DETERMINANT AND SPECTRUM PRESERVING MAPS ON Mn
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Let Mn be the algebra of all complex n × n matrices and ϕ : Mn → Mn 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 ∈ Mn. 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.
Abstract
Let Mn be the algebra of all complex n × n matrices and ϕ : Mn → Mn 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 ∈ Mn. 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.
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