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

Abstract

Let $M_n$ be the algebra of $n\times n$ complex matrices. We consider arbitrary subalgebras $A$ of $M_n$ which contain the algebra of all upper-triangular matrices (i.e. block upper-triangular subalgebras), and their Jordan embeddings. We first describe Jordan embeddings $\varphi :A\to M_n$ as maps of the form $\varphi \left(X\right)=TX{T}^{-1}$ or $\varphi \left(X\right)=T{X}^t{T}^{-1}$, where $T\in M_n$ is an invertible matrix, and then we obtain a simple criteria 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 $\varphi :A\to M_n$ (when $n\ge 3$) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless $A=M_n$ when injectivity is superfluous).