Regularity questions in free probability ask what can be learned about a tracial von Neumann algebra from probabilistic-flavoured qualities of a set of generators. Broadly speaking there are two approaches — one based in microstates, one in free derivations — which with the failure of Connes Embedding are now known to be distinct. The non-microstates approach is not obstructed by non-embeddable variables, but can be more difficult to work with for other reasons. I will speak on recent work with Brent Nelson, where we introduce a quantity called the free Stein dimension, which measures how readily derivations may be defined on a collection of variables. I will spend some time placing it in the context of other non-microstates quantities, and sketch a proof of the exciting fact that free Stein dimension is a *-algebra invariant.
I will discuss ε-independence, which is an interpolation of classical and free independence originally studied by Młotkowski and later by Speicher and Wysoczanski. To be ε-independent, a family of algebras in particular must satisfy pairwise classical or free independence relations prescribed by a $\{0, 1\}$-matrix ε, as well as more complicated higher order relations. I will discuss how matrix models for this independence may be constructed in a suitably-chosen tensor product of matrix algebras. This is joint work with Benoît Collins.
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.
Bi-free probability is a generalization of free probability to study pairs of left and right faces in a non-commutative probability space. In this talk, I will demonstrate a characterization of bi-free independence inspired by the "vanishing of alternating centred moments" condition from free probability. I will also show how these ideas can be used to introduce a bi-free unitary Brownian motion and a liberation process which asymptotically creates bi-free independence.