(Sub)factor Seminar
Fall 2008 - Spring 2009

• Date: 8/25/08
  • Organizational meeting.

• Date: 9/5/08
  • no meeting, Wabash Modern Analysis Seminar

• Date: 9/12/08
  • Romain Tessera, Vanderbilt University
  • Title: A characterization of relative property T (joint work with Yves de Cornulier)
  • Abstract: We prove that a semidirect product of a group G with an abelian group V does not have relative property T with respect to V if and only if there exists a mean on the dual of V, which is
    -G-invariant,
    -supported on the trivial representation,
    -distinct from the dirac measure.

• Date: 9/19/08
  • Pinhas Grossman, Vanderbilt University
  • Title: Strong Singularity for Subfactors
  • Abstract: We will describe an example of a subfactor of the hyperfinite II₁ factor which is singular but not strongly singular (with constant one). This is joint work with Alan Wiggins.

• Date: 9/26/08
  • Thomas Sinclair, Vanderbilt University
  • Title: Superrigidity of Bernoulli Actions of Some Product Groups
  • Abstract: Following Sorin Popa ("On the Superrigidity of Malleable Actions with Spectral Gap," available on arXiv) I will demonstrate that the Bernoulli action of G = H x K is cocycle superrigid, where H is a nonamenable group and K is an arbitrary infinite group. As a consequence, any group L which has an ergodic action orbit equivalent to the Bernoulli action of G is isomorphic to G and the actions are conjugate.

• Date: 10/3/08
  • Shamindra Ghosh, Vanderbilt University
  • Title: From planar algebras to subfactors
  • Abstract: Starting from a planar algebra satisfying suitable conditions, we will describe the construction (by Jones, Shlyakhtenko and Walker) of a subfactor whose planar algebra is the one with which we started. The first construction was given by Popa, and then by Guionnet, Shlykhtenko and Jones.

• Date: 10/10/08
  • Jesse Peterson, Vanderbilt University
  • Title: Derivations and quantum Dirichlet forms on von Neumann algebras, an introduction.
  • Abstract: We will give in introductory talk to the theory of derivations and quantum Dirichlet forms on von Neumann algebras. Emphasis will be placed on the association between closable real derivations, closable quantum Dirichlet forms, and semigroups of completely positive maps, à la Jean-Luc Sauvageot. We will then describe how this association can be applied to various situations such as those involving group and group-measure space von Neumann algebras.

• Date: 10/17/08
  • Junhao Shen, University of New Hampshire
  • Title: Topological Free Entropy Dimension for Blackadar and Kirchberg's MF algebras.
  • Abstract: We will start the talk with introduction to MF C*-algebras in the sense of Blackadar and Kirchberg. Then we will indicate the connection between MF C*-algebras and Brown-Douglas-Fillmore's extension semigroup. Basing on the work by Haagerup and Thorbjornsen on the reduced group C*-algebras of free groups, we will give several new examples of Blackadar and Kirchberg's MF algebras, followed by several new examples of C*-algebras whose BDF-extension semigroup is not group.
    
    In the second half of the lecture, we will introduce Voiculescu's topological free entropy theory for unital C*-algebras. We will see the reason why the definition of Voiculescu's free entropy is based on the properties of MF C*-algebras. Then we will present some calculation of topological free entropy dimension for several important classes of MF C*-algebras.

• Date: 10/24/08
  • Junsheng Fang, Texas A&M University
  • Title: The radial (Laplacian) masa in a free group factor is maximal injective
  • Abstract: The radial (or Laplacian) masa in a free group factor is the abelian von Neumann algebra generated by the sum of the generators (of the free group) and their inverses. We prove that the Laplacian masa has an asymptotic orthogonality property and therefore is maximal injective in the free group factor. Combining with Popa's intertwining technique and our recent results of groupoid normalizers of tensor product von Neumann algebras, we are able to prove that the tensor product of a type I maximal injective von Neumann subalgebra which has the asymptotic orthogonality property with an arbitrary type I maximal injective von Neumann sublagebra is maximal injective. This is joint work with Jan Cameron, Mohan Ravichandran and Stuart White.

• Date: 10/31/08
  • no meeting, Von Neumann Algebras and Ergodic Theory of Group Actions

• Date: 11/7/08
  • Shamindra Ghosh, Vanderbilt University
  • Title: From planar algebras to subfactors, Part II
  • Abstract: Starting from a planar algebra satisfying suitable conditions, we will describe the construction (by Jones, Shlyakhtenko and Walker) of a subfactor whose planar algebra is the one with which we started. The first construction was given by Popa, and then by Guionnet, Shlykhtenko and Jones.

• Date: 11/14/08
  • Alan Wiggins, Vanderbilt University
  • Title: Groupoid Normalizers of Tensor Products
  • Abstract: Given a unital subalgebra B of a II_1 factor M , define the groupoid normalizers G_N(B) of B in M to be all partial isometries v \in M with vBv* , v*Bv \subset B. We show that when B_i' \cap M_i = Z (B_i), i = 1, 2, then G_N(B_1)'' \otimes G_N(B_2)'' = G_N(B_1 \otimes B_2)'' . This is joint work with Roger Smith, Stuart White, and Junsheng Fang.
    pdf version

• Date: 11/21/08
  • Ionut Chifan, UC Los Angeles
  • Title: Deformation/spectral gap rigidity principle for von Neumann algebras and some applications to ergodic theory
  • Abstract: In this talk I will discuss Popa's deformation/spectral gap rigidity technique for von Neumann algebras and I will present some new applications to solidity and to ergodic theory. For instance, I will prove the folowing result: Suppose that G \curvearrowright [0,1]^G is the Bernoulli action of a countable infinite group G and denote by R_{G \curvearrowright [0,1]^G} the induced equivalence relation. Then for every subequivalence relation S \subset R_{G \curvearrowright [0,1]^G} there exists a measurable partition {X_i}, i \geq 0 of [0,1]^G formed of R-invariant sets such that R_{|X_0} is hyperfinite and R_{|X_i} is strongly ergodic (hence non-hyperfinite and ergodic) for every i \geq 1. This talk is based on two papers I have written jointly with A. Ioana respectively C. Houdayer.
    pdf version

• Date: 11/28/08
  • no meeting, Thanksgiving break

• Date: 12/5/08
  • No meeting.

  Winter Break.

• 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
  • von Neumann Algebras and Ergodic Theory, at UCLA.

• 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

Past NCGOA and Subfactor seminars