Publications de l'Institut Mathématique (Beograd) Vol. 72(86), pp. 2938 (2002) 

THE CATEGORY OF COMPACT METRIC SPACES AND ITS FUNCTIONAL ANALYTIC DUALSBranka Pavlovi\'cMatemati\v cki institut, Kneza Mihaila 35, 11001 Beograd, p.p. 367, Yugoslavia and Griffith University, Nathan Qld. 4111, Brisbane, AustraliaAbstract: A Lipschitz algebra $\operatorname{Lip}(X,d_X)$ over a compact metric space $(X,d_X)$ consists of all complex valued continuous functions on $(X,d_X)$ which are Lipschitz with respect to $d_X$ and the standard metric on the complex plane ${\mathbb C}$ (absolute value). The norm on $\operatorname{Lip}(X,d_X)$ is given by $\f\=\sup\{f(x):x\in X\}+\sup\{f(x)f(y)/d_X(x,y): x,y\in X\;&\; x\ne y\}$. We show that the category $\operatorname{CLip}$ in which objects are Lipschitz algebras and morphisms are algebra homomorphisms is dual to the category $\operatorname{CMet}$ in which objects are compact metric spaces and morphisms are Lipschitz maps. Let $(X,d)$ be any metric space, and let $Y=\{(x,y)\in X\times X: x\ne y\}$. De Leeuw derivation defined by the metric $d$ is the operator $D:C_b(X)\to C_b(Y)$ be defined by $(Df)(x,y)=(f(y)f(x))/d(x,y)$ for $(x,y)\in Y$. We consider the category $\operatorname{CDer}$ in which objects are pairs $(C(X),D_X)$, where $(X,d_X)$ is a compact metric space and $D_X$ is the correspoding de Leeuw derivation, and morphisms are all homomorphisms $\nu: C(X)\to C(Y)$ for which $f\in\operatorname{Domain}(D_X)$ implies $\nu f\in\operatorname{Domain}(D_Y)$. We show that $\operatorname{CDer}$ is equivalent to $\operatorname{CLip}$, and that $\operatorname{CDer}$ is dual to $\operatorname{CMet}$. Keywords: Lipschitz algebras; de Leeuw derivations; dual and equivalent categories Classification (MSC2000): 18B99; 18B30;46J10;46L89;46M15. Full text of the article:
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