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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Institution: | 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{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered}
\widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\
\widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\
\end{gathered} \right.
$
\widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered}
\widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\
\widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\
\end{gathered} \right.
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Keywords: | |
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