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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.
Date17 July 2013
CreatorsJohnson, Jeffrey Verlyn
ContributorsThomas Tonev, Karel Stroethoff, Jennifer Halfpap, Eric Chesebro, Eijiro Uchimoto
PublisherThe University of Montana
Source SetsUniversity of Montana Missoula
Detected LanguageEnglish
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