On classical adjoint-commuting mappings between matrix algebras - Chooi, W.L.; Ng, W.S. (Elsevier, 2010)
Let F be a field and let m and n be integers with m, n >= 3. Let M(n) denote the algebra of n x n matrices over F. In this note, we characterize mappings psi : M(n) -> M(m) that satisfy one of the following conditions: 1. vertical bar F vertical bar = 2 or vertical bar F vertical bar > n + 1, and psi (adj (A + alpha B)) = adj (psi (A) + alpha psi (B)) for all A, B is an element of M(n) and alpha is an element of F with psi (I(n)) not equal 0. 2. psi is surjective and psi (adj (A - B)) = adj (psi (A) - psi (B)) for every A, B is an element of M(n). Here, adj A denotes the classical adjoint of the matrix A, and I(n) is the identity matrix of order n. We give examples showing the indispensability of the assumption psi (I(n)) not equal 0 in our results. (C) 2009 Elsevier Inc. All rights reserved.