# Functional Analysis Seminar

## 2018-2019

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 2018

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" 2019

Date Speaker Title + Abstract
January 15
Joint with Geometry
Artem Pulemotov
University of Queensland
The Ricci iteration on homogeneous spaces
The Ricci iteration is a discrete analogue of the Ricci flow. Introduced in 2007, it has been studied extensively on Kähler manifolds, providing a new approach to uniformisation. In the talk, we will define the Ricci iteration on compact homogeneous spaces and discuss a number of existence, convergence and relative compactness results. This is largely based on joint work with Timothy Buttsworth (Queensland), Yanir Rubinstein (Maryland) and Wolfgang Ziller (Penn).
February 5
Joint with Probability
Brian Hall
University of Notre Dame
Eigenvalues of random matrices in the general linear group
I will consider random matrices in the general linear group GL(N;C) distributed according to a heat kernel measure. This may also be described as the distribution of Brownian motion in GL(N;C) starting at the identity. Numerically, the eigenvalues appear to cluster into a certain domain $\Sigma_t$ as $N$ tends to infinity. A natural candidate for the limiting eigenvalue distribution is the “Brown measure” of the limiting object, which is Biane’s "free multiplicative Brownian motion." I will describe recent work with Driver and Kemp in which we compute this Brown measure. The talk will be self contained and will have lots of pictures.
February 8
Note unusual day
Ben Hayes
University of Virginia
Quotients of Bernoulli shifts associated to operators with an $\ell^{2}$-inverse
Let G be a countable, discrete, group and f an element of the integral group ring over G. It is well known how to associate to f an action of G on a compact, metrizable, abelian group. It turns out to be particularly interesting to consider those f with an $\ell^{2}$-inveres: i.e. a vector $\xi\in \ell^{2}(G)$ so that $f*\xi=\delta_{1}$. Many nice ergodic theoretic properties of the corresponding action have been established in this context. I will give certain examples of f,G for which we can say that this action is a quotient of a Bernoulli shift. When G is amenable, this implies that it *is* a Bernoulli shift.
February 19 Daniel Drimbe
University of Regina
On the tensor product decomposition of II$_1$ factors arising from groups and group actions
In a joint work with D. Hoff and A. Ioana, we have discovered the following product rigidity phenomenon: if $\Gamma$ is an icc group measure equivalent to a product of non-elementary hyperbolic groups, then any tensor product decomposition of the II$_1$ factor $L(\Gamma)$ arises only from the canonical direct product decomposition of $\Gamma$. Subsequently, I. Chifan, R. de Santiago and W. Sucpikarnin classified all the tensor product decompositions for group von Neumann algebras arising from a large class of amalgamated free products. In this talk we will give an overview of these results and discuss about a similar rigidity phenomenon that appears in the context of von Neumann algebras arising from actions. More precisely, we prove that if $\Gamma$ is a product of certain groups and $\Gamma\curvearrowright (X,\mu)$ is an arbitrary free ergodic measure preserving action, then we show that any tensor product decomposition of the II$_1$ factor $L^\infty(X)\rtimes\Gamma$ arises only from the canonical direct product decomposition of the underlying action $\Gamma\curvearrowright X.$
February 26 Sayan Das
University of Iowa
On the generalized Neshveyev-Stormer conjecture
The study of group actions on probability measure spaces plays a central role in modern mathematics. The (generalized) Neshveyev-Stormer conjecture states that the group action on a probability measure space can be completely understood by studying the inclusion of the group von Neumann algebra inside the group measure space construction. In my talk I shall show that the Neshveyev-Stormer conjecture is true for a large class of actions. This talk is based on a joint work with Ionut Chifan.

### Spring 2019

Date Speaker Title + Abstract
April 9 Rolando de Santiago
UCLA
$L^2$ Betti numbers and $s$-malleable deformations.
A major theme in the study of von Neumann algebras is to investigate which structural aspects of the group extend to its von Neumann algebra. I present recent progress made by Dan Hoff, Ben Hayes, Thomas Sinclair and myself in the case where the group has positive first $L^2$ Betti number. I will also expand on our analysis of $s$-malleable deformations and their relation to cocylces which forms the foundation of our work.
April 11
11am in APM 6402
Ian Charlesworth
UC Berkeley
Free Stein Information
I will speak on recent joint work with Brent Nelson, where we introduce a free probabilistic regularity quantity we call the free Stein information. The free Stein information measures in a certain sense how close a system of variables is to admitting conjugate variables in the sense of Voiculescu. I will discuss some properties of the free Stein information and how it relates to other common regularity conditions.
April 23
11am in APM 7321
Todor Tsankov
Universite Paris Diderot
Bernoulli disjointness
The concept of disjointness of dynamical systems (both topological and measure-theoretic) was introduced by Furstenberg in the 60s and has since then become a fundamental tool in dynamics. In this talk, I will discuss disjointness of topological systems of discrete groups. More precisely, generalizing a theorem of Furstenberg (who proved the result for the group of integers), we show that for any discrete group $G$, the Bernoulli shift $2^G$ is disjoint from any minimal dynamical system. This result, together with techniques of Furstenberg, some tools from the theory of strongly irreducible subshifts, and Baire category methods, allows us to answer several open questions in topological dynamics: we solve the so-called "Ellis problem" for discrete groups and characterize the underlying topological space for the universal minimal flow of discrete groups. This is joint work with Eli Glasner, Benjamin Weiss, and Andy Zucker.
April 30 Amine Marrakchi
RIMS, Kyoto
Tensor product decompositions and rigidity of full factors
A central theme in the theory of von Neumann algebras is to determine all possible tensor product decompositions of a given factor. I will present a recent joint work with Yusuke Isono where we use the rigidity of full factors and a new flip automorphism approach in order to study this problem. Among other things, we show that a separable full factor admits at most countably many tensor product decompositions (up to stable unitary conjugacy). We also establish new primeness and Unique Prime Factorization results for crossed products coming from compact actions of irreducible higher rank lattices (e.g. $\mathrm{SL}_n(\mathbb{Z})$ for $n > 2$) as well as noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable).
May 6
11am in APM 6402
David Sherman
University of Virginia
Old and new theorems on closures and convex hulls of unitary orbits of self-adjoint operators
This will be a selective survey about unitary orbits of self-adjoint operators. I will start with convex hulls of matrices, where the key tool is majorization. Then I will discuss closures associated to infinite-dimensional operators, and I will give a new description of the weak* closure. Finally I will discuss new results about weak* closures for self-adjoint elements in von Neumann factors. Perhaps most interesting is a noncommutative Lyapunov phenomenon": the type I (atomic) case turns out to be qualitatively different from types II and III, in which the closure is automatically convex and again described by majorization. This is joint work with Chuck Akemann.
May 7 Matthew Wiersma
University of Alberta
Hermitian groups are amenable
A locally compact group $G$ is Hermitian if the spectrum $\sigma_{L^1(G)}(f)$ is contained in $\mathbb R$ for every $f=f^*\in L^1(G)$. Examples of Hermitian groups include all abelian locally compact groups. A question from the 1960s asks whether every Hermitian group is amenable. I will speak on the history and recent affirmative solution to this problem.
May 28 Mike Hartglass
Santa Clara University
Free products of finite-dimensional von Neumann algebras in terms of free Araki-Woods factors
A landmark result by Dykema in 1993 classified free products of finite-dimensional von Neumann algebras equipped with tracial states. In 1997, Shlyakhtenko constructed the almost periodic free Araki-Woods factors, a natural non-tracial analogue to free group factors. He asked whether free products of finite-dimensional von Neumann algebras with respect to non-tracial states can be described in terms of free Araki-Woods factors. In this talk, I will answer Shlyakhtenko's question in the affirmative, therefore providing a complete classification of free products of finite dimensional von Neumann algebras. This is joint work with Brent Nelson.