An extension of Petek-Šemrl preserver theorems for Jordan embeddings of structural matrix algebras

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 $n\ge 3$, as (injective in the case of ${\mathcal{T}}_n$) continuous commutativity and spectrum preserving maps $\varphi :M_n\to M_n$ and $\varphi :{\mathcal{T}}_n\to {\mathcal{T}}_n$. Recently, in a joint work with Petek, the authors extended this characterization to the maps $\varphi :\mathcal{A}\to M_n$, where $\mathcal{A}$ is an arbitrary subalgebra of $M_n$ that contains ${\mathcal{T}}_n$. In particular, any such map $\varphi$ is a Jordan embedding and hence of the form $\varphi (X)=TX{T}^{-1}$ or $\varphi (X)=T{X}^t{T}^{-1}$, 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 $\varphi :\mathcal{A}\to M_n$ is necessarily a Jordan embedding. In contrast to the previous cases, such maps $\varphi$ no longer need to be multiplicative/antimultiplicative, nor rank-one preservers.

22 pages, to appear in J. Math. Anal. Appl

References in corpus (3)

• Potent preservers of incidence algebras
• Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties
• Jordan embeddings and linear rank preservers of structural matrix algebras

Cited by in corpus (1)

• Multiplicative and Jordan multiplicative maps on structural matrix algebras