Maps between Banach function algebras satisfying certain norm conditions

Maliheh Hosseini; Fereshteh Sady

Open Mathematics (2013)

  • Volume: 11, Issue: 6, page 1020-1033
  • ISSN: 2391-5455

Abstract

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Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let A ¯ and B ¯ be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ℂ{0, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)−α‖Y=‖ρ(p)τ(p′) − α‖x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M B and a homeomorphism ϕ: M B → M A between the maximal ideal spaces of B and A such that for all f ∈ A, [...] where · ^ denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from A ¯ onto B ¯ inducing a homeomorphism between M B ¯ and M A ¯ . We also show that under an additional assumption related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider the case where α = 0 and X and Y are locally compact.

How to cite

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Maliheh Hosseini, and Fereshteh Sady. "Maps between Banach function algebras satisfying certain norm conditions." Open Mathematics 11.6 (2013): 1020-1033. <http://eudml.org/doc/269472>.

@article{MalihehHosseini2013,
abstract = {Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let $\bar\{A\}$ and $\bar\{B\}$ be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ℂ\{0, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)−α‖Y=‖ρ(p)τ(p′) − α‖x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M B and a homeomorphism ϕ: M B → M A between the maximal ideal spaces of B and A such that for all f ∈ A, [...] where $\hat\{\cdot \}$ denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from $\bar\{A\}$ onto $\bar\{B\}$ inducing a homeomorphism between $M_\{\bar\{B\}\}$ and $M_\{\bar\{A\}\}$ . We also show that under an additional assumption related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider the case where α = 0 and X and Y are locally compact. },
author = {Maliheh Hosseini, Fereshteh Sady},
journal = {Open Mathematics},
keywords = {Banach function algebras; Uniform algebras; Norm-preserving; Peripheral range; Choquet boundary; uniform algebras; norm-preserving; peripheral range},
language = {eng},
number = {6},
pages = {1020-1033},
title = {Maps between Banach function algebras satisfying certain norm conditions},
url = {http://eudml.org/doc/269472},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Maliheh Hosseini
AU - Fereshteh Sady
TI - Maps between Banach function algebras satisfying certain norm conditions
JO - Open Mathematics
PY - 2013
VL - 11
IS - 6
SP - 1020
EP - 1033
AB - Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let $\bar{A}$ and $\bar{B}$ be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ℂ{0, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)−α‖Y=‖ρ(p)τ(p′) − α‖x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M B and a homeomorphism ϕ: M B → M A between the maximal ideal spaces of B and A such that for all f ∈ A, [...] where $\hat{\cdot }$ denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from $\bar{A}$ onto $\bar{B}$ inducing a homeomorphism between $M_{\bar{B}}$ and $M_{\bar{A}}$ . We also show that under an additional assumption related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider the case where α = 0 and X and Y are locally compact.
LA - eng
KW - Banach function algebras; Uniform algebras; Norm-preserving; Peripheral range; Choquet boundary; uniform algebras; norm-preserving; peripheral range
UR - http://eudml.org/doc/269472
ER -

References

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