Classification of Jordan multiplicative maps on matrix algebras

2025/03/31 by Gogić, Ilja, Tomašević, Mateo · 1 citation
#16S50 #16W20 #20M25 #47B49 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2503.24094

Abstract

Let M_n(\mathbbF) be the algebra of n × n matrices over a field \mathbbF of characteristic not equal to 2. If n≥ 2, we show that an arbitrary map ϕ: M_n(\mathbbF) → M_n(\mathbbF) is Jordan multiplicative, i.e. it satisfies the functional equation ϕ(XY+YX)=ϕ(X)ϕ(Y)+ϕ(Y)ϕ(X), for all X,Y ∈ M_n(\mathbbF) if and only if one of the following holds: either ϕ is constant, equal to P/2 for some idempotent P ∈ M_n(\mathbbF), or there exists an invertible matrix T ∈ M_n(\mathbbF) and a ring monomorphism ω: \mathbbF → \mathbbF such that ϕ(X)=Tω(X)T^-1 or ϕ(X)=Tω(X)^tT^-1, for all X ∈ M_n(\mathbbF), where ω(X) denotes the matrix obtained by applying ω entrywise to X. In particular, any Jordan multiplicative map ϕ: M_n(\mathbbF) → M_n(\mathbbF) with ϕ(0)=0 is automatically additive. The analogous characterization fails when \mathbbF has characteristic 2.

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