The projection constant for the trace class

Abstract

We study the projection constant of the space of operators on $n$-dimensional Hilbert spaces, with the trace norm, ${\mathcal{S}}_1(n)$. We show an integral formula for the projection constant of ${\mathcal{S}}_1(n)$; namely $\hat{\mathbf{I}}»({\mathcal{S}}_1(n))=n{\int }_{{\mathcal{U}}_n}|\text{tr}(U)|dU,$ where the integration is with respect to the Haar probability measure on the group ${\mathcal{U}}_n$ of unitary operators. Using a probabilistic approach, we derive the limit formula ${\lim }_{n\to \infty }\hat{\mathbf{I}}»({\mathcal{S}}_1(n))/n=\sqrt{\ddot{I}}€/2.$

arXiv admin note: substantial text overlap with arXiv:2208.06467