Korean J. Math.  Vol 20, No 3 (2012)  pp.
DOI: https://doi.org/10.11568/kjm.2012.20.3.

DETERMINANT AND SPECTRUM PRESERVING MAPS ON M_n

Sang Og Kim

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. 


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ISSN: 1976-8605 (Print), 2288-1433 (Online)

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