Boundaries of weak peak points in noncommutative algebras of Lipschitz functions |
| |
Authors: | Kassandra Averill Ann Johnston Ryan Northrup Robert Silversmith Aaron Luttman |
| |
Institution: | 1.Department of Mathematics,SUNY Potsdam,Potsdam,USA;2.Department of Mathematics,University of Southern California,Los Angeles,USA;3.Department of Mathematics,University of Kentucky,Lexington,USA;4.Department of Mathematics,University of Michigan,Ann Arbor,USA;5.Department of Mathematics,Clarkson University,Potsdam,USA |
| |
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,
\mathbbF\mathbb{F}) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where
\mathbbF\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,
\mathbbF\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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|