Pre-Riesz Spaces

Kalauch, Anke

doi:10.1515/9783110476293

Cited by 13 publications

(41 citation statements)

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“…In particular, every directed Archimedean partially ordered vector space is a pre-Riesz space. For the theory of pre-Riesz spaces, see also [10]. Abramovich's question extends naturally to the pre-Riesz setting.…”

Section: Introductionmentioning

confidence: 99%

Disjointness preserving operators on normed pre-Riesz spaces: extensions and inverses

2019

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We explore inverses of disjointness preserving bijections in infinite dimensional normed pre-Riesz spaces by several methods. As in the case of Banach lattices, our aim is to show that such inverses are disjointness preserving. One method is extension of the operator to the Riesz completion, which works under suitable denseness and continuity conditions. Another method involves a condition on principle bands. Examples illustrate the differences to the Riesz space theory.

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Section: Introductionmentioning

confidence: 99%

Disjointness preserving operators on normed pre-Riesz spaces: extensions and inverses

2019

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“…The cone X + is called the four ray cone in R 3 . All bands in X are computed in [15,Example 4.4.18]. Besides the two trivial bands {0} and X there are four directed bands -namely the lines spanned by v 1 , .…”

Section: Bandsmentioning

confidence: 99%

“…For a more detailed discussion of this notion and of its origin, we refer to [15,Section 4.1.3] and [18,Defintion 8 and Proposition 9].…”

Section: 3mentioning

confidence: 99%

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A short note on band projections in partially ordered vector spaces

Glück

2019

Indagationes Mathematicae

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Disjointness, bands, and band projections are a classical and essential part of the structure theory of vector lattices. If X is such a lattice, those notions seem -at first glance -intimately related to the lattice operations on X. The last fifteen year, though, have seen an extension of all those concepts to a much larger class of ordered vector spaces.In fact if X is an Archimedean ordered vector space with generating cone, or a member of the slightly larger class of pre-Riesz spaces, then the notions of disjointness, bands and band projections can be given proper meaning and give rise to a non-trivial structure theory.The purpose of this note is twofold: (i) We show that, on any pre-Riesz space, the structure of the space of all band projections is remarkably close to what we have in the case of vector lattices. In particular, this space is a Boolean algebra. (ii) We give several criteria for a pre-Riesz space to already be a vector lattice. These criteria are coined in terms of disjointness and closely related concepts, and they mark how lattice-like the order structure of pre-Riesz spaces can get before the theory collapses to the vector lattice case.

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“…In[9, Example 4.1.19] the authors considers a pre-Riesz space X which is fordable, but not pervasive. One can show that X is weakly pervasive.…”

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confidence: 99%

Pervasive and weakly pervasive pre-Riesz spaces

Kalauch

Malinowski

2019

Indagationes Mathematicae

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Pervasive pre-Riesz spaces are defined by means of vector lattice covers. To avoid the computation of a vector lattice cover, we give two distinct intrinsic characterizations of pervasive pre-Riesz spaces. We introduce weakly pervasive pre-Riesz spaces and observe that this property can be easily checked in examples. We relate weakly pervasive pre-Riesz spaces to pre-Riesz spaces with the Riesz decomposition property.

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Pre-Riesz Spaces

Cited by 13 publications

References 0 publications

Disjointness preserving operators on normed pre-Riesz spaces: extensions and inverses

Disjointness preserving operators on normed pre-Riesz spaces: extensions and inverses

A short note on band projections in partially ordered vector spaces

Pervasive and weakly pervasive pre-Riesz spaces

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