Fall 2021

- Date:
**9/3/21****Srivatsav Kunnawalkam Elayavalli, Vanderbilt University**- Title: Strong 1-boundedness for Property T von Neumann algebras
- Abstract: Ben Hayes, David Jekel and myself showed recently that all von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. I will present this result and its proof.

- Date:
**9/10/21****Ishan Ishan, Vanderbilt University**- Title: Von Neumann equivalence and weak forms of amenability
- Abstract: The notion of von Neumann equivalence, which generalizes both measure equivalence and W*-equivalence, was introduced recently by Jesse Peterson, Lauren Ruth, and myself. We showed that many analytic properties, such as amenability, property (T), the Haagerup property, and proper proximality are preserved under von Neumann equivalence. In this talk, I will present my new work expanding the list of properties stable under von Neumann equivalence. In particular, we will discuss the stability of weak amenability, weak Haagerup property, and the approximation property (of Haagerup and Kraus) under von Neumann equivalence.

- Date:
**10/1/21****Michael Montgomery, Vanderbilt University**- Title: Spin model subfactors
- Abstract: Complex Hadamard matrices generate a class of irreducible hyperfinite subfactors with integer Jones index coming from spin model commuting squares. I will prove a theorem that establishes a criterion implying that these subfactors have infinite depth. I then show that Paley type II and Petrescu's continuous family of Hadamard matrices yield infinite depth subfactors. Furthermore, infinite depth subfactors are a generic feature of continuous families of complex Hadamard matrices. In this talk I will present an outline for the proof of these results.

- Date:
**10/8/21****Hrvoje Stojanovic, Vanderbilt University**(11:30am -12:50pm central)- Title: New examples of irreducible hyperfinite subfactors with rational, non-integer Jones index
- Abstract: In his thesis, Schou showed that certain infinite-dimensional commuting squares could be used to construct irreducible hyperfinite subfactors. Bisch used this approach to construct a subfactor with index 4.5 that was the first example of an irreducible hyperfinite subfactor with rational, non-integer index. In this talk I will present new examples of irreducible hyperfinite subfactors with rational, non-integer indices. These examples are constructed from infinite-dimensional commuting squares, and the smallest index among them is 5+1/3.

- Date:
**10/15/21****No meeting, Fall break**

- Date:
**10/22/21****Dan-Virgil Voiculescu, UC Berkeley**- Title: Miscellaneous about Commutants mod
- Abstract: I will discuss some general aspects of commutants modulo normed ideals. This will include iteration of the construction, the commutant mod associated with a smooth manifold and an analogy with capacity in nonlinear potential theory.

- Date:
**10/29/21**(11:30am -12:50pm central)**Mikael Rørdam, University of Copenhagen**- Title: Irreducible inclusions of simple C*-algebras
- Abstract: There are several naturally occurring interesting examples of inclusions of simple C*-algebras with the property that all intermediate C*-algebras likewise are simple. By an analogy to von Neumann algebras, we refer to such inclusions as being C*-irreducible. We give an intrinsic characterization of C*-irreducible inclusions, and use this characterization to exhibit (and revisit) such inclusions, both known ones and new ones, arising from groups and dynamical systems. By a theorem of Popa, an inclusion of II_1-factors is C*-irreducible if and only if it is irreducible with finite Jones index. We explain how one can construct C*-irreducible inclusions from inductive limits. In a recent joint work with Echterhoff we consider when inclusions of the form $A^H \subseteq A \rtimes G$ are C*-irreducible, where G and H are groups acting on a C*-algebra A. Such inclusions in the setting of II_1 factors were considered by Bisch and Haagerup.

- Date:
**11/5/21****Sorin Popa, UCLA**- Title: On the notions of amenability and weak-amenability for subfactors.
- Abstract. I'll first recall the notion of
*representation*of a II$_1$ subfactor $N\subset M$ (always assumed of finite Jones index and extremal), in particular of*standard representation*of $N\subset M$. Then I'll recall the definition of*amenability/injectivity*of a subfactor, requiring that $N\subset M$ is the range of a norm-1 projection from its standard rep, and which I've showed in the 1990s to be equivalent to the fact that $M\simeq R$ and $\|\Gamma_{N\subset M}\|^2=[M:N]$ (Kesten type condition), and also to the fact that $N\subset M$ can be exhausted by the higher relative commutants of a $(N\subset M)$-compatible tunnel. We provide a new equivalent characterization, showing amenability implies $N\subset M$ is the range of a norm-1 projection in ANY of its reps. I'll define*weak amenability/injectivity*for $N\subset M$ by requiring it merely has one rep with norm-1 projection. Note this implies $N, M \simeq R$, but not all hyperfinite subfactors are weakly amenable (e.g. $N_{\sigma}\subset R$ locally trivial/diagonal from a Bernoulli action of a simple property (T) group). The Haagerup-Schou hyperfinite subfactor $N\subset M$ of index $\|E_{10}\|^2=4.0262...$ is weakly amenable, but since it has standard (principal) graph $\Gamma_{N\subset M}=A_\infty$ and $\|A_\infty\|^2=4 < [M:N]$, it is not amenable. The main result we present shows that any weakly amenable subfactor has index equal to the square norm of a bipartite graph.

- Date:
**11/12/21****Corey Jones, North Carolina State University**- Title: A Categorical Connes' $\chi(M)$
- Abstract: For a finite von Neumann algebra $M$ with separable predual, we construct a braiding on the unitary tensor category $\tilde{\chi}(M)$ of dualizable approximately inner and centrally trivial $M$-$M$ bimodules, generalizing the usual notions for automorphisms and extending Connes' $\chi(M)$. Our unitary braiding on $\tilde{\chi}(M)$ extends Jones' $\kappa$ invariant. Given a finite depth inclusion $M_{0}\subseteq M_{1}$ of non-Gamma $\rm{II}_1$ factors, we show that the braided unitary tensor category $\tilde{\chi}(M_{\infty})$ is equivalent to the Drinfeld center of the standard invariant, where $M_{\infty}$ is the inductive limit of the associated Jones tower. This implies that for any pair of finite depth non-Gamma inclusions $N_{0}\subseteq N_{1}$ and $M_{0}\subseteq M_{1}$, if the standard invariants are not Morita equivalent, then the inductive limit factors $N_{\infty}$ and $M_{\infty}$ are not Morita equivalent. This talk is based on joint work with Quan Chen and David Penneys.

- Date:
**11/19/21****Changying Ding, Vanderbilt University**- Title: Proper proximality for groups acting on trees
- Abstract: The class of properly proximal groups was introduced by Boutonnet, Ioana, and Peterson. In this talk, I will present a joint work with Srivatsav Kunnawalkam Elayavalli, where we show that many groups acting on trees are properly proximal.

- Date:
**11/26/21****No meeting, Thanksgiving break**

- Date:
**12/3/21****James Tener, ANU Mathematical Sciences Institute**

- Date:
**12/10/21** - End of Fall Semester.