Subfactor Seminar
Fall 2021
Organizers: Dietmar Bisch and Jesse Peterson
Fridays, 4:10-5:30pm central (For European speakers: 11:30am-12:50pm central)
In SC 1431 for local speakers (For external speakers: Zoom Meeting ID: 943 9395 6397)
- 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
- 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
- Title: Conformal nets are geometric
- Abstract: In this talk I'll present joint work in progress with André Henriques which
shows that any conformal net (i.e. a net of factors corresponding to
intervals of the unit circle) has a geometric origin. More precisely, I'll
explain how the factors are generated by insertion operators built from a
two-dimensional geometric field theory, or alternatively from a vertex
operator algebra. A similar analysis is possible for representations of
a conformal net, which correspond to subfactors.
- End of Fall Semester.
Past NCGOA and Subfactor seminars
NCGOA home page
VU math department's calendar
Dietmar Bisch's home page
Jesse Peterson's home page