The projection constant for the trace class

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 $\mathit{\lambda }({\mathcal{S}}_1(n))=n{\int }_{{\mathcal{U}}_n}\mathrm{\mid }\text{tr}(U)\mathrm{\mid }dU,$ 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 ${\lim }_{n\to \mathrm{\infty }}\mathit{\lambda }({\mathcal{S}}_1(n))\mathrm{/}n=\sqrt{\pi }\mathrm{/}2\mathrm{.}$