An extension of Petek-\v{S}emrl preserver theorems for Jordan embeddings of structural matrix algebras

TL;DR

This paper extends Petek-emrl's theorems on Jordan automorphisms from matrix algebras to structural matrix algebras, characterizing when spectrum and commutativity preserving maps are Jordan embeddings.

Contribution

It provides necessary and sufficient conditions for injective spectrum and commutativity preserving maps on structural matrix algebras to be Jordan embeddings, broadening previous results.

Findings

Characterization of maps as Jordan embeddings under new conditions

Extension of previous theorems to structural matrix algebras containing all diagonals

Maps need not be multiplicative or rank-one preservers in this setting

Abstract

Let $M_n$ be the algebra of $n \times n$ complex matrices and $\mathcal{T}_n \subseteq M_n$ the corresponding upper-triangular subalgebra. In their influential work, Petek and \v{S}emrl characterize Jordan automorphisms of $M_n$ and $\mathcal{T}_n$, when $n \geq 3$, as (injective in the case of $\mathcal{T}_n$) continuous commutativity and spectrum preserving maps $\phi : M_n \to M_n$ and $\phi : \mathcal{T}_n \to \mathcal{T}_n$. Recently, in a joint work with Petek, the authors extended this characterization to the maps $\phi : \mathcal{A} \to M_n$, where $\mathcal{A}$ is an arbitrary subalgebra of $M_n$ that contains $\mathcal{T}_n$. In particular, any such map $\phi$ is a Jordan embedding and hence of the form $\phi(X)=TXT^{-1}$ or $\phi(X)=TX^tT^{-1}$, for some invertible matrix $T\in M_n$. In this paper we further extend the aforementioned results in the context of structural matrix algebras (SMAs), i.e. subalgebras $\mathcal{A}$ of $M_n$ that contain all diagonal matrices. More precisely, we provide both a necessary and sufficient condition for an SMA $\mathcal{A}\subseteq M_n$ such that any injective continuous commutativity and spectrum preserving map $\phi: \mathcal{A} \to M_n$ is necessarily a Jordan embedding. In contrast to the previous cases, such maps $\phi$ no longer need to be multiplicative/antimultiplicative, nor rank-one preservers.