Norm conditions for real-algebra isomorphisms between uniform algebras

Rumi Shindo

Open Mathematics (2010)

  • Volume: 8, Issue: 1, page 135-147
  • ISSN: 2391-5455

Abstract

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Let A and B be uniform algebras. Suppose that α ≠ 0 and A 1 ⊂ A. Let ρ, τ: A 1 → A and S, T: A 1 → B be mappings. Suppose that ρ(A 1), τ(A 1) and S(A 1), T(A 1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A 1, S(e 1)−1 ∈ S(A 1) and S(e 1) ∈ T(A 1) for some e 1 ∈ A 1 with ρ(e 1) = 1, then there exists a real-algebra isomorphism S ˜ : A → B such that S ˜ (ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications of this result.

How to cite

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Rumi Shindo. "Norm conditions for real-algebra isomorphisms between uniform algebras." Open Mathematics 8.1 (2010): 135-147. <http://eudml.org/doc/269806>.

@article{RumiShindo2010,
abstract = {Let A and B be uniform algebras. Suppose that α ≠ 0 and A 1 ⊂ A. Let ρ, τ: A 1 → A and S, T: A 1 → B be mappings. Suppose that ρ(A 1), τ(A 1) and S(A 1), T(A 1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A 1, S(e 1)−1 ∈ S(A 1) and S(e 1) ∈ T(A 1) for some e 1 ∈ A 1 with ρ(e 1) = 1, then there exists a real-algebra isomorphism \[ \tilde\{S\} \] : A → B such that \[ \tilde\{S\} \] (ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications of this result.},
author = {Rumi Shindo},
journal = {Open Mathematics},
keywords = {Banach algebra; Uniform algebra; Norm-preserving; uniform algebra; isomorphisms; norm preserving},
language = {eng},
number = {1},
pages = {135-147},
title = {Norm conditions for real-algebra isomorphisms between uniform algebras},
url = {http://eudml.org/doc/269806},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Rumi Shindo
TI - Norm conditions for real-algebra isomorphisms between uniform algebras
JO - Open Mathematics
PY - 2010
VL - 8
IS - 1
SP - 135
EP - 147
AB - Let A and B be uniform algebras. Suppose that α ≠ 0 and A 1 ⊂ A. Let ρ, τ: A 1 → A and S, T: A 1 → B be mappings. Suppose that ρ(A 1), τ(A 1) and S(A 1), T(A 1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A 1, S(e 1)−1 ∈ S(A 1) and S(e 1) ∈ T(A 1) for some e 1 ∈ A 1 with ρ(e 1) = 1, then there exists a real-algebra isomorphism \[ \tilde{S} \] : A → B such that \[ \tilde{S} \] (ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications of this result.
LA - eng
KW - Banach algebra; Uniform algebra; Norm-preserving; uniform algebra; isomorphisms; norm preserving
UR - http://eudml.org/doc/269806
ER -

References

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