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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)].

##### MSC:
 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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