Minicourses

Ben Hayes

University of Virginia

Invitation to Microstates Free Entropy Theory  

I will present a brief overview of 1-bounded entropy, which is a modification of Voiculescu's free entropy dimension, and is known to be an invariant of the generated von Neumann algebra. In particular, I plan to discuss:

•  its recent relevance to long-standing and frequently promoted problems in II1-factor theory (e.g. nonelementarily equivalent full factors, the Peterson-Thom property),

•  its connection to random matrix theory, and

• some of the historical context behind its definition and the most important recent developments in this theory.

Kristin Courtney

University of Southern Denmark 

Local Lifting and Weak Expectations 

In the 90's, Kirchberg tied together several open problems, famously including Connes' Embedding Problem, by leveraging deep connections between his Local Lifting Property and Lance's Weak Expectation Property. In the ensuing years, these properties have garnered significant interest in their own right.

In this course, I will introduce these two concepts as well as several different ways to view them. We shall see how pitting them against one another reveals deep connections between seemingly unrelated concepts, and in the end, we will touch on more recent progress on understanding which C*-algebras have or lack these properties. 

Gandalf Lechner

Friedrich-Alexander-University

Standard Subspaces  

A standard subspace of a complex Hilbert space is a closed real linear

subspace with dense complex span that contains no non-trivial complex subspace. This definition is quite elementary and does not refer to operator algebras. Nonetheless, standard subspaces have a surprisingly rich structure that is in particular fundamental in the Tomita–Takesaki modular theory of von Neumann algebras and its applications. In these lectures, the theory of standard subspaces will first be introduced as a subject of interest in its own right and then compared to von Neumann algebras in standard form. Also some new developments related to inclusions of standard subspaces, twisted Araki-Woods factors, and the internal structure of standard subspaces will be presented.