Free Probability and Operators

Date: November 18, 2022

Time: 4:00PM - 5:00PM

Location: BLOC 306

Speaker: Zhiyuan Yang, TAMU

Title: Free Poisson von Neumann algebras

Abstract: We would like to introduce a free Poisson type functor for left Hilbert algebras similar to the Voiculescu's free Gaussian functor and the free Araki-Woods functor. For any left Hilbert algebra A, we consider the von Neumann algebra Γ(A) generated by the operators X(a)=l^*(a)+l(a)+p(a) acting on the full Fock space F(A), where a is in A, l^*(a), l(a) are the creation and annihilation operators, and p(a) is an preservation operator. Using a simple combinatorial method, we will show that when A is a W^* algebra with a finite weight, Γ(A) is isomorphic to the free product of L(Z) and an algebra determined by A (with a possible extra atom). In particular, this implies that the filtration W^* algebras of a free Poisson process (over a time interval) are the interpolated free group factors. We will also describe the behavior of completely positive maps under this functor.