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Homomorphisms on function lattices. (English) Zbl 1058.54008
Let $X$ be a completely regular space, $C\left(X\right)$ the set of all continuous functions from $X$ into the real numbers $ℝ$, and $L\subset C\left(X\right)$ a unital vector lattice. The structural space $H\left(L\right)$ is the set of all lattice homomorphisms from $L$ into $ℝ$. For each $x\in X$ the point evaluation map ${\delta }_{x}$ belongs to $H\left(L\right)$, where ${\delta }_{x}\left(f\right)=f\left(x\right)$ for each $f\in L$. If $L$ separates points and closed sets, then $X$ is a topolopical subspace of $H\left(L\right)$, and $H\left(L\right)$ is a realcompactification of $X$. Indeed, all realcompactifications of $X$ have the form $H\left(L\right)$ for some $L\subset C\left(X\right)$. In the present paper, the authors study the realcompactifications $H\left(L\right)$ and compactifications $H\left({L}^{*}\right)$ of $X$, where ${L}^{*}$ denotes the set of bounded elements in $L$, and $H\left({C}^{*}\left(X\right)\right)$ is the Stone-Čech compactification $\beta X$ of $X$. They call $X$ $L$-realcompact in case $H\left(L\right)=X$. They show, among other things, that for each $\varphi \in H\left(L\right)$ there exists $\xi \in \beta X$ such that, for all $f\in L$, ${f}^{\beta }\left(\xi \right)\ne \infty$ and $\varphi \left(f\right)={f}^{\beta }\left(\xi \right)$. They study the case where $L=\text{Lip}\left(X\right)$ 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 $\text{Lip}\left(X\right)$ and $\text{Lip}\left(Y\right)$ 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 ${\text{Lip}}^{*}\left(X\right)$ and ${\text{Lip}}^{*}\left(Y\right)$ was obtained by N. Weaver [Pac. J. Math. 164, No. 1, 179–193 (1994; Zbl 0797.46007)].
##### MSC:
 54C40 Algebraic properties of function spaces (general topology) 46E05 Lattices of continuous, differentiable or analytic functions 54D35 Extensions of topological spaces (compactifications, supercompactifications, completions, etc.) 54D60 Realcompactness and real compactification (general topology)