Homomorphisms on Lattices of Continuous Functions

Abstract

This paper deals with lattices of continuous functions and their homomorphisms, with emphasis on isomorphisms. As usual, we write C(X) for the lattice of all real-valued continuous functions on a topological space X with the order induced by that of R, that is, f ≤ g meaning f(x) ≤ g(x) for all x ∈ X. The sublattice of bounded functions is denoted C∗(X). Until further notice X and Y will denote compact Hausdorff spaces. Suppose we are given an isomorphism T : C(Y ) → C(X), that is, bijection satisfying T (f ∨ g) = Tf ∨ Tg and (this is equivalent for bijections) T (f ∧ g) = Tf ∧ Tg. What can be said about T? In particular, how to represent it? We emphasize that T is not assumed to be linear. As far as I know, these problems were first considered by Kaplansky in his venerable oldies [16] and [17]. In the former he showed that if the lattices C(Y ) and C(X) are isomorphic, then X and Y are homeomorphic. The proof is of Stonian style and proceeds by duality (the points of X are identified as equivalence classes of prime ideals of C(X), a prime ideal being the kernel of some homomorphism onto the lattice {0, 1} and so on; see also [2, pp. 227–228] and [23, pp. 129–130]). The papers [14, 24] contain extensions to noncompact spaces. The sequel [17] studies continuity properties of isomorphisms. For instance, it is proved that, referring to the usual sup norm topology, T is continuous if and only if it admits a representation as Tf(x) = t(x, f(τ(x))) (f ∈ C(Y ), x ∈ X),