Evaluating characterizations of truncation homomorphisms on truncated vector lattices of functions
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Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00096-4 ORIGINAL PAPER
Evaluating characterizations of truncation homomorphisms on truncated vector lattices of functions Karim Boulabiar1 · Sameh Bououn1 Received: 13 May 2020 / Accepted: 10 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract Let X be a nonempty set. A vector sublattice L of ℝX is said to be truncated if L contains with any function f the function f ∧ 𝟏X . A nonzero linear functional 𝜓 ( on L )is called a truncation homomorphism if it preserves truncation (i.e., 𝜓 f ∧ 𝟏X = min {𝜓(f ), 1} for all f ∈ L ). These concepts generalize the notion of unital lattice homomorphisms on unital vector sublattices of ℝX . Via a unitization process, we extend the different evaluating characterizations of unital lattice homomorphisms, previously obtained by Garrido and Jaramillo, to the truncation homomorphisms on truncated vector sublattices of functions. Keywords Evaluation · Net · Lattice homomorphism · Realcompact · Stone property · Stone-Čech compactification · Truncation homomorphism · Truncated vector sublattice · Unitization Mathematics Subject Classification 46A40
1 Introduction It is well-known that a linear functional 𝜓 on the lattice-ordered algebra C(X) of all real-valued continuous functions on a compact Hausdorff space X is a unital lattice homomorphism if and only if it is an evaluation at some point x of X, i.e.,
𝜓(f ) = f (x)
for all f ∈ C(X)
Dedicated to the memory of Professor Abdelmajid Triki.
Communicated by Denny Leung. * Karim Boulabiar [email protected] 1
Laboratoire de Recherche LATAO, Département de Mathematiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092 El Manar, Tunisia Vol.:(0123456789)
K. Boulabiar and S. Bououn
(see, e.g., Theorem 2.33 in [1]). In their remarkable papers [8, 9], Garrido and Jaramillo investigated the extent to which such a representation can be generalized to a wider class of unital vector sublattices of C(X) . In this regard, they have mainly proved that if 𝜓 is a linear functional on a unital vector sublattice L of C(X) , then 𝜓 is a unital lattice homomorphism if and only if 𝜓 is an evaluation at some point in the Stone-Čech compactification 𝛽X of X. They obtained, as consequences, some necessary and sufficient conditions on L for X to be L-realcompact, i.e., any unital lattice homomorphism on L is an evaluation at some point in X. They also used their aforementioned representation theorem to establish the equivalence between unital lattice homomorphisms and positive algebra homomorphisms on unital latticeordered subalgebras of C(X) . Although they cover a quite large spectrum of function lattices, these results, relevant as they are, cannot deal with the non-unital case. It seems to be natural therefore to look beyond the framework of lattices containing the constant functions. From this point of view, we have thought about vector sublattices possessing the so-called Stone property. Recall here that a vector subspace E of the l
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