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 $T_n\subseteq M_n$ the corresponding upper-triangular subalgebra. In their influential work, Petek and Šemrl characterize Jordan automorphisms of $M_n$ and $T_n$, when $n\ge 3$, as (injective in the case of $T_n$) continuous commutativity and spectrum preserving maps $\varphi :M_n\to M_n$ and $\varphi :T_n\to T_n$. Recently, in a joint work with Petek, the authors extended this characterization to the maps $\varphi :A\to M_n$, where $A$ is an arbitrary subalgebra of $M_n$ that contains $T_n$. In particular, any such map ϕ is a Jordan embedding and hence of the form $\varphi \left(X\right)=TX{T}^{-1}$ or $\varphi \left(X\right)=T{X}^t{T}^{-1}$, 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 $A$ of $M_n$ that contain all diagonal matrices. More precisely, we provide both a necessary and sufficient condition for an SMA $A\subseteq M_n$ such that any injective continuous commutativity and spectrum preserving map $\varphi :A\to M_n$ 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.