Functional Analysis Seminar

2018-2019

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

Fall 2019

Date Speaker Title + Abstract
October 15 David Jekel
UCLA
Triangular Transport of Measure for Non-commutative Random Variables
We study tuples $(X_1,\dots,X_d)$ of self-adjoint operators in a tracial $W^*$-algebra whose non-commutative distribution is the free Gibbs law for a (sufficiently regular) convex potential $V$. Such tuples model the large $N$ behavior of random matrices $(X_1^{(N)}, \dots, X_d^{(N)})$ chosen according to the measure $e^{-N^2 V(x)}\,dx$ on $M_N(\mathbb{C})_{sa}^d$. Previous work showed that $W^*(X_1,\dots,X_d)$ is isomorphic to the free group factor $L(\mathbb{F}_d)$. In a recent preprint, we showed that an isomorphism $\phi: W^*(X_1,\dots,X_d)$ can be chosen so that $W^*(X_1,\dots,X_k)$ is mapped to the canonical copy of $L(\mathbb{F}_k)$ inside $L(\mathbb{F}_d)$ for each $k$. The idea behind the proof is to apply PDE methods for constructing transport to Gaussian to the conditional density of $X_j^{(N)}$ given $X_1^{(N)}, \dots, X_{j-1}^{(N)}$. Then we analyze the asymptotic behavior of these transport maps as $N \to \infty$ using a new type of functional calculus, which applies certain $\|\cdot\|_2$-continuous functions to tuples of self-adjoint operators to self-adjoint tuples in (Connes-embeddable) tracial $W^*$-algebras.
November 12 Matthew Wiersma
UCLA
$L^p$-representations and C*-algebras
A unitary representation $\pi\colon G\to B(H)$ of a locally compact group $G$ is an \emph{$L^p$-representation} if $H$ admits a dense subspace $H_0$ so that the matrix coefficient $$ G\ni s\mapsto \langle \pi(s)\xi,\xi\rangle$$ belongs to $L^p(G)$ for all $\xi\in H_0$. The \emph{$L^p$-C*-algebra} $C^*_{L^p}(G)$ is the C*-completion $L^1(G)$ with respect to the C*-norm $\|f\|_{C^*_{L^p}}$ defined to be the supremum of $\|\pi(f)\|$ over all $L^p$-representations $\pi$ of $G$, and all $f\in L^1(G)$. Surprisingly, the C*-algebra $C^*_{L^p}(G)$ is intimately related to the enveloping C*-algebra of the Banach $*$-algebra $PF^*_p(G)$ ($2\leq p\leq \infty$). Consequently, we characterize the states of $C^*_{L^p}(G)$ as corresponding to positive definite functions that ``almost'' belong to $L^p(G)$ in some suitable sense for ``many'' $G$ possessing the Haagerup property, and either the rapid decay property or Kunze-Stein phenomenon. It follows that the canonical map $$ C^*_{L^p}(G)\to C^*_{L^{p'}}(G)$$ is not injective for $2\leq p'< p\leq \infty$ when $G$ is non-amenable and belongs to the class of groups mentioned above. As a byproduct of our techniques, we give a near solution to a 1978 conjecture of Cowling. This is primarily based on joint work with E. Samei.

"Winter" 2020

Date Speaker Title + Abstract
January 28 Pieter Spaas
UCLA
What can central sequences in von Neumann algebras look like?
We will discuss central sequence algebras of von Neumann algebras, and provide a class of $\mathrm{II}_1$ factors whose central sequence algebra is not the "tail" algebra associated to any decreasing sequence of von Neumann subalgebras. This settles a question of McDuff from 1969. We will also discuss an application of these techniques to the notion of tracial stability for groups. This is based on joint work with Adrian Ioana.
March 3 Michael Brannan
Texas A&M University
TBA

Spring 2020

Date Speaker Title + Abstract
March 31 Ching Wei Ho
Indiana University
TBA
May 5 Matt Kennedy
University of Waterloo
TBA
May 19 Jorge Garza Vargas
UC Berkeley
TBA