Maps on matrices preserving idempotency of a triadic relation

Abstract

Let M_n be the algebra of all n×n real or complex matrices. In this paper, we give a full description of continuous maps on M_n such that Φ(A)(Φ(B)− Φ(C)) is idempotent if and only if A(B − C) is idempotent for all A, B, C ∈ M_n.