Boundaries of weak peak points in noncommutative algebras of Lipschitz functions

• Autorzy: Kassandra Averill , Ann Johnston , Ryan Northrup , Robert Silversmith , Aaron Luttman
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Lipschitz algebra; Shilov boundary; Real function algebra; Quaternions; Weak peak points; Choquet boundary

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 2
• Strony: 646-655

Daty

wydano

2012-04-01

online

2012-01-18

Twórcy

autor

Kassandra Averill

• SUNY Potsdam

autor

Ann Johnston

• University of Southern California

autor

Ryan Northrup

• University of Kentucky

autor

Robert Silversmith

• University of Michigan

autor

Aaron Luttman

• Clarkson University

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Identyfikatory

DOI

10.2478/s11533-011-0133-9