# Multiplicatively and non-symmetric multiplicatively norm-preserving maps

Open Mathematics (2010)

• Volume: 8, Issue: 5, page 878-889
• ISSN: 2391-5455

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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, $Tf\left(y\right)=\left\{\begin{array}{c}\eta \left(y\right)f\left(\phi \left(y\right)\right)y\in K,\\ -\frac{\alpha }{\left|\alpha \right|}\eta \left(y\right)\overline{f\left(\phi \left(y\right)\right)}y\in c\left(B\right)\setminus K\end{array}\right$/extract_itex] . In particular, if T satisfies the stronger condition R π(fg + α) = R π(Tf Tg + α), where R π(.) denotes the peripheral range of algebra elements, then Tf(y) = T1(y)f(φ(y)), y ∈ c(B), for some homeomorphism φ: c(B) → c(A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying ‖Tf Tg‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations. ## How to cite top Maliheh Hosseini, and Fereshteh Sady. "Multiplicatively and non-symmetric multiplicatively norm-preserving maps." Open Mathematics 8.5 (2010): 878-889. <http://eudml.org/doc/269046>. @article{MalihehHosseini2010, 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\lbrace \{\begin\{array\}\{c\}\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\{array\}\} \right.$ . In particular, if T satisfies the stronger condition R π(fg + α) = R π(Tf Tg + α), where R π(.) denotes the peripheral range of algebra elements, then Tf(y) = T1(y)f(φ(y)), y ∈ c(B), for some homeomorphism φ: c(B) → c(A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying ‖Tf Tg‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations.},
author = {Maliheh Hosseini, Fereshteh Sady},
journal = {Open Mathematics},
keywords = {Banach function algebra; Range-preserving map; Multiplicatively norm-preserving; Peaking function; Peripheral range; Peripheral spectrum; range-preserving map; multiplicatively norm preserving; peaking function; peripheral range; peripheral spectrum},
language = {eng},
number = {5},
pages = {878-889},
title = {Multiplicatively and non-symmetric multiplicatively norm-preserving maps},
url = {http://eudml.org/doc/269046},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Maliheh Hosseini
TI - Multiplicatively and non-symmetric multiplicatively norm-preserving maps
JO - Open Mathematics
PY - 2010
VL - 8
IS - 5
SP - 878
EP - 889
AB - 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, $Tf\left( y \right) = \left\lbrace {\begin{array}{c}\eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\ - \frac{\alpha }{{\left| \alpha \right|}}\eta \left( y \right)\overline{f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\ \end{array}} \right.$ . In particular, if T satisfies the stronger condition R π(fg + α) = R π(Tf Tg + α), where R π(.) denotes the peripheral range of algebra elements, then Tf(y) = T1(y)f(φ(y)), y ∈ c(B), for some homeomorphism φ: c(B) → c(A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying ‖Tf Tg‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations.
LA - eng
KW - Banach function algebra; Range-preserving map; Multiplicatively norm-preserving; Peaking function; Peripheral range; Peripheral spectrum; range-preserving map; multiplicatively norm preserving; peaking function; peripheral range; peripheral spectrum
UR - http://eudml.org/doc/269046
ER -

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