Real-linear isometries between function algebras

Takeshi Miura

Open Mathematics (2011)

  • Volume: 9, Issue: 4, page 778-788
  • ISSN: 2391-5455

Abstract

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Let A and B be uniformly closed function algebras on locally compact Hausdorff spaces with Choquet boundaries Ch A and ChB, respectively. We prove that if T: A → B is a surjective real-linear isometry, then there exist a continuous function κ: ChB → z ∈ ℂ: |z| = 1, a (possibly empty) closed and open subset K of ChB and a homeomorphism φ: ChB → ChA such that T(f) = κ(f ∘φ) on K and T f = κ f o φ ¯ on ChB K for all f ∈ A. Such a representation holds for surjective real-linear isometries between (not necessarily uniformly closed) function algebras.

How to cite

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Takeshi Miura. "Real-linear isometries between function algebras." Open Mathematics 9.4 (2011): 778-788. <http://eudml.org/doc/269446>.

@article{TakeshiMiura2011,
abstract = {Let A and B be uniformly closed function algebras on locally compact Hausdorff spaces with Choquet boundaries Ch A and ChB, respectively. We prove that if T: A → B is a surjective real-linear isometry, then there exist a continuous function κ: ChB → z ∈ ℂ: |z| = 1, a (possibly empty) closed and open subset K of ChB and a homeomorphism φ: ChB → ChA such that T(f) = κ(f ∘φ) on K and $T\left( f \right) = \kappa \overline\{fo\phi \}$ on ChB K for all f ∈ A. Such a representation holds for surjective real-linear isometries between (not necessarily uniformly closed) function algebras.},
author = {Takeshi Miura},
journal = {Open Mathematics},
keywords = {Commutative Banach algebra; Function algebra; Isometry; Isomorphism; Uniform algebra; function algebra; Choquet boundary; surjection; isometry; isomorphism; homeomorphism},
language = {eng},
number = {4},
pages = {778-788},
title = {Real-linear isometries between function algebras},
url = {http://eudml.org/doc/269446},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Takeshi Miura
TI - Real-linear isometries between function algebras
JO - Open Mathematics
PY - 2011
VL - 9
IS - 4
SP - 778
EP - 788
AB - Let A and B be uniformly closed function algebras on locally compact Hausdorff spaces with Choquet boundaries Ch A and ChB, respectively. We prove that if T: A → B is a surjective real-linear isometry, then there exist a continuous function κ: ChB → z ∈ ℂ: |z| = 1, a (possibly empty) closed and open subset K of ChB and a homeomorphism φ: ChB → ChA such that T(f) = κ(f ∘φ) on K and $T\left( f \right) = \kappa \overline{fo\phi }$ on ChB K for all f ∈ A. Such a representation holds for surjective real-linear isometries between (not necessarily uniformly closed) function algebras.
LA - eng
KW - Commutative Banach algebra; Function algebra; Isometry; Isomorphism; Uniform algebra; function algebra; Choquet boundary; surjection; isometry; isomorphism; homeomorphism
UR - http://eudml.org/doc/269446
ER -

References

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