Positivity 12 (2008), 341–362 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1385-1292/020341-22, published online January 11, 2008 Positivity DOI 10.1007/s11117-007-2114-6 Homomorphisms on Lattices of Continuous Functions F´ elix Cabello S´ anchez Mathematics Subject Classiﬁcation (2000). 11H56. Keywords. Lattice of continuous functions, homomorphism, representation. 1. Introduction 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 func- tions on a topological space X with the order induced by that of R,thatis, 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. Sup- pose 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  and . In
Positivity – Springer Journals
Published: Jan 11, 2008
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