THE CATEGORY OF COMPACT METRIC SPACES AND ITS FUNCTIONAL ANALYTIC DUALS

Branka Pavlovi\'c

Abstract: 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.

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