Multiplicative and Jordan multiplicative maps on structural matrix algebras

TL;DR

The paper addresses when injective maps between structural matrix algebras and $M_n$ that preserve either standard multiplication or Jordan-type products are automatically additive. Building on the Jodeit–Lam and Molnar results, it extends the analysis to SMAs via a central decomposition into central classes and a detailed study of idempotents, rank-one projections, and associated endomorphisms. The main contribution is a complete equivalence: the absence of central rank-one idempotents guarantees additivity, and such maps admit an explicit form involving a global invertible conjugation, a transitive scaling $g^*$, and, for each central class, a ring endomorphism $\omega_C$ with a local parity choice $\dagger_C$. The work also clarifies the limits of extending these results to arbitrary unital subalgebras and discusses potential extensions to other fields, anchoring the results in a robust SMA framework with central decompostion and rigid structure-preserving maps.

Abstract

Let $M_n$ denote the algebra of $n \times n$ complex matrices and let $\mathcal{A}\subseteq M_n$ be an arbitrary structural matrix algebra, i.e. a subalgebra of $M_n$ that contains all diagonal matrices. We consider injective maps $φ: \mathcal{A}\to M_n$ that satisfy the condition $$ φ(X \bullet Y) = φ(X) \bullet φ(Y), \quad \text{for all } X,Y \in \mathcal{A}, $$ where $\bullet$ is either the standard matrix multiplication $(X,Y)\mapsto XY$, the Jordan product $(X,Y) \mapsto XY+YX$, or the normalized Jordan product $(X,Y) \mapsto \frac{1}{2}(XY+YX)$. We show that all such maps $φ$ are automatically additive if and only if $\mathcal{A}$ does not contain a central rank-one idempotent. Moreover, in this case, we fully characterize the form of these maps.

Theorems & Definitions (25)

• Lemma 2.1
• proof
• Theorem 2.2: GogicTomasevic
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof