Notes on Free Probability Theory

关于自由量子群的量子子群的一个注记

A note on quantum subgroups of free quantum groups

本文分类了自由酉量子群对偶的有限自由积中的量子子群，将其刻画为满足不变性、包含性和连通性的子群与子集对。

Notes on microstate free entropy of projections

We study the microstate free entropy of projections, and establish its basic properties similar to the self-adjoint variable case. Our main contribution is to characterize the pair-block freeness of projections by the additivity of their free entropy (Theorem 4.1), in the proof of which a transportation cost inequality plays an important role. We also briefly discuss the free pressure in relation to the microstate free entropy of projections.

A note on weak amenability for reduced free products of discrete quantum groups

We prove that the Cowling-Haagerup constant of a reduced free product of weakly amenable discrete quantum groups with Cowling-Haagerup constant equal to 1 is again equal to 1.

A note on full free product C*-algebras, lifting and quasidiagonality

We study lifting properties for full product C*-algebras with amalgamation over ${\mathbb C}1$ and give new proofs for some results of Kirchberg and Pisier. We extend the result of Choi on the quasidiagonality of $C^*({\mathbb F}_n)$, proving that the free product with amalgamation over ${\mathbb C}1$ of a family of unital quasidiagonal C*-algebras is quasidiagonal.

A note on unital full amalgamated free products of quasi-diagonal C*-algebras

In the paper, we consider the question whether a unital full amalgamated free product of quasidiagonal C*-algebras is quasidiagonal again. We give a sufficient condition such that a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite dimensional C*- algebra is quasidiagonal. Applying this result, we conclude that a unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there are faithful tracial states on each of these two AF algebras such that the restrictions on the common subalgebra agree.

A Note about proving non-$Γ$ under a finite non-microstates free Fisher information Assumption

We prove that if $X_{1},...,X_{n} (n >1)$ are selfadjoints in a $W^{*}$-probability space with finite non-microstates free Fisher information, then the von Neumann algebra $W^{*}(X_{1},...,X_{n})$ they generate doesn't have property $Γ$ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.