Matrix Algebras with a Certain Compression Property I

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.

24 pages

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• Matrix Algebras with a Certain Compression Property II

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• Matrix Algebras with a Certain Compression Property II