On the shape of correlation matrices for unitaries - Mori, Michiya (MATHEMATICA SCANDINAVICA, 2024)
For a positive integer $n$, we study the collection $\mathcal {F}_{fin}(n)$ formed of all $n× n$ matrices whose entries $a_{ij}$, $1≤ i,j≤ n$, can be written as $a_{ij}=τ (U_j^*U_i)$ for some $n$-tuple $U_1, U_2, …, U_n$ of unitaries in a finite-dimensional von Neumann algebra $\mathcal {M}$ with tracial state τ. We show that $\mathcal {F}_{fin}(n)$ is not closed for every $n≥ 8$. This improves a result by Musat and Rørdam which states the same for $n≥ 11$.