Spring 2017

- Date:
**1/13/17****Phillip Wesolek, Binghamton University**- Title: Chief series and chief factors in locally compact groups.
- Abstract: (Joint work with C. Reid.) The class of locally compact groups is very large and extends well beyond the familiar connected Lie groups. Moreover, a general theory for locally compact groups has emerged in recent years; this theory considers the interaction between topological structure and geometric structure. We add to this theory by showing that compactly generated locally compact groups admit a finite chief series, up to compact factors and discrete factors. Using the existence of this series, we then show that chief factors necessarily arise in all locally compact groups with sufficiently rich topological structure. Indeed, we describe explicitly the structure of locally compact groups without chief factors.

- Date:
**1/20/17****Ben Hayes, Vanderbilt University**- Title: Weak equivalence to Bernoulli shifts for some algebraic actions.
- Abstract: Let G be a countable discrete group. Given two pmp actions of G on (X,\mu),(Y,\nu). There is a way to say that the action of G on (X,\mu) is weakly contained in the action of G on (Y,\nu) due to Kechris (this is formulated in a way similar to weak containment of representations and has similarities to finite representability of Banach spaces). We say that two actions are weakly equivalent if each if weakly contained in the other. An algebraic action of G is an action by automorphisms on a compact, metrizable, abelian group X. This is a pmp action if we give X the Haar measure. I'll discuss weak equivalence for certain algebraic actions which are related to convolution operators on G. I'll show that when such operators are invertible (equivalently when they are invertible in the group von Neumann algebra), then these actions are weakly contained in Bernoulli shifts. No knowledge of weak containment or algebraic actions will be assumed.

- Date:
**2/17/17****Koichi Shimada, Kyoto University**- Title: A classification of R-actions with faitful Connes--Takesaki modules on hyperfinite factors
- Abstract: We classify certain R-actions on (type III) hyperfinite factors, up to cocycle conjugacy. More precisely, we show that an invariant called the Connes--Takesaki module completely distinguishs actions which are not approximately inner at any non-trivial point. Our classification result is related to the uniqueness of the hyperfinite type $\mathrm{III}_1$ factor, shown by Haagerup. More precisely, the uniqueness of the hyperfinite type $\mathrm{III}_1$ factor is equivalent to the uniquness of $\mathbf{R}$-actions with a certain condition on the hyperfinite type $\mathrm{II_{\infty}}$ factor. We classify actions on hyperfinite type III factors with an analogous property. The proof is based on Masuda--Tomatsu's recent work and the uniqueness of the hyperfinite type III_1 factor.

- Date:
**2/24/17****David Penneys, The Ohio State University**

- Date:
**3/3/17****Sayan Das, Vanderbilt University**- Title: Poisson Boundaries of finite von Neumann algebras (Joint with J. Peterson).
- Abstract: In my talk, I shall discuss Izumi's notion of noncommutative Poisson boundary, in the setting of finite von Neumann algebras. I shall also talk about a noncommutative generalization of Kaimanovich's "Double Ergodicity of the boundary", and provide some applications to the study bounded derivations on a finite von Neumann algebras.

- Date:
**3/10/17****No Meeting, Spring Break.**

- Date:
**3/24/17****Scott Atkinson, Vanderbilt University**- Title: Unitary dilation of freely independent contractions
- Abstract: In this talk, we will show that n contractions on a Hilbert space that are freely independent with respect to some state can be simultaneously dilated to n unitaries on a larger Hilbert space that are freely independent with respect to a natural extension of the original state. This is a free version of the well-known Sz.-Nagy-Foias dilation theorem. We will survey some basics of dilation theory including the Sz.-Nagy dilation theorem, Ando's dilation theorem, and Parrott's counterexample. If time permits, we will discuss some other contexts in which we can ask a similar dilation question. This is based on joint work with Chris Ramsey.

- Date:
**3/31/17****Ben Hayes, Vanderbilt University**- Title: Applications of 1-Bounded Entropy to von Neumann algebras
- Abstract: I will discuss the 1-bounded entropy, a certain way to measure ``how many" embeddings there are of a certain tracial von Neumann algebra into R^{\omega}. I will discuss how to use the 1-bounded entropy to attack interesting problems about the structure of subalgebras of tracial von Neumann algebras.

- Date:
**4/14/17****Pieter Naaijkens, UC Davis**- Title: Subfactors and wiretapping channels
- Abstract: Quantum information theory studies how one can use quantum mechanical systems to send (classical or quantum) information. Although there are notable exceptions, most of the theory considers the setting of finite dimensional quantum systems. In this talk we will consider infinite systems modeled by von Neumann algebras. In particular, we consider subfactors, and ask how well we can distinguish normal states on the bigger algebra, compared to when we restrict these states to the smaller algebra. I will argue how the Jones index can be used to give a quantitative answer, and how this can be interpreted as a "wiretap channel". I will also discuss two examples of physical systems where such subfactors naturally occur: rational CFT and Kitaev's quantum double models. Partly based on joint work with Leander Fiedler and Tobias Osborne.

- Date:
**4/21/17****Michael Hartglass, UC Riverside**- Title: Free transport for interpolated free group factors
- Abstract: A few years ago in a landmark paper, Guionnet and Shlyakhtenko proved the existence of free transport maps from the free group factors to von Neumann algebras generated by elements which have a joint law "close" to that of the free semicircular law. In this talk, I will discuss how to modify their idea to obtain similar results for interpolated free group factors using an operator-valued framework. This is joint work with Brent Nelson.

- Date:
**4/28/17****Brandon Seward, Courant Institute of Mathematical Sciences**- Title: Bernoullicity and Treeability
- Abstract: A classic theorem of Ornstein states that all factors of Bernoulli shifts over Z are isomorphic to Bernoulli shifts. On the other hand, a surprising result of Popa and Popa-Sasyk shows that property (T) groups admit factors of Bernoulli shifts which are not Bernoulli. Popa's factor construction can easily be applied to any countable group, though its not at all clear when the resulting factor is Bernoulli. I will discuss the Bernoullicity of Popa's factors and present some partial results. This is joint work with Damien Gaboriau.

- Date:
**5/5/17 - 5/11/17****The Fifteenth Annual NCGOA Spring Institute, at Vanderbilt University**

- End of Spring Semester.