Spring 2019

- Date:
**1/11/19****Alex Bearden, University of Texas at Tyler**- Title: An equivariant weak expectation property and amenable actions
- Abstract: We introduce a module version of the weak expectation property (WEP) for operator modules over completely contractive Banach algebras $A$. In the case when $A=L^1(G)$ for a locally compact group $G$, this yields a natural notion of group equivariant WEP (which has also been studied recently by Buss--Echterhoff--Willett). A notion dual to this arises by taking $A$ to be the Fourier algebra $A(G)$. These dual notions are related in the setting of $C^*$-dynamical systems, where we show that an action $G \curvearrowright X$ of an exact locally compact group is topologically amenable if and only if $C_0(X)$ has the $L^1(G)$-WEP, and if and only if the reduced crossed product $G \rtimes C_0(X)$ has the $A(G)$-WEP. Restricting to the case of discrete groups $\Gamma$, we also provide a detailed study of the $\Gamma$-WEP of $\Gamma$-$C^*$-algebras. This is joint work with Jason Crann and Mehrdad Kalantar.

- Date:
**2/8/19****Maria Gerasimova, TU Dresden**- Title: Unitarisability of discrete groups
- Abstract: Let $\Gamma$ be a discrete group. A group $\Gamma$ is called
*unitarisable*if for any Hilbert space $H$ and any uniformly bounded representation $\pi: \Gamma \to B(H)$ of $\Gamma$ on $H$ there exists an operator $S: H\to H$ such that $S^{-1}\pi(g)S$ is a unitary representation for any $g \in \Gamma$. It is well known that amenable groups are unitarisable. It has been open ever since whether amenability characterises unitarisability of groups.

**Dixmier**: Are all unitarisable groups amenable?

One of the approaches to study unitarisability is related to the space of Littlewood functions $T_1(\Gamma)$. We define the**Littlewood exponent**$\rm{Lit}(\Gamma)$ of a group $\Gamma$:

$\rm{Lit}(\Gamma)=\inf\big\{ p : T_1(\Gamma) \subseteq \ell^p(\Gamma) \big\}.$

We will show that, on the one hand, $\rm{Lit}(\Gamma)$ is related to unitarisability and amenability and, on the other hand, it is related to some geometry of $\Gamma$.

We believe that the most natural way to study groups with ${\rm Lit}(\Gamma) \leq p$ is to consider not only the actions on Hilbert spaces, but also the actions on a wider classes of spaces, for example, on $p$-spaces. We will define $p$-isometrisability of groups and discuss some results and open questions.

- Date:
**2/22/19****Brent Nelson, Vanderbilt University**- Title: Covariant Derivations
- Abstract: Let M be a von Neumann algebra equipped with an action of a locally compact group G. In this talk I will discuss how derivations on M can be lifted to derivations on the crossed product of M by G, and what properties are inherited by this new derivation. As an application, I will show how a result of Davies and Lindsay (about the closure of a derivation on a tracial von Neumann algebra) can be extended to the non-tracial case by way of the continuous core.

- Date:
**3/1/19****Ben Hayes, University of Virginia**- Title: Pivots of malleable deformations
- Abstract: I will discuss ongoing joint work with Rolando de Santiago, Daniel Hoff, and Thomas Sinclair. In it, we investigate malleable deformations in the sense of Popa. We show that (under some mixing conditions) any diffuse subalgebra on which the deformation is rigid is uniquely contained in a maximal rigid subalgebra. We establish permanence properties of these subalgebras, discuss some applications, along with applications to L^{2}-rigidity.

- Date:
**3/8/19****No Meeting, Spring Break.**

- Date:
**3/15/19****Robert McRae, Vanderbilt University**- Title: Commutants of compact groups actions on vertex operator algebras
- Abstract: A (unitary) vertex operator algebra (VOA) can be viewed as part of the Hilbert space of states for a two-dimensional conformal quantum field theory, with each vector in the VOA corresponding to a set of fields in the theory parametrized by a complex variable, called a vertex operator. Symmetries of the theory may be reflected in the action of a compact group on the VOA by unitary automorphisms, and the vertex operators commuting with the group action generate a vertex operator subalgebra called the orbifold subalgebra. In this talk, I will discuss the definition and basic examples of unitary VOAs, including the rank N Heisenberg/N free bosons VOA (which admits a unitary action of O(N, R)) and the sl_2-root lattice VOA (which admits an action of SO(3, R)). I will also discuss a recent result of mine that under sufficiently nice conditions, the orbifold subalgebra admits a tensor category of representations equivalent to the tensor category of finite-dimensional continuous representations of the compact group.

- Date:
**3/22/19****Sayan Das, University of Iowa**- Title: On the (extended) Neshveyev-Stormer rigidity phenomenon
- Abstract: In this talk I shall provide a fairly large class of examples of group actions $\Gamma \ca X$ satisfying the extended Neshveyev-Stormer rigidity phenomenon : whenever $\Lambda \ca Y$ is a free ergodic pmp action and there is a $\ast$-isomorphism $\Theta:L^\infty(X)\rtimes \Gamma \rightarrow L^\infty(Y)\rtimes \Lambda$ such that $\Theta(L(\Gamma))=L(\Lambda)$ then the actions $\Gamma \ca X$ and $\Lambda \ca Y$ are conjugate.

The proof involves a symbiosis of Popa's deformation/rigidity theory, and Jones' Subfactor theory. If time permits, I will mention some consequences towards the classification of intermediate subfactors. This talk is based on a recent joint work with Prof. Ionut Chifan.

- Date:
**3/29/19****Ionut Chifan, University of Iowa**- Title: Rigidity in group von Neumann algebra
- Abstract: In the thirties F. J. Murray and J. von Neumann found a natural way to associate a von Neumann algebra L(G) to every countable discrete group G. Classifying L(G) in terms of G emerged overtime as a natural yet quite challenging problem as these algebras tend to have very limited "memory" of the underlying group. This is perhaps best illustrated by Connes' famous result asserting that all icc amenable groups give rise to isomorphic von Neumann algebras; thus in this case, besides amenability, the algebra has no recollection of the usual group invariants like torsion, rank, or generators and relations. However, in the non-amenable regime the situation is far more complex; examples where the von Neumann algebraic structure is sensitive to various algebraic group properties have been discovered via Popa's deformation/rigidity theory. In this talk I will present new instances when the von Neumann algebra completely retains canonical algebraic constructions in group theory such as (infinite) direct product, amalgamated free product, or wreath product. In addition, I will discuss several applications to the study of rigidity in the C*-setting.

- Date:
**4/5/19****David Jekel, UCLA**- Title: Free Entropy for Convex Free Gibbs States
- Abstract: Voiculescu defined several types of entropy for non-commutative probability distributions. We present a new proof that the microstates and non-microstates free entropy agree for a free Gibbs state given by a sufficiently regular potential (similar to a result of Dabrowski from 2017). This proof proceeds by viewing both types of entropy as the large N limit of classical quantities for probability measures defining N x N random matrix models, and constructing solutions to certain PDE associated to the random matrix models in a dimension-independent way. As background, we will review the definition and motivation for each type of free entropy and some of their applications to von Neumann algebras.

- Date:
**4/12/19****Lara Ismert, University of Nebraska, Lincoln**- Title: A Covariant Stone-von Neumann Theorem
- Abstract: In this talk, we formulate and prove a version of the Stone-von Neumann Theorem for every C*-dynamical system of the form (G, K(H), a) where G is a locally compact Hausdorff abelian group and H is a Hilbert space. The novelty of our work stems from our covariant representation of the Weyl Commutation Relation on Hilbert K(H)-modules instead of just Hilbert spaces. This is joint work with Leonard Huang (University of Nevada, Reno).

- Date:
**4/19/19****Lauren Ruth, Vanderbilt University**- Title: On measure equivalence and properly proximal groups
- Abstract: In 2018, Boutonnet, Ioana, and Peterson introduced the class of properly proximal groups and studied their von Neumann algebras. The class of properly proximal groups is wide, containing all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semisimple Lie groups. We prove that proper proximality is a measure equivalence invariant. This is joint work with Ishan Ishan and Jesse Peterson.

- Date:
**5/3/19 - 5/9/19****The Seventeenth Annual NCGOA Spring Institute at Vanderbilt University**

- End of Spring Semester.