# Free Probability and Operators

## Fall 2023

Date: | September 22, 2023 |

Time: | 4:00pm |

Location: | BLOC 506A |

Speaker: | Srivatsav Kunnawalkam Elayavalli, UCSD |

Title: | How to construct two non Gamma II_1 factors with non isomorphic ultrapowers |

Abstract: | I will describe a construction due to myself, Chifan and Ioana of a II_1 factor N such that N^omega is not isomorphic to L(F_2)^omega for any free ultrafilter omega. This is the first known such examples. The proof uses Voiculescu's free entropy theory and Popa's deformation rigidity theory. |

Date: | October 23, 2023 |

Time: | 4:00pm |

Location: | BLOC 117 |

Speaker: | Nico Spronk, University of Waterloo |

Title: | Traces on group C*-algebras |

Abstract: | We consider traces on the reduced and full C*-algebras, $C^*_r(G)$ and $C^*(G)$, of a locally compact group $G$. These traces can be understood very well for compactly generated $G$, in particular connected and almost connected $G$. We have for compactly generated $G$ an alternate proof of the result of Kennedy and Raum that $C^*_r(G)$ admits a trace exactly when $G$ has an open amenable radical. We also have a simple proof of Ng’s result that $G$ is amenable exactly when $C^*_r(G)$ is nuclear and admits a trace. We can then exploit a recent result of Schafhauser on AF-embeddability to characterize for which second countable almost connected $G$, $C^*(G)$ is AF-embeddable, generalizing some results of Beltita and Beltita; and we show that any tracially separated amenable group admits quasidaigonal $C^*_r(G)$. This is joint work with B. Forrest (Waterloo) and M. Wiersma (Winnipeg). |

Date: | November 3, 2023 |

Time: | 4:00pm |

Location: | BLOC 306 |

Speaker: | Rafael Morales, Baylor University |

Title: | Real Roots of Hypergeometric Polynomials Via Finite Free Convolution |

Abstract: | We discuss the newly discovered applications of the notion of the free finite convolution of polynomials (that is being developed in the framework of free probability theory) to the study of properties of zeros of some hypergeometric polynomials. Part of the talk deals with the use of classical family of orthogonal polynomials in order to obtain the said properties. |

Date: | November 10, 2023 |

Time: | 4:00pm |

Location: | BLOC 306 |

Speaker: | Adrian Celestino, TU Graz |

Title: | Antipode formulas, Schröder trees and cumulants in non-commutative probability |

Abstract: | In a series of recent papers, Ebrahimi-Fard and Patras developed an algebraic approach for cumulants in non-commutative probability based on a combinatorial Hopf algebra of words on words on an alphabet. In particular, they showed that the combinatorial moment-cumulants formulas, expressed in terms of non-crossing partitions, can be retrieved from specific fixed-point equations involving linear functionals on a Hopf algebra. In this talk, we discuss a combinatorial formula for the antipode in this Hopf algebra, which is represented in terms of Schröder trees, which have recently appeared in the context of non-commutative probability theory. Finally, we will see the implications of the antipode formula in non-commutative probability, namely, cumulant-moment formulas and free Wick polynomials in terms of Schröder trees. Based on an ongoing joint work with Yannic Vargas. |

Date: | November 17, 2023 |

Time: | 4:00pm |

Location: | BLOC 306 |

Speaker: | Ryo Toyota, TAMU |

Title: | Complete Haagerup inequality for Gromov hyperbolic groups |

Abstract: | In 1978, U Haagerup showed that if f is a function of the free group F_r which is supported on words with length exactly k, then the operator norm of the left regular representation |lambda(f)| is bounded by (k+1) times l^2-norm of f. Now this is called the Haagerup inequality, and its operator valued analogue was proved by Buchholz. In the operator valued case, the above (k+1)-l^2-norms is replaced by different (k+1)-operator norms associated to word decompositions. We will discuss how to generalize it for Gromov hyperbolic groups. This is a joint work with Zhiyuan Yang. |

Date: | December 8, 2023 |

Time: | 4:00pm |

Location: | BLOC 306 |

Speaker: | Antonio Ismael Cano-Marmol, Baylor University |

Title: | Xp inequalities on free groups |

Abstract: | Naor and Schechtman recently introduced the so-called metric Xp inequalities, an obstruction for embeddings of Lq into Lp whenever 2 < q < p . This invariant was refined by Naor via a fundamental inequality in the Hamming cube which strongly relies on Fourier analysis. In this talk, we will show that this latter result can be understood within the frame of noncommutative harmonic analysis. In particular, the case over the group von Neumann algebra of the free group will be described. Moreover, we will briefly discuss some metric consequences and possible further work. This is joint work with José M. Conde-Alonso and Javier Parcet. |