Consider the matrix algebra $\mathbb{M}_n(\mathbb{C})$ with H-S inner producr ($\langle a, b\rangle =tr (a^*b)$). What is the minimal dimension of any operator system $\mathcal{A}$ in $\mathbb{M}_n(\mathbb{C})$ such that it always contains an one dimensional projection.
Further, if for such an $\mathcal{A} $ with dimension $m$, can we guarantee that there will be a set of mutually orthogonal projections spanning $\mathcal{A}$?
Advanced thanks for any comments, suggestions, references, and etc..