On the shape of correlation matrices for unitaries

Abstract

For a positive integer $n$, we study the collection ${\mathcal{F}}_{\mathrm{fin}}(n)$ formed of all $n\times n$ matrices whose entries $a_{ij}$, $1\le i,j\le n$, can be written as $a_{ij}=\tau (U_j^{\ast }{U}_i)$ for some $n$-tuple $U_1,U_2,\dots ,U_n$ of unitaries in a finite-dimensional von Neumann algebra $\mathcal{M}$ with tracial state $\tau$. We show that ${\mathcal{F}}_{\mathrm{fin}}(n)$ is not closed for every $n\ge 8$. This improves a result by Musat and Rørdam which states the same for $n\ge 11$.

4 pages