An extension of Petek-Šemrl preserver theorems for Jordan embeddings of structural matrix algebras
Ilja Gogić, Mateo Tomašević
TL;DR
This work extends the Petek–Šemrl preserver theorems from full matrix algebras and upper-triangular subalgebras to structural matrix algebras (SMAs) that contain all diagonal matrices. It proves a precise, testable condition (i) on an SMA $_ ho$ that guarantees every injective continuous commutativity- and spectrum-preserving map $_ ho o M_n$ is a Jordan embedding, thereby generalizing Jordan-embedding characterizations to a broad SMA class. The approach relies on analyzing the norm-closure of rank-one diagonalizable matrices in SMAs, employing the $(ullet)^{ lat S}$ and $(ullet)^{ lat S^c}$ constructions, and leveraging existing SMA Jordan-embedding descriptions to extend parabolic and rank-preservation arguments. The results delineate when Jordan-embedding structure persists in SMAs and illustrate the necessity of the hypotheses via counterexamples, highlighting the boundary between classical preservers and more general SMA mappings.
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 Šemrl 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 $φ: M_n \to M_n$ and $φ: \mathcal{T}_n \to \mathcal{T}_n$. Recently, in a joint work with Petek, the authors extended this characterization to the maps $φ: \mathcal{A} \to M_n$, where $\mathcal{A}$ is an arbitrary subalgebra of $M_n$ that contains $\mathcal{T}_n$. In particular, any such map $φ$ is a Jordan embedding and hence of the form $φ(X)=TXT^{-1}$ or $φ(X)=TX^tT^{-1}$, 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 $φ: \mathcal{A} \to M_n$ is necessarily a Jordan embedding. In contrast to the previous cases, such maps $φ$ no longer need to be multiplicative/antimultiplicative, nor rank-one preservers.