Multiplicatively Spectrum-preserving and Norm-preserving Maps between Invertible Groups of Commutative Banach Algebras
نویسندگان
چکیده
Let A and B be unital semisimple commutative Banach algebras and T a map from the invertible group A onto B. Linearity and multiplicativity of the map are not assumed. We consider the hypotheses on T : (1) σ(TfTg) = σ(fg); (2) σπ(TfTg−α)∩σπ(fg−α) 6= ∅; (3) r(TfTg−α) = r(fg−α) hold for some non-zero complex number α and for every f, g ∈ A, where σ(·) (resp. σπ(·)) denotes the (resp. peripheral) spectrum and r(·) denotes the spectral radius. Under each of the hypotheses we show representations for T and under additional assumptions we show that T is extended to an algebra isomorphism. In particular, if T is a surjective group homomorphism such that T preserves the spectrum or T is a surjective isometry with respect to the spectral radius, then T is extended to an algebra isomorphism. Similar results holds for maps from A onto B.
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