Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras

• Autorzy: Osamu Hatori , Go Hirasawa , Takeshi Miura
• Języki publikacji: EN

Abstrakty

EN

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 $$ \widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right. $$ for all a ∈ A, where e is unit element of A. If, in addition, $$ \widehat{T\left( e \right)} = 1 $$ and $$ \widehat{T\left( {ie} \right)} = i $$ on M B, then T is an algebra isomorphism.

Słowa kluczowe

EN

Uniform algebra; Commutative Banach algebra; Maximal ideal space; Shilov boundary; Algebra isomorphism; Norm-additive operator; Norm-linear operator

Kategorie tematyczne

• 46J10: Banach algebras of continuous functions, function algebras

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 3
• Strony: 597-601

Daty

wydano

2010-06-01

online

2010-05-30

Twórcy

autor

Osamu Hatori

• Niigata University

autor

Go Hirasawa

• Ibaraki University

autor

Takeshi Miura

• Yamagata University

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Identyfikatory

DOI

10.2478/s11533-010-0025-4