## Peripherally-Multiplicative Spectral Preservers Between Function Algebras

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.

 Identifer oai:union.ndltd.org:MONTANA/oai:etd.lib.umt.edu:etd-07082013-125525 Date 17 July 2013 Creators Johnson, Jeffrey Verlyn Contributors Thomas Tonev, Karel Stroethoff, Jennifer Halfpap, Eric Chesebro, Eijiro Uchimoto Publisher The University of Montana Source Sets University of Montana Missoula Language English Detected Language English Type text Format application/pdf Source http://etd.lib.umt.edu/theses/available/etd-07082013-125525/ Rights unrestricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to University of Montana or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report.

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