Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties

Abstract

Let Mn be the algebra of n×n complex matrices. We consider arbitrary subalgebras A of Mn which contain the algebra of all upper-triangular matrices, and their Jordan embeddings. We first describe Jordan embeddings ϕ:A→Mn as maps of the formϕ(X)=TXT−1orϕ(X)=TXtT−1, where T∈Mn is an invertible matrix, and then we obtain a simple criterion of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings ϕ:A→Mn (when n≥3) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless A=Mn when injectivity is superfluous).