MATRIX ALGEBRAS WITH A CERTAIN COMPRESSION PROPERTY I

Abstract

An algebra A of n × n complex matrices is said to be projection compressible if P AP is an algebra for all orthogonal projections P ∈ Mn(C). Analogously, A is said to be idempotent compressible if EAE is an algebra for all idempotents E in Mn(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 M3(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 M3(C) is projection compressible if and only if it is idempotent compressible. Our findings are extended to algebras of arbitrary size in [2]