(Sub)factor Seminar

Spring 2009

Fridays, 4:10-5:30pm in SC 1310

• 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