Functional Analysis Seminar

2017-2018

Time Location Seminar Organizers
Tuesday — 11:00am–12:00pm AP&M 6402 (unless otherwise specified) Adrian Ioana (aioana@ucsd.edu)
Todd Kemp (tkemp@ucsd.edu)

Fall 2017

Date Speaker Title + Abstract
October 9 Christopher Schafhauser
York University
Subalgebras of AF-Algebras
A long-standing open question, formalized by Blackadar and Kirchberg in the mid 90's, asks for an abstract characterization of C$^*$-subalgebras of AF-algebras. I will discuss some recent progress on this question: every separable, exact C$^*$-algebra which satisfies the UCT and admits a faithful, amenable trace embeds into an AF-algebra. Moreover, the AF-algebra may be chosen to be simple and unital with unique trace and the embedding may be taken to be trace-preserving. Modulo the UCT, this characterizes C$^*$-subalgebras of simple, unital AF-algebras. As an application, for any countable, discrete, amenable group $G$, the reduced C$^*$-algebra of $G$ embeds into a UHF-algebra.
November 6
2:30pm in APM 6402
David Jekel
UCLA
An Elementary Approach to Free Gibbs Laws Given by Convex Potentials
We present an alternative approach to the theory of free Gibbs laws with convex potentials developed by Dabrowski, Guionnet, and Shlyakhtenko. Instead of solving SDE's, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose $\mu_N$ is a probability measure on on $M_N(\mathbb{C})_{sa}^m$ given by uniformly convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$ is asymptotically approximable by trace polynomials in a certain sense. Then the moments of $\mu_N$ converge to a non-commutative law $\lambda$. Moreover, the free entropies $\chi(\lambda)$, $\underline{\chi}(\lambda)$, and $\chi^*(\lambda)$ agree and equal the limit of the normalized classical entropies of $\mu_N$. An upcoming paper will use the same techniques to obtain transport maps from $\lambda$ to a free semicircular family as the limit of transport maps for the matrix models $\mu_N$.
November 8
1pm in APM 6402
Pooya Vahidi Ferdowsi
Caltech
Classification of Choquet-Deny Groups
A countable discrete group is said to be Choquet-Deny if it has a trivial Poisson boundary for every non-degenerate probability measure on the group. In other words, a countable discrete group is Choquet-Deny if non-degenerate random walks on the group have trivial behavior at infinity. For example, all abelian groups are Choquet-Deny. It has been long known that all Choquet-Deny groups are amenable. I will present a recent result classifying countable discrete Choquet-Deny groups: a countable discrete group is Choquet-Deny if and only if none of its quotients have the infinite conjugacy class property. As a corollary, a finitely generated group is Choquet-Deny if and only if it is virtually nilpotent. This is a joint work with Joshua Frisch, Yair Hartman, and Omer Tamuz.
November 13 Scott Atkinson
Vanderbilt University
Tracial stability and graph products
A unital $C^\ast$-algebra $A$ is tracially stable if maps on $A$ that are approximately (in trace) unital $\ast$-homomorphisms can be approximated (in trace) by honest unital $\ast$-homomorphisms on $A$. Tracial stability is closed under free products and tensor products with abelian $C^\ast$-algebras. In this talk we expand these results to show that for a graph from a certain class, the corresponding graph product (a simultaneous generalization of free and tensor products) of abelian $C^\ast$-algebras is tracially stable. We will then discuss two applications of this result: a selective version of Lin’s Theorem and a characterization of the amenable traces on certain right-angled Artin groups.
November 14
3pm in APM 6402
Anush Tserunyan
UIUC
A pointwise ergodic theorem for quasi-pmp graphs
We prove a pointwise ergodic theorem for locally countable ergodic quasi-pmp (nonsingular) graphs, which gives an increasing sequence of Borel subgraphs with finite connected components, averages over which converge a.e. to the expectations of $L^1$-functions. This can be viewed as a random analogue of pointwise ergodic theorems for group actions: instead of taking a (deterministic) sequence of subsets of the group and using it at every point to compute the averages, we allow every point to coherently choose such a sequence at random with a strong condition that the sets in the sequence determine aconnected subgraph of the Schreier graph of the action.
December 4 Daniel Hoff
UCLA
Rigid Components of s-Malleable Deformations

In the theory of von Neumann algebras, fundamental unsolved problems going back to the 1930s have seen remarkable progress in the last two decades due to Sorin Popa's breakthrough deformation/rigidity theory. Popa's discovery hinges on the fact that, just as stirring a soup allows one to locate its most rigid (and desirable) hidden components, ``deformability" of an algebra $M$ allows one to precisely locate ``rigid" subalgebras known to exist only via a supposed isomorphism $M \cong N$.

This talk will focus on joint work with Rolando de Santiago, Ben Hayes, and Thomas Sinclair, which shows that any diffuse subalgebra which is rigid with respect to a mixing $s$-malleable deformation is in fact contained in subalgebra which is uniquely maximal with respect to that rigidity. In particular, an algebra generated by a family of rigid subalgebras which intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members answers a question of Jesse Peterson asked at the American Institute of Mathematics (AIM), but the result is most striking when the family is infinite.


Winter 2018

Date Speaker Title + Abstract
January 15 Artem Pulemotov
University of Queensland
TBA
February 19 Daniel Drimbe
University of Regina
TBA
February 26 Sayan Das
University of Iowa
TBA
April 30 Amine Marrakchi
RIMS, Kyoto
TBA

Spring 2018

Date Speaker Title + Abstract