On Voiculescu's Semicircular Matrices

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