Matrix algebras with a certain compression property I

An algebra $\mathcal{A}$ of $n\times n$ complex matrices is said to be \textit{idempotent compressible} if $E\mathcal{A}E$ an for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. Analogously, \textit{projection $P\mathcal{A}P$ orthogonal projections $P$ in $\mathbb{M}_n(\mathbb{C})$. In this paper we construct several examples unital algebras that admit these properties. addition, a complete classification the idempotent compressible subalgebras $\mathbb{M}_3(\mathbb{C})$ obtained up similarity and transposition. It shown setting, two notions compressibility agree: subalgebra projection only it compressible. Our findings are extended arbitrary size sequel paper.