# Matrix Algebras with a Certain Compression Property II

## TL;DR

This paper classifies unital projection compressible subalgebras of complex matrix algebras for dimensions four and higher, showing they are also idempotent compressible and extending previous results to all unital matrix algebras.

## Contribution

It provides a detailed classification of projection compressible subalgebras in higher dimensions and proves the equivalence of projection and idempotent compressibility for all unital matrix algebras.

## Key findings

- All projection compressible subalgebras are idempotent compressible.
- Classification of such subalgebras in $ ext{M}_n( ext{C})$, $n extgreater{}4$.
- Equivalence of projection and idempotent compressibility for all unital matrix algebras.

## Abstract

A subalgebra $\mathcal{A}$ of $\mathbb{M}_n(\mathbb{C})$ is said to be projection compressible if $P\mathcal{A}P$ is an algebra for all orthogonal projections $P\in\mathbb{M}_n(\mathbb{C})$. Likewise, $\mathcal{A}$ is said to be idempotent compressible if $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. In this paper, a case-by-case analysis is used to classify the unital projection compressible subalgebras of $\mathbb{M}_n(\mathbb{C})$, $n\geq 4$, up to transposition and unitary equivalence. It is observed that every algebra shown to admit the projection compression property is, in fact, idempotent compressible. We therefore extend the findings of Cramer, Marcoux, and Radjavi (arXiv:1904.06803 [math.RA]) in the setting of $\mathbb{M}_3(\mathbb{C})$, proving that the two notions of compressibility agree for all unital matrix algebras.