Jordan embeddings and linear rank preservers of structural matrix algebras

TL;DR

This paper characterizes Jordan embeddings and rank preservers of structural matrix algebras, extending known automorphism results and providing a complete description of linear rank preservers within these algebras.

Contribution

It introduces new characterizations of Jordan embeddings and automorphisms of SMAs, and fully describes linear rank preservers as specific algebraic maps.

Findings

Jordan automorphisms of SMAs are characterized and generalized.

Any unital linear rank-one preserver on an SMA is a Jordan embedding.

Complete description of linear rank preservers as maps involving invertible matrices and central idempotents.

Abstract

We consider subalgebras $\mathcal{A}$ of the algebra $M_n$ of $n \times n$ complex matrices that contain all diagonal matrices, known in the literature as the structural matrix algebras (SMAs). Let $\mathcal{A} \subseteq M_n$ be an arbitrary SMA. We first show that any commuting family of diagonalizable matrices in $\mathcal{A}$ can be intrinsically simultaneously diagonalized (i.e. the corresponding similarity can be chosen from $\mathcal{A}$). Using this, we then characterize when one SMA Jordan-embeds into another and in that case we describe the form of such Jordan embeddings. As a consequence, we obtain a description of Jordan automorphisms of SMAs, generalizing Coelho's result on their algebra automorphisms. Next, motivated by the results of Marcus-Moyls and Molnar-\v{S}emrl, connecting the linear rank-one preservers with Jordan embeddings $M_n \to M_n$ and $\mathcal{T}_n \to M_n$ (where $\mathcal{T}_n$ is the algebra of $n \times n$ upper-triangular matrices) respectively, we show that any linear unital rank-one preserver $\mathcal{A} \to M_n$ is necessarily a Jordan embedding. As the converse fails in general, we also provide a necessary and sufficient condition for when it does hold true. Finally, we obtain a complete description of linear rank preservers $\mathcal{A} \to M_n$, as maps of the form $X\mapsto S\left(PX + (I-P)X^t\right)T$, for some invertible matrices $S,T \in M_n$ and a central idempotent $P\in\mathcal{A}$.