Maximality of the microstates free entropy for R-diagonal elements

Absence of algebraic relations and of zero divisors under the assumption of full non-microstates free entropy dimension

We show that in a tracial and finitely generated $W^\ast$-probability space existence of conjugate variables excludes algebraic relations for the generators. Moreover, under the assumption of maximal non-microstates free entropy dimension, we prove that there are no zero divisors in the sense that the product of any non-commutative polynomial in the generators with any element from the von Neumann algebra is zero if and only if at least one of those factors is zero. In particular, this shows that in this case the distribution of any non-constant self-adjoint non-commutative polynomial in the g…

Fractal entropies and dimensions for microstate spaces, II

For a selfadjoint element x in a tracial von Neumann algebra and $α= δ_0(x)$ we compute bounds for $\mathbb H^α(x),$ where $\mathbb H^α(x)$ is the free Hausdorff $α$-entropy of $x.$ The bounds are in terms of $\int \int_{\mathbb R^2 -D} \log |y-z| dμ(y) dμ(z)$ where $μ$ is the Borel measure on the spectrum of x induced by the trace and $D \subset \mathbb R^2$ is the diagonal. We compute similar bounds for the free Hausdorff entropy of a free family of selfadjoints.

关于完全正映射的双自由熵

Bi-Free Entropy with Respect to Completely Positive Maps

本文构造了关于完全正映射的非微观态双自由熵概念，通过扩展算子值双自由结构建立共轭变量，并将自由Fisher信息和熵扩展至该设置，证明在双R对角元素处取得极值。

The free entropy dimension of some $L^{\infty}[0,1]$-circular operators

We find the microstates free entropy dimension of a large class of $L^{\infty}[0,1]$-circular operators, in the presence of a generator of the diagonal subalgebra.

Rectangular random matrices, related free entropy and free Fisher's information

We prove that independent rectangular random matrices, when embedded in a space of larger square matrices, are asymptotically free with amalgamation over a commutative finite dimensional subalgebra $D$ (under an hypothesis of unitary invariance). Then we consider elements of a finite von Neumann algebra containing $D$, which have kernel and range projection in $D$. We associate them a free entropy with the microstates approach, and a free Fisher's information with the conjugate variables approach. Both give rise to optimization problems whose solutions involve freeness with amalgamation over $…

强1-有界性、L²-Betti数、代数soficity与图积

Strong 1-boundedness, $L^2$-Betti numbers, algebraic soficity, and graph products

本文证明当底层*-代数的第一L²-Betti数为零时，非平凡有限维von Neumann代数的图积具有强1-有界性，方法结合代数soficity概念与置换模型的概率构造。