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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 , and LC(X) a unital vector lattice. The structural space H(L) is the set of all lattice homomorphisms from L into . For each xX the point evaluation map δ x belongs to H(L), where δ x (f)=f(x) for each fL. 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 LC(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 βX of X. They call X L-realcompact in case H(L)=X. They show, among other things, that for each ϕH(L) there exists ξβX such that, for all fL, f β (ξ) and ϕ(f)=f β (ξ). They study the case where L=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 Lip(X) and 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 Lip * (X) and Lip * (Y) was obtained by N. Weaver [Pac. J. Math. 164, No. 1, 179–193 (1994; Zbl 0797.46007)].
54C40Algebraic properties of function spaces (general topology)
46E05Lattices of continuous, differentiable or analytic functions
54D35Extensions of topological spaces (compactifications, supercompactifications, completions, etc.)
54D60Realcompactness and real compactification (general topology)