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Homomorphisms on function lattices. (English) Zbl 1058.54008
Let \(X\) be a completely regular space, \(C(X)\) the set of all continuous functions from \(X\) into the real numbers \(\mathbb R\), and \(L\subset C(X)\) a unital vector lattice. The structural space \(H(L)\) is the set of all lattice homomorphisms from \(L\) into \(\mathbb R\). For each \(x\in X\) the point evaluation map \(\delta_x\) belongs to \(H(L)\), where \(\delta_x(f)=f(x)\) for each \(f\in L\). If \(L\) separates points and closed sets, then \(X\) is a topolopical subspace of \(H(L)\), and \(H(L)\) is a realcompactification of \(X\). Indeed, all realcompactifications of \(X\) have the form \(H(L)\) for some \(L\subset C(X)\). In the present paper, the authors study the realcompactifications \(H(L)\) and compactifications \(H(L^*)\) of \(X\), where \(L^*\) denotes the set of bounded elements in \(L\), and \(H(C^* (X))\) is the Stone-Čech compactification \(\beta X\) of \(X\). They call \(X\) \(L\)-realcompact in case \(H(L)=X\). They show, among other things, that for each \(\varphi\in H(L)\) there exists \(\xi\in\beta X\) such that, for all \(f\in L\), \(f^\beta(\xi)\neq\infty\) and \(\varphi(f)=f^\beta(\xi)\). They study the case where \(L=\text{Lip}(X)\) is the set of real valued Lipschitz functions on a metric space \(X\) and show that \(X\) is \(L\)-realcompact if, and only if, every closed ball in \(X\) is compact. Other results concern a vector lattice isomorphism \(T\) between \(\text{Lip}(X)\) and \(\text{Lip} (Y)\) which maps unit element to unit element. They show that if \(X,Y\) are complete, then \(T\) exists if, and only if, \(X\) and \(Y\) are Lipschitz homeomorphic. A similar result for \(\text{Lip}^*(X)\) and \(\text{Lip}^* (Y)\) was obtained by N. Weaver [Pac. J. Math. 164, No. 1, 179–193 (1994; Zbl 0797.46007)].

54C40 Algebraic properties of function spaces in general topology
46E05 Lattices of continuous, differentiable or analytic functions
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54D60 Realcompactness and realcompactification
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