Subfactor Seminar
Spring 2015

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Organizers: Dietmar Bisch, Vaughan Jones, and Jesse Peterson

Fridays, 4:10-5:30pm in SC 1432

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• Date: 1/23/15
  • Corey Jones, Vanderbilt University
  • Title: Representations and Universal Norm for the Tube Algebra of Rigid C*-tensor categories
  • Abstract: In the finite depth case, the tube algebra of a rigid C*-tensor category is finite dimensional, and its representation category is known to be equivalent to the Drinfeld center of the category. In the infinite depth case, the tube algebra is infinite dimensional, but the category of Hilbert space representations of the tube algebra retains the structure of a braided weakly rigid monoidal category. In analogy with discrete groups, we show that one can define a universal norm on the tube algebra and that Hilbert space representations of the tube algebra are given by representations of the C* closure of the tube algebra with respect to this norm. Recently, Popa and Vaes defined a universal norm on the fusion algebra of a rigid C*-tensor category, allowing them to define various approximation properties (Amenability, Property T, Haageruup) for rigid C*-tensor categories which they show agree with previously existing definitions from the subfactor case. The fusion algebra is naturally identified as a sub-algebra of the tube algebra, and we will show that the restriction of our norm to this subalgebra agrees with the universal norm defined by Popa and Vaes. This is joint work with Shamindra Ghosh.

• Date: 1/30/15
  • Corey Jones, Vanderbilt University
  • Title: Representations and Universal Norm for the Tube Algebra of Rigid C*-tensor categories, II
  • Abstract: In the finite depth case, the tube algebra of a rigid C*-tensor category is finite dimensional, and its representation category is known to be equivalent to the Drinfeld center of the category. In the infinite depth case, the tube algebra is infinite dimensional, but the category of Hilbert space representations of the tube algebra retains the structure of a braided weakly rigid monoidal category. In analogy with discrete groups, we show that one can define a universal norm on the tube algebra and that Hilbert space representations of the tube algebra are given by representations of the C* closure of the tube algebra with respect to this norm. Recently, Popa and Vaes defined a universal norm on the fusion algebra of a rigid C*-tensor category, allowing them to define various approximation properties (Amenability, Property T, Haageruup) for rigid C*-tensor categories which they show agree with previously existing definitions from the subfactor case. The fusion algebra is naturally identified as a sub-algebra of the tube algebra, and we will show that the restriction of our norm to this subalgebra agrees with the universal norm defined by Popa and Vaes. This is joint work with Shamindra Ghosh.

• Date: 2/13/15
  • Ben Hayes, Vanderbilt University
  • Title: Polish Models and Sofic Entropy
  • Abstract: Suppose G is a countable discrete group with a pmp action on a standard probability space (X,m). A topological model for the action is an action of G on a metrizable, separable topological space X’ preserving a Borel probability measure m’ and such that pmp action of G on (X’,m’) is isomorphic to the action of G on (X,m). We call the model compact (or Polish) if X’ is compact (or Polish). In 2009, Lewis Bowen extended entropy for pmp actions of an amenable group to the much larger class of sofic groups, provided the action has a finite generating partition. David Kerr and Hanfeng Li removed the assumption of a finite generating partition shortly after, and moreover described how one can compute the entropy in terms of a compact model. It turns out that compact models always exist so this may be considered a satisfactory result. Nevertheless, there are examples where it easy to describe a Polish model and not so easy to describe a compact model, for example for Gaussian actions. Because of this, we describe how one computes the entropy in terms of a given Polish model. Polish models also turn out to be a natural way to handle generating families of functions which are unbounded. Using our Polish model formalism we are able to deduce properties of the Koopman representation of an action from positive entropy assumptions. For example, we are able to show that compact actions must have entropy at most zero. We can further show that entropy decreases under compact extensions. We will discuss the techniques which are mostly “linear” and rely on representations of C*-algebras. Time permitting, we will explain how our techniques can be used to recover a Theorem due to Voiculescu stating that if a von Neumann algebra has microstates free entropy dimension >1 with respect to some set of generators, then this algegra has no Cartan subalgebra.

• Date: 2/20/15
  • Arnaud Brothier, Vanderbilt University
  • Title: Approximation properties for subfactors

• Date: 3/6/15
  • No Meeting, Spring Break.