1 Commutativity preserving maps on real matrices

By Ajda Fošner.

Let M_{n}(ℝ) be the algebra of all n × n real matrices. A map ϕ : M_{n}(ℝ) → M_{n}(ℝ) preserves commutativity if ϕ(A)ϕ(B) = ϕ(B)ϕ(A) whenever AB = BA, A, B ∈ M_{n}(ℝ). If ϕ is bijective and both ϕ and ϕ^{−1} preserve commutativity, then we say that ϕ preserves commutativity in both directions. We will talk about non-linear maps on M_{n}(ℝ) that preserve commutativity in both directions or in one direction only.