DETERMINANT AND SPECTRUM PRESERVING MAPS ON M_n

Abstract

Let M_n be the algebra of all complex n × n matrices and φ : M_n → M_n a surjective map (not necessarily additive or multiplicative) satisfying one of the following equations:

det(φ(A)φ(B) + φ(X)) = det(AB + X), A, B, X ∈ Mn, σ(φ(A)φ(B) + φ(X)) = σ(AB + X), A, B, X ∈ M_n.

Then it is an automorphism, where σ(A) is the spectrum of A ∈ Mn. We also show that if A be a standard operator algebra, B is a unital Banach algebra with trivial center and if φ : A → B is a multiplicative surjection preserving spectrum, then φ is an algebra isomorphism.