Homomorphisms on Lattices of Continuous Functions

Sánchez, Félix Cabello

doi:10.1007/s11117-007-2114-6

Cited by 20 publications

(10 citation statements)

References 18 publications

(18 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 3.1 shows that if X and Y are realcompact spaces such that C(X) and C(Y ) are order isomorphic, then the associated homeomorphism obtained in Theorem 2.11 restricts to a homeomorphism between X and Y . It generalizes the previously known result of F. Cabello Sanchez for the case of compact X and Y [8].…”

supporting

confidence: 87%

“…The class includes spaces of continuous, uniformly continuous, Lipschitz, little Lipschitz and differentiable functions and their "local" versions. In particular, they include as special cases all the main results of [8,9,10,11]. It is shown that an order isomorphism between any two such spaces must be a nonlinear weighted composition operator.…”

mentioning

confidence: 99%

“…See also the work of Jiménez-Vargas and Villegas-Vallecillos concerning linear order isomorphisms on spaces of Lipschitz functions [19]. Within the last decade, a number of papers by F. and J. Cabello Sanchez have appeared that characterize nonlinear order isomorphisms on various function spaces [8,9,10,11]. In this paper, our aim is to present a unified and thorough study of nonlinear order isomorphisms between function spaces.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear order isomorphisms on function spaces

Leung

Tang

2016

Dissertationes Math.

View full text Add to dashboard Cite

Let X be a topological space. A subset of C(X), the space of continuous real-valued functions on X, is a partially ordered set in the pointwise order. Suppose that X and Y are topological spaces, and A(X) and A(Y ) are subsets of C(X) and C(Y ) respectively. We consider the general problem of characterizing the order isomorphisms (order preserving bijections) between A(X) and A(Y ). Under some general assumptions on A(X) and A(Y ), and when X and Y are compact Hausdorff, it is shown that existence of an order isomorphism between A(X) and A(Y ) gives rise to an associated homeomorphism between X and Y . This generalizes a classical result of Kaplansky concerning linear order isomorphisms between C(X) and C(Y ) for compact Hausdorff X and Y . The class of near vector lattices is introduced in order to extend the result further to noncompact spaces X and Y . The main applications lie in the case when X and Y are metric spaces. Looking at spaces of uniformly continuous functions, Lipschitz functions, little Lipschitz functions, spaces of differentiable functions, and the bounded, "local" and "bounded local" versions of these spaces, characterizations of when spaces of one type can be order isomorphic to spaces of another type are obtained.

show abstract

supporting

confidence: 87%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Nonlinear order isomorphisms on function spaces

Leung

Tang

2016

Dissertationes Math.

View full text Add to dashboard Cite

show abstract

“…In type I 1 cases, we can completely understand the general form of order isomorphisms between effect algebras essentially by the measure theoretic idea of Cabello Sánchez in [3]. Suppose that φ : E(M ) → E(N ) is an order isomorphism between effect algebras of commutative von Neumann algebras.…”

Section: It Follows By Lemma 33 Thatmentioning

confidence: 99%

“…There are several studies on nonlinear order isomorphisms between self-adjoint parts of commutative C *algebras (e.g. [3]). As we can see in the case A = B = C, such mappings can be far from linear.…”

Section: Introductionmentioning

confidence: 99%