## Graduate Student Theses, Dissertations, & Professional Papers

2013

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, Eric Chesebro, Eijiro Uchimoto

#### Keywords

Peripherally-Multiplicative Spectral Preservers Be

#### Publisher

University of Montana

#### Abstract

General sufficient conditions are established for maps between function algebras to be composition or weighted composition operators, which extend previous results regarding spectral conditions for maps between uniform algebras. Let X and Y be a locally compact Hausdorff spaces, where A \subset C(X) and B \subset C(Y) are function algebras, not necessarily with unit. Also let \partial A be the Shilov boundary of A, \delta A the Choquet boundary of A, and p(A) the set of p-points of A. A map T \colon A \to B is called weakly peripherally-multiplicative if the peripheral spectra of fg and TfTg have non-empty intersection for all f,g in A. (i.e. \sigma_{pi}( fg ) \cap \sigma_{pi}(TfTg ) \neq \emptyset for all f,g in A) The map is said to be almost peripherally-multiplicative if the peripheral spectrum of fg is contained in the peripheral spectrum of TfTg (or if the peripheral spectrum of TfTg is contained in the peripheral spectrum of fg) for all f,g in A. Let X be a locally compact Hausdorff space and A \subset C(X) be a dense subalgebra of a function algebra, not necessarily with unit, such that \delta A = p(A). We show that if T\colon A \to B is a surjective map onto a function algebra B\subset C(Y) that is almost peripherally-multiplicative, then there is a homeomorphism \psi\colon \delta B\to\delta A and a function \alpha on \delta B so that (Tf)(y)=\a(y)\,f(\psi(y)) for all f \in A and y \in\delta B, i.e. T is a weighted composition operator where the weight function is a signum function. We also show that if T is weakly peripherally-multiplicative, and either \sigma_{pi}(f)\subset \sigma_{pi}(Tf) for all f in A, or, alternatively, \sigma_{pi}(Tf) \subset \sigma_{pi}(f) for all f in A, then (Tf)(y)=f(\psi(y)) for all f \in A and y \in \delta B. In particular, if A and B are uniform algebras and T \colon A \to B is a weak peripherally-multiplicative operator, that has a limit, say b, at some a in A with a^2=1, then (Tf)(y)=b(y)\,a(\psi(y))\, f(\psi(y)) for every f in A and y in \delta B. Also, we show that if a weak peripherally-multiplicative map preserving peaking functions in the sense \mathcal{P}(B) \subset T[ \mathbb{T} \cdot \mathcal{P}(A)] or T[\mathcal{P}(A)] \subset \mathbb{T} \cdot \mathcal{P}(B) then T is a weighted composition operator with a signum weight function. Finally, for function algebras containing sufficiently many peak functions, including function algebras on metric spaces, it is shown that weak peripherally-multiplicative maps are necessarily composition operators.

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