Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras |
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| Authors: | Osamu Hatori Go Hirasawa Takeshi Miura |
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| Affiliation: | 1.Department of Mathematics, Faculty of Science,Niigata University,Niigata,Japan;2.Faculty of Engineering,Ibaraki University,Hitachi,Japan;3.Department of Applied Mathematics and Physics,Yamagata University,Yonezawa,Japan |
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| Abstract: | Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B , respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M B → M A and a closed and open subset K of M B such that | $
widehat{Tleft( a right)}left( y right) = left{ begin{gathered}
widehat{Tleft( e right)}left( y right)hat aleft( {phi left( y right)} right) y in K hfill
widehat{Tleft( e right)}left( y right)overline {hat aleft( {phi left( y right)} right)} y in M_mathcal{B} backslash K hfill
end{gathered} right.
$
widehat{Tleft( a right)}left( y right) = left{ begin{gathered}
widehat{Tleft( e right)}left( y right)hat aleft( {phi left( y right)} right) y in K hfill
widehat{Tleft( e right)}left( y right)overline {hat aleft( {phi left( y right)} right)} y in M_mathcal{B} backslash K hfill
end{gathered} right.
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