M. Ali Asadi-Vasfi, Purdue University
The Rank Ratio Function and the Radius of Comparison Function
In this talk, we introduce the Rank Ratio Function, a tool for studying Cuntz semigroups and their relative radii of comparison. We also introduce the Radius of Comparison Function, which can be used to construct non-classifiable C*-algebras with different relative radii of comparison.
Ionut Chifan, University of Iowa
McDuff $W^*$-rigidity for group II$_1$ factors
We develop new techniques at the rich interface between geometric group theory and von Neumann algebras to identify the first examples of ICC groups $G$ whose factors $\mathcal{L}(G)$ are McDuff and exhibit a new rigidity phenomenon, which we term \emph{McDuff $W^*$-superrigidity}. Specifically, for such groups $G$, an arbitrary group $H$ satisfies $\mathcal{L}(H) \cong \mathcal{L}(G)$ if and only if $H \cong G \times A$ for some ICC amenable group $A$. Our examples of $G$ arise as infinite direct sums of property (T) wreath-like product extensions whose natural 2-cocycles satisfy a novel boundedness condition.
Patrick DeBonis, Purdue University
The W* and C*-algebras of Similarity Structure Groups
I will give an overview of Countable Similarity Structure (CSS) groups and the subclass of CSS* groups, which can be viewed as generalizations of Thompson's group V. Then I will highlight how their group von Neumann algebras are prime but not solid, and their reduced group C*-algebras are simple. This is joint work with Eli Bashwinger.
Changying Ding, Univeristy of California-Los Angeles
Structure and non-isomorphisms of q-Araki-Woods factors
Hiai’s construction of q-Araki-Woods factors generalizes both Shlyakhtenko’s free Araki-Woods factors and Bozejko-Speicher’s q-Gaussian algebras. In this talk, I will present that these q-Araki-Woods factors are strongly solid when almost periodic. Under a certain spectrum condition of the associated representation, I will show that q-Araki-Woods with infinite variables are not isomorphic to free Araki-Woods factors. This is a joint work with Hui Tan.
Priyanga Ganesan, University of California-San Diego
Quantum Non-Local Games
In recent years, nonlocal games have received significant attention in operator algebras and resulted in highly fruitful interactions, including the resolution of the Connes Embedding Problem. Nonlocal games describe scenarios that test the ability of two non-communicating parties to give correlated responses. In this talk, I will provide an overview of the theory of non-local games and their connections to operator algebras. I will then present a generalization of non local games to allow for quantum inputs and outputs. We will see some applications of these games and discuss how the existence of perfect strategies in this quantum input/output framework can be characterised.
Mayha Ghandehari, University of Delaware
Amenability (constants) of the Fourier algebras of locally compact groups
A major trend in non-commutative harmonic analysis is to investigate function spaces related to Fourier analysis (and representation theory) of non-abelian groups. The Fourier algebra and the Fourier-Steiltjes algebra, which are associated with the regular representation and the universal representation of the ambient group respectively, are important examples of such function spaces. One important structural property of Banach algebras is the notion of amenability, whose study has inspired some deep work within particular classes of Banach algebras such as $L^1$ algebras or Fourier algebras. In this talk, we study the amenability constant of Fourier algebras. If a group is finite then the amenability constant of its Fourier algebra admits an explicit formula [Johnson,1994]. Such a formula cannot be obtained in the infinite case. Using non-abelian Fourier analysis, we obtain an upper bound for this constant when the group is discrete.
This talk is based on joint work with Y. Choi.
Ishan Ishan, University of Nebraska -- Lincoln
Von Neumann Orbit Equivalence
Recently, Wu and I introduced the notion of von Neumann orbit equivalence, which is a non-commutative generalization of the notion of orbit equivalence. In this talk, I will discuss the stability of this equivalence relation under taking free products and graph products.
Junhwi Lim, Vanderbilt University
Planar algebras associated to cocommuting squares
The ‘generalized symmetries’ of subfactors are encoded by their planar algebras. A natural foundational question is “What minimal structure can be universally expected from these symmetries?” For arbitrary subfactors, Jones showed that the associated planar algebra contains the Temperley-Lieb-Jones algebra. When an intermediate subfactor is present, the planar algebra contains the Fuss-Catalan-Bisch-Jones algebra, introduced by Bisch and Jones. However, the situation with two intermediate subfactors remains an open problem. As a natural special case, we study cocommuting squares of factors with ‘group-like’ properties. We exhibit the skein relations of their associated planar algebras and show that these algebras extend the partition algebras. This is based on a joint work with Dietmar Bisch.
Melody Molander, The Ohio State University
Subfactor Planar Algebras at Index 4
Subfactor planar algebras were introduced by Vaughan Jones as a diagrammatic way to view the standard invariant of subfactors. The Kuperberg program seeks to describe all presentations of subfactor planar algebras in order to classify them and prove results about their corresponding categories purely diagrammatically. This program has been completed for index less than 4 and remains an area of ongoing research for index greater than 4. This talk will discuss the program at index 4. At this index, Popa proved that planar algebras other than Temperley-Lieb have an affine A, D, or E principal graph. In this talk, I will describe an explicit presentation by generators and relations for some of the subfactor planar algebras at index 4.
Dimitri Shlyakhtenko, Univeristy of California -- Los Angeles
Strong 1-boundedness via non-microstates entropy
Strong 1-boundedness (introduced by K. Jung) is a property of a tracial von Neumann algebra that can be checked on sets of generators. Free group factors (and, more generally, many free products) do not have this property; on the other hand, certain structural properties (property Gamma, having a Cartan subalgebra, non-primeness, etc.) have been shown, through the works of Voiculescu, Ge, Jung, Hayes, and others, to be strongly one-bounded. In a joint work with B. Major, we give new proofs of several strong one-boundedness results using non-microstates free entropy, and utilizing a relative version of the Biane-Capitaine-Guionnet free entropy inequality recently proved by Jekel and Pi.
Dan Ursu, York University
Non-conventional averaging in C*-algebras
Several important averaging properties have shown up in the theory of operator algebras, most notably the Dixmier averaging property and its variants, which deals with convex averages of elements in some unitary orbit. In joint work with Matthew Kennedy, expanding upon the work of Magajna in the theory of C*-convex averages, we develop a strong new averaging property and separation theorem, and use it to characterize when the intermediate subalgebra structure of a crossed product is entirely canonical. Progress-permitting, we will also give a sneak peek at some preliminary results using these same averaging techniques applied to the ideal structure of crossed products.
Rufus Willett, University of Hawaii
Secondary pairings and group representations
As part of the basic definitions there is a ('primary') pairing between K-theory and K-homology of a C*-algebra A taking values in Z. This pairing must vanish on the torsion subgroups; however, as a result of its vanishing, there is a 'secondary' pairing between the torsion subgroups of K-theory and K-homology (in opposite parities) that takes values in Q/Z.
When A=C*(G) is a group C*-algebra, I'll discuss applications to group representations, and briefly describe three ways to understand the secondary pairing (each with a different flavor): in terms of flat bundles (topology); relative eta invariants (geometry); and the Thomsen exact sequence and zeta map (C*-algebras).
I will not assume any background in K-homology or the topological or geometric inputs.