will look at the historical development of this problem. Then we will talk about some recent work to generalize this result to the case where $R$ is a finite Riemann surface. In this case the problem involves studying the algebra of holomorphic functions that are fixed by the action of a Fuchsian group.

- Date:
**2/27/09****Nate Brown, Penn State University**- Title:
**Classifying Hilbert modules** - Abstract: I'll explain how the Cuntz semigroup serves as a replacement for K_0 when one attempts to classify all countably generated Hilbert module over a C*-algebra.

- Date:
**3/6/09****no meeting, Spring break.**

- Date:
**3/13/09****Thomas Sinclair, Vanderbilt University**- Title:
**Cocycle superrigidity for Gaussian actions.** - Abstract: I will talk about how Popa's cocycle superrigidity theorem for Bernoulli actions of Propoerty(T) groups and product groups can be extended to so-called s-L^2-rigid groups. I will also talk about a method for constructing Gaussian actions of groups which are not cocycle superrigid. These results were obtained in collaboration with Jesse Peterson.

- Date:
**3/15/09 - 3/18/09** - Date:
**3/20/09****Jon Bannon, Siena College**- Title:
**On the closability of certain L^2 derivations.** - Abstract: In this talk we consider the following question: Given a densely defined derivation d from a type II_1 factor $M$ into its coarse correspondence $L^2(M)\otimes L^2(M^op)$, and p a projection in $M^{op}\otimesM$, is the cut-down derivation $pd$ closable? We present an example that is evidence against this always happening.

- Date:
**3/27/09****Weihua Li, University of New Hampshire**- Title:
**Topological free entropy dimension of approximately divisible C*-algebras.** - Abstract: The class of approximately divisible C*-algebras was first introduced and studied by B. Blackadar, A. Kumjian and M. Rordam. D. Voiculescu introduced the notion of topological free entropy dimension of elements in a unital C*-algebra as an analogue of free entropy dimension in the context of C*-algebra. Let A be a separable, unital, approximately divisible C*-algebra. We show that A is generated by two self-adjoint elements, and the topological free entropy dimension of any finite generating set of A is less than or equal to 1. In addition, we show that the similarity degree of A is at most 5. Thus an approximately divisible C*-algebra has an affirmative answer to Kadison's similarity problem.

- Date:
**4/3/09****Alan Wiggins, Vanderbilt University**- Title:
**The CB Isomorphic Classification of Hyperfinite Type III Preduals.** - Abstract: The isomorphism class (either complete or Banach) of the predual of a von Neumann algebra carries some information about the isomorphism class (as a W$^*$-algebra) of its dual. I will discuss how much information can be extracted. Namely, the complete embeddability of the trace-class operators or Pisier's operator Hilbert space OH into the predual can aid in distinguishing whether a given von Neumann algebra is finite, semifinite non-finite, or purely infinite. However, work of Haagerup, Rosenthal, and Sukochev shows that all hyperfinite factors of type III$_{\lambda}$ where $0<\lambda\leq 1$ have completely isomorphic preduals. Finally, a recent result of Haagerup and Musat shows that this cannot extend to $\lambda=0$, even if we only focus on the ITPF1 case. If time permits, I may also discuss Connes' bicentralizer problem for separable type III$_1$ factors.

- Date:
**4/10/09****Richard Burstein, University of Ottawa**- Title:
**Automorphisms of bipartite graph planar algebras.** - Abstract:
A planar algebra is a graded vector space along with a certain graphical
calculus, namely an associative action of the planar operad. These
algebras were first developed by Jones for use in the classification of
$II_1$ subfactors, but they have since been used in other areas such as
category theory. The standard invariant of every (finite-index, extremal)
$II_1$ subfactor may be described as a planar algebra; conversely, every
planar algebra obeying certain additional conditions ("of subfactor type")
is in fact the standard invariant of a subfactor.

Planar algebras may also be obtained from bipartite graphs. These bipartite graph planar algebras are rarely of subfactor type, but they may have subfactor-type planar subalgebras. In fact every subfactor planar algebra may be embedded in the bipartite graph planar algebra on the subfactor's principal graph.

In general, it is difficult to show that a graded subspace of a bipartite graph planar algebra $P$ is closed under the planar operad. However, if we consider a group $G$ of automorphisms of $P$ (i.e., invertible graded linear maps on $P$ which commute with the planar operad), then the set of fixed points $P^G$ is closed under the operad action.

I will describe the automorphism group of an arbitary bipartite graph planar algebra, and give conditions for the fixed points $P^G$ to be of subfactor type. I will describe several examples of this construction, including some new infinite-depth subfactors.

- Date:
**4/17/09****Jan Cameron, Texas A&M University**- Title:
**Normalizers of subfactors.** - Abstract:
The normalizer
N_M(B) of a subalgebra B of a type II_1 factor M has been studied
in various contexts. In this work, we study the relationship between the structure of
the group of normalizing unitaries and the von Neumann algebra it generates. We
show that N_M(B) imposes a discrete crossed product structure on the generated
von Neumann algebra. By analyzing the structure of weakly closed bimodules in
N_M(B)'', this leads to a "Galois-type" theorem for normalizers, in which we find a
description of the subalgebras of N_M(B)'' in terms of a unique countable subgroup
of N_M(B). Implications for inclusions B \subset M arising from the group von Neumann
algebra and tensor product constructions will also be addressed. Our work also
yields new examples of norming subalgebras in finite von Neumann algebras: If
B \subset M is a regular inclusion of II_1 factors, then B norms M.

pdf version

**End of Academic Year**