Multiplicative and Jordan multiplicative maps on structural matrix algebras

2025/03/18 by Gogić, Ilja, Tomašević, Mateo · 1 citation
#16S50 #16W20 #20M25 #47B49 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2503.14116

Abstract

Let M_n denote the algebra of n × n complex matrices and let A⊆ 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: A→ M_n that satisfy the condition ϕ(X \bullet Y) = ϕ(X) \bullet ϕ(Y), for all X,Y ∈ A, where \bullet is either the standard matrix multiplication (X,Y)↦ XY, the Jordan product (X,Y) ↦ XY+YX, or the normalized Jordan product (X,Y) ↦ (1)/(2)(XY+YX). We show that all such maps ϕ are automatically additive if and only if A does not contain a central rank-one idempotent. Moreover, in this case, we fully characterize the form of these maps.

Cited by

• The structure of k-potents and mixed Jordan-power preservers on matrix algebras

Related

• Classification of Jordan multiplicative maps on matrix algebras
• An approach between the multiplicative and additive structure of a Jordan ring
• The structure of k-potents and mixed Jordan-power preservers on matrix algebras
• Multiplicative maps on matrix algebras
• Jordan maps on standard operator algebras