On Voiculescu's Semicircular Matrices

Abstract

Assume $\mathbb{N}$ is a von Neumann algebra of type II${}_1$ with a tracial state ${\tau }_{\mathbb{N}}$, and $\M$ is the von Neumann algebra of the $n\times n$ matrices over $\mathbb{N}$ with the canonical tracial state $τ_{\M}$. Let ${\mathcal{D}}_n$ be the subalgebra of $\M$ consisting of scalar diagonal matrices in $\M$. In this article, we study the properties of semicircular elements in $\M$ that are free from ${\mathcal{D}}_n$ with respect to $τ_{\M}$. Then we define a new concept "matricial distance" of two elements in $\M$ and compute the matricial distance between two free semicircular elements in $\M$.