## Graduate Student Theses, Dissertations, & Professional Papers

2008

Dissertation

#### Degree Type

Doctor of Philosophy (PhD)

#### Other Degree Name/Area of Focus

Mathematical Sciences

#### Department or School/College

Department of Mathematics

Thomas Tonev

#### Commitee Members

Jennifer Halfpap, Karel Stroethoff, Keith Yale, Eijiro Uchimoto

#### Keywords

Banach algebras, uniform algebras, functional analysis, preserver problems

#### Publisher

University of Montana

#### Abstract

There has been much interest in characterizing maps between Banach algebras that preserve a certain equation or family of elements. There is a rich history in such problems that assume the map to be linear, so called linear preserver problems. More recently, there has been an interest in not assuming the map is linear a priori and instead to assume it preserves some equation involving the spectrum, a portion of the spectrum, or the norm.

After a brief introduction to uniform algebras, we give a rigorous development of the theory of boundaries. This includes a new alternative proof of the famous Shilov Theorem. Also a generalization of Bishop's Lemma is given and proved. Two spectral preserver problems are introduced and solved for the class of uniform algebras. One of these problems is given in terms of a portion of the spectrum called the peripheral spectrum. The other is given by a norm condition.

The first spectral preserver problem concerns weakly-peripherally multiplicative maps between uniform algebras. These are maps T from A to B such that the intersection of the peripheral spectra of TfTg and fg is not empty for all f and g in A. It is proven that if T is a weakly-peripherally multiplicative map (not necessarily linear) that preserves the family of peak functions then it is an isometric algebra isomorphism.

The second of these preserver problems shows that if T is a map (not necessarily linear) between uniform algebras, A and B, such that the norm of TfTg + 1 equals the norm of fg + 1 for all f, g in A, then T is a weighted composition operator composed with a conjugation operator. In particular, if T1 = 1 and Ti = i then T is an isometric algebra isomorphism.

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