Year of Award
2009
Document Type
Dissertation
Degree Type
Doctor of Philosophy (PhD)
Degree Name
Mathematics
Department or School/College
Department of Mathematical Sciences
Committee Chair
Thomas Tonev
Commitee Members
Jennifer Halfpap, Karel Stroethoff, Nikolaus Vonessen, Nancy Hinman
Keywords
uniform algebra, peripheral spectrum, peaking function, isomorphism, norm, Banach algebra
Abstract
Let A ⊂ C(X) and B ⊂ C(Y ) be uniform algebras with Choquet boundaries δA and δB. We establish sufficient conditions for a surjective map T: A → B to be an algebra isomorphism. In particular, we show that if T: A → B is a surjection that preserves the norm of the sums of the moduli of algebra elements, then T induces a homoemorphism between the Choquet boundaries of A and B such that |Tf| = |f| on the Choquet boundary of B. If, in addition, T preserves the norms of all linear combinations of algebra elements and either preserves both 1 and i or the peripheral spectra of C{peaking functions, then T is a composition operator and thus an algebra isomorphism. We also show that if a surjection T that preserves the norm of the sums of the moduli of algebra elements also preserves the norms of sums of algebra elements as well as either preserving both 1 and i or preserving the peripheral spectra of C{peaking functions, then T is a composition operator and thus an algebra isomorphism. In the process, we generalize the additive analog of Bishop's Lemma.
Recommended Citation
Yates, Rebekah, "Norm-Preserving Criteria for Uniform Algebra Isomorphisms" (2009). Graduate Student Theses, Dissertations, & Professional Papers. 10620.
https://scholarworks.umt.edu/etd/10620
© Copyright 2009 Rebekah Yates