CS50: Advances in operator algebras and free probability

date: 7/17/2025, time: 14:00-15:30, room: ICS 140

Organizer: Andrej Srakar (University of Ljubljana)

Chair: Andrej Srakar (University of Ljubljana)

Cyclic conditional freeness

Nicolas GILLIERS (NYUAD)

Voiculescu’s freeness emerges when analysing polynomials on random matrices with eigenspaces in generic positions. I will present a result extending Voiculescu’s asymptotic freeness by proposing a random matrix model, the Vortex model, which displays asymptotic conditional freeness introduced by Bozejko, Leinert, and Speicher. I will define a new non-commutative independence called cyclic-conditional freeness, unifying three independences: infinitesimal freeness, cyclic-monotone independence (Collins, Hasebe, Sakuma), and cyclic-Boolean independence (Arizmendi, Hasebe, Lehner). Fluctuations in the Vortex model can be computed with the help of this new independence. This is a joint work with O. Arizmendi, G. Cebron

Bibliography

\([1]\) Arizmendi, Octavio, Guillaume Cébron, and Nicolas Gilliers. "Combinatorics of cyclic-conditional freeness." arXiv preprint arXiv:2311.13178 (2023).

\([3]\) Cébron, Guillaume, and Nicolas Gilliers. "Asymptotic cyclic-conditional freeness of random matrices." Random Matrices: Theory and Applications 13.01 (2024): 2350014.

\([2]\) Fujie, Katsunori, and Takahiro Hasebe. "Free probability of type B prime." Transactions of the American Mathematical Society (2025).

\([4]\) Arizmendi, Octavio, Takahiro Hasebe, and Franz Lehner. "Cyclic independence: Boolean and monotone." arXiv preprint arXiv:2204.00072 (2022).

Khintchine inequality for mixtures of free and independent semicircles

Raghavendra Tripathi (NYU Abu Dhabi)

The classical Khintchine’s inequality provides a fundamental comparison between the \(L_p\) and \(L_2\) norms of sums of independent Rademacher variables. Over the decades, Khintchine-type inequalities have been extended in multiple directions. A profound and structurally rich generalization of Khintchine’s inequality arises in the non-commutative setting, where the classical scalar-valued theory is extended to the operator algebra setting. In this setting, one considers sums of the form \(\sum_{i=1}^L a_i\otimes s_i\) where the \(a_i\)’s are bounded linear operators on a Hilbert space and \(s_i\)’s are free semicircle random variables and goal is to estimates the corresponding \(p\)-Schatten norm. More generally, one can take \(s_i\) to be \(G\)-independent collection of semicircles for some finite graph \(G\). We prove a sharp Khintchine-type estimate for (scalar) linear combinations of \(G\)-independent semicircles.

Theorem.\([\)Scalar Khintchine inequality for mixtures\(]\) Let \(G\) be a graph on \([L]\) and let \((s_{i})_{i=1}^{L}\) be a collection of \(G\)-independent semicircles. For any collection \((\alpha_{i})_{i=1}^{L}\) of complex numbers and any \(p\ge 1\), we have \[\begin{aligned} \left\|\sum_{i=1}^{L}\alpha_is_{i}\right\|_{2p}\leq C_p^{\frac{1}{2p}}\min\left\{\sum_{i=1}^{L}|\alpha_i|^2 |c^*(i)|,\;\; p\sum_{i=1}^{L}|\alpha_i|^2\right\}^{1/2}, \end{aligned}\] where \(|c^*(i)|\) is the size of the largest clique in \(G\) containing \(i\). In particular, \[\begin{aligned} \left\|\sum_{i=1}^{L}\alpha_is_{i}\right\|_{2p}\leq C_p^{\frac{1}{2p}}\sqrt{\omega(G)\wedge p}\, \big(\sum_{i=1}^{L}|\alpha_i|^2\big)^{1/2}, \end{aligned}\] where \(\omega(G)\) is the clique number of \(G\).

Our next result extends the scalar valued case to the operator valued coefficients.

Theorem.\([\)Operator-valued Khintchine inequality for mixtures\(]\) Let \(G\) be a graph on \([L]\) with clique number \(\omega(G)\) and let \((s_{i})_{i=1}^{L}\) be a collection of \(G\)-independent semicircles. Let \(\mathcal{H}\) be a Hilbert space and let \((a_i)_{i=1}^{L}\subseteq B(\mathcal{H})\). Then, \[\begin{aligned} \left\|\sum_{i=1}^{L} a_i\otimes s_i\right\|_{B(\mathcal{H})\otimes \mathcal{A}} \le 2 \sqrt{\omega(G)} \max\left\{\left\|\sum_{i=1}^{L}a_ia_i^*\right\|_{B(\mathcal{H})}^{1/2}, \left\|\sum_{i=1}^{L}a_i^*a_i\right\|_{B(\mathcal{H})}^{1/2}\right\}\;. \end{aligned}\] Here, \(B(\mathcal{H})\otimes \mathcal{A}\) denotes the minimal tensor product. This strengthens a recent result by Collins and Miyagawa.

Multi-Algebra Independences Arising in Bi-Free Probability

Daniel Pepper (York University)

In non-commutative probability, there are five main types of independence between algebras, each arising from actions of operators on corresponding product spaces. Voiculescu noticed that by studying both left and right actions on the free product simultaneously, and thus forming the notion of bi-free independence for pairs of algebras, one could study both classical (tensor) and free independence from this one viewpoint. Later it was found that in fact, all five types of independence could be studied through bi-free probability.

The advent of bi-free independence spawned other notions of multi-algebra independences. Among these are Liu’s notions of free-Boolean independence for pairs of algebras, and free-free-Boolean independence for triples of algebras. The question naturally arose whether these too could be studied through bi-free probability.

This talk, in an attempt to answer affirmatively to the aforementioned question, will sketch how free-free-Boolean independent operators can be embedded into, and therefore studied through bi-free probability.