Boundaries of weak peak points in noncommutative algebras of Lipschitz functions

Kassandra Averill; Ann Johnston; Ryan Northrup; Robert Silversmith; Aaron Luttman

Open Mathematics (2012)

  • Volume: 10, Issue: 2, page 646-655
  • ISSN: 2391-5455

Abstract

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It has been shown that any Banach algebra satisfying ‖f 2‖ = ‖f‖2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, 𝔽 ) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where 𝔽 = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X, 𝔽 ) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary.

How to cite

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Kassandra Averill, et al. "Boundaries of weak peak points in noncommutative algebras of Lipschitz functions." Open Mathematics 10.2 (2012): 646-655. <http://eudml.org/doc/269272>.

@article{KassandraAverill2012,
abstract = {It has been shown that any Banach algebra satisfying ‖f 2‖ = ‖f‖2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, $\mathbb \{F\}$ ) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where $\mathbb \{F\}$ = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X, $\mathbb \{F\}$ ) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary.},
author = {Kassandra Averill, Ann Johnston, Ryan Northrup, Robert Silversmith, Aaron Luttman},
journal = {Open Mathematics},
keywords = {Lipschitz algebra; Shilov boundary; Real function algebra; Quaternions; Weak peak points; Choquet boundary; real function algebras; weak peak points},
language = {eng},
number = {2},
pages = {646-655},
title = {Boundaries of weak peak points in noncommutative algebras of Lipschitz functions},
url = {http://eudml.org/doc/269272},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Kassandra Averill
AU - Ann Johnston
AU - Ryan Northrup
AU - Robert Silversmith
AU - Aaron Luttman
TI - Boundaries of weak peak points in noncommutative algebras of Lipschitz functions
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 646
EP - 655
AB - It has been shown that any Banach algebra satisfying ‖f 2‖ = ‖f‖2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, $\mathbb {F}$ ) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where $\mathbb {F}$ = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X, $\mathbb {F}$ ) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary.
LA - eng
KW - Lipschitz algebra; Shilov boundary; Real function algebra; Quaternions; Weak peak points; Choquet boundary; real function algebras; weak peak points
UR - http://eudml.org/doc/269272
ER -

References

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