# Matrix Algebras with a Certain Compression Property I

## TL;DR

This paper investigates special properties of matrix algebras related to their behavior under compression by idempotents and projections, providing examples and a complete classification for 3x3 matrices.

## Contribution

It constructs examples of unital algebras with these compression properties and classifies all unital idempotent compressible subalgebras of 3x3 matrices up to similarity and transposition.

## Key findings

- Unital algebras with compression properties are constructed.
- Complete classification of 3x3 unital idempotent compressible subalgebras.
- Projection and idempotent compressibility are equivalent in 3x3 case.

## Abstract

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.