#### 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

#### Publisher

The University of Montana

#### 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