Ian Charlesworth

Lecturer

Cardiff University School of Mathematics

Email: charlesworthi@cardiff.ac.uk

I have been a lecturer with the Department of Mathematics at Cardiff University since 2022. I have been a postdoc with the Department of Mathematics at KU Leuven since 2021. Prior to that I held a postdoctoral position at KU Leuven from 2021-2022, an NSF postdoctoral position at UC Berkeley from 2018-2021, and an S.E. Warschawski Visiting Assistant Professorship at UCSan Diego from 2017-2018. My Ph.D. studies took place from 2012-2017 at the University of California, Los Angeles under the supervision of Professor Dimitri Shlyakhtenko; my thesis was titled 'On bi-free probability and free entropy.' I was an undergraduate at the University of Waterloo in Waterloo, Ontario, Canada studying Pure Mathematics and Computer Science.

My research interests lie mostly in the field of free probability and non-commutative probability theory; the field attempts to apply probabilistic techniques to operator algebras, drawing useful analogues from well-known results in probability. I have also dabbled in the study of subfactors and quantum symmetries. In the distant past I have attacked problems in database query optimization.

## On the structure of graph product von Neumann algebras, with R. de Santiago, B. Hayes, D. Jekel, S. Kunnawalkam Elayavalli, and B. Nelson; arXiv:2404.08150 (2024)

We undertake a comprehensive study of structural properties of graph products of von Neumann algebras equipped with faithful, normal states, as well as properties of the graph products relative to subalgebras coming from induced subgraphs. Among the technical contributions in this paper include a complete bimodule calculation for subalgebras arising from subgraphs. As an application, we obtain a complete classification of when two subalgebras coming from induced subgraphs can be amenable relative to each other. We also give complete characterizations of when the graph product can be full, diffuse, or a factor. Our results are obtained in a broad generality, and we emphasize that they are new even in the tracial setting. They also allow us to deduce new results about when graph products of groups can be amenable relative to each other.## Random permutation matrix models for graph products, with R. de Santiago, B. Hayes, D. Jekel, S. Kunnawalkam Elayavalli, and B. Nelson; arXiv:2404.07350 (2024)

Graph independence (also known as $\varepsilon$-independence or $\lambda$-independence) is a mixture of classical independence and free independence corresponding to graph products or groups and operator algebras. Using conjugation by certain random permutation matrices, we construct random matrix models for graph independence with amalgamation over the diagonal matrices. This yields a new probabilistic proof that graph products of sofic groups are sofic.## Strong 1-boundedness, L

^{2}-Betti numbers, algebraic soficity, and graph products, with R. de Santiago, B. Hayes, D. Jekel, S. Kunnawalkam Elayavalli, and B. Nelson; arXiv:2305.19463 (2023)We show that graph products of non trivial finite dimensional von Neumann algebras are strongly 1-bounded when the underlying $*$-algebra has vanishing first $L^2$-Betti number. The proof uses a combination of the following two key ideas to obtain lower bounds on the Fuglede-Kadison determinant of matrix polynomials in a generating set: a notion called 'algebraic soficity' for $*$-algebras allowing for the existence of Galois bounded microstates with asymptotically constant diagonals; a probabilistic construction of the authors of permutation models for graph independence over the diagonal.

## On Free Stein Dimension, with B. Nelson; arXiv:2201.00062 (2022).

We establish several properties of the free Stein dimension, an invariant for finitely generated unital tracial $*$-algebras. We give formulas for its behaviour under direct sums and tensor products with finite dimensional algebras. Among a given set of generators, we show that (approximate) algebraic relations produce (non-approximate) bounds on the free Stein dimension. Particular treatment is given to the case of separable abelian von Neumann algebras, where we show that free Stein dimension is a von Neumann algebra invariant. In addition, we show that under mild assumptions $L^2$-rigidity implies free Stein dimension one. Finally, we use limits superior/inferior to extend the free Stein dimension to a von Neumann algebra invariant---which is substantially more difficult to compute in general---and compute it in several cases of interest.

## Analogues of Entropy in Bi-Free Probability Theory: Microstates, with P. Skoufranis; IMRN.[arXiv]

In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are developed, such as the behaviour of the entropy when transformations on the left variables or on the right variables are performed. In addition, the microstate bi-free entropy is demonstrated to be additive over bi-free collections and is computed for all bi-free central limit distributions.## Free Stein irregularity and dimension, with B. Nelson; J. Operator Theory

**85**(2021), no. 1, 101-133.[arXiv]We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to the Murray--von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu's free difference quotients. We call this dimension the free Stein dimension, and show that it is a $*$-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In the one-variable case, we show that the free Stein dimension agrees with the free entropy dimension, and in the multivariable case compute it in a number of examples.## Matrix models for ε-free independence, with B. Collins; Arch. Math.

**116**(2021), no. a, 585-605.[arXiv]We investigate tensor products of random matrices, and show that independence of entries leads asymptotically to ε-free independence, a mixture of classical and free independence studied by Młotkowski and by Speicher and Wysoczański. The particular ε arising is prescribed by the tensor product structure chosen, and conversely, we show that with suitable choices an arbitrary ε may be realized in this way. As a result we obtain a new proof that $\mathcal{R}^\omega$-embeddability is preserved under graph products of von Neumann algebras, along with an explicit recipe for constructing matrix models.## Analogues of Entropy in Bi-Free Probability Theory: Non-Microstate, with P. Skoufranis; Adv. Math.

**375**(2020), 107367.[arXiv]In this paper, we extend the notion of non-microstate free entropy to the bi-free setting. Using a diagrammatic approach involving bi-non-crossing diagrams, bi-free difference quotients are constructed as analogues of the free partial derivations. Adjoints of bi-free difference quotients are discussed and used to define bi-free conjugate variables. Notions of bi-free Fisher information and non-microstate entropy are defined and properties of free entropy are extended to the bi-free setting.

## Free Stein Dimension. A talk recorded at the

*Wales MPPM seminar*.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.

## Asymptotic ε-independence. A presentation recorded at the Banff International Research Station in October 2019 during the workshop

*Classification Problems in von Neumann Algebras*.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.

## Free Stein Information. A presentation recorded at the Fields Institute in February 2019 during the

*Southern Ontario Operator Algebras Seminar*.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.

## An alternating moment condition and liberation for bi-freeness. A presentation recorded at the Banff International Research Station in December 2016 during the workshop

*Analytic Versus Combinatorial in Free Probability*.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.