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