Multiplicatively and non-symmetric multiplicatively norm-preserving maps |
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Authors: | Maliheh Hosseini Fereshteh Sady |
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Institution: | 1. Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran
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Abstract: | Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖
X
and ‖.‖
Y
denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖
Y
= ‖fg‖
X
, for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖
X
= ‖Tf Tg + α‖
Y
, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each f ∈ A,
$
Tf\left( y \right) = \left\{ \begin{gathered}
\eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\
- \frac{\alpha }
{{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\
\end{gathered} \right.
$
Tf\left( y \right) = \left\{ \begin{gathered}
\eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\
- \frac{\alpha }
{{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\
\end{gathered} \right.
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Keywords: | |
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