  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.
