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On Voiculescu's Semicircular Matrices

Junhao Shen
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Abstract

Assume $\N$ is a von Neumann algebra of type II$_1$ with a tracial state $τ_{\N}$, and $\M$ is the von Neumann algebra of the $n\times n$ matrices over $\N$ 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 $\M$.

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Junhao Shen. 2005-10-03. On Voiculescu's Semicircular Matrices. https://arxiv.org/abs/math/0508306

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