Maps on matrix algebras preserving idempotents

摘要

Let M-n be the algebra of all n x n complex matrices and P-n the set of all idempotents in M-n. Suppose phi : M-n --> M-n is a surjective map satisfying A - lambdaB is an element of P-n if and only if phi(A) - lambdaphi(B) is an element of P-n, A, B is an element of M-n, lambda is an element of C. Then either phi is of the form phi(A) = TAT(-1), A is an element of M-n, or phi is of the form phi(A) = TA(t)T(-1), A is an element of M-n, where T is an element of M-n is a nonsingular matrix. (C) 2003 Elsevier Inc. All rights reserved. [References: 15]