Smallest order closed sublattices and option spanning

Gao, Niushan; Leung, Denny H.

doi:10.1090/proc/13820

“…These are of interest in their own right. Other such results are [16,Theorem 8.8], [17,Theorem 2.13], and [26,Proposition 2.12].…”

Section: Equality Of Adherences Of Vector Sublatticesmentioning

confidence: 98%

“…It is known that the o-adherence of a vector sublattice of a Dedekind complete Banach lattice E with a strong order unit can be a proper sublattice of its uo-adherence; see [17,Lemma 2.6] for details. When the vector sublattice contains a strong order unit of E, however, then this cannot occur, not even in general vector lattices.…”

Section: Corollary 65 Let E Be a Dedekind Complete Vector Lattice Suc...mentioning

confidence: 99%

Convergence structures and Hausdorff uo-Lebesgue topologies on vector lattice algebras of operators

Deng

¹

,

Jeu

²

2022

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A vector sublattice of the order bounded operators on a Dedekind complete vector lattice can be supplied with the convergence structures of order convergence, strong order convergence, unbounded order convergence, strong unbounded order convergence, and, when applicable, convergence with respect to a Hausdorff uo-Lebesgue topology and strong convergence with respect to such a topology. We determine the general validity of the implications between these six convergences on the order bounded operator and on the orthomorphisms. Furthermore, the continuity of left and right multiplications with respect to these convergence structures on the order bounded operators, on the order continuous operators, and on the orthomorphisms is investigated, as is their simultaneous continuity. A number of results are included on the equality of adherences of vector sublattices of the order bounded operators and of the orthomorphisms with respect to these convergence structures. These are consequences of more general results for vector sublattices of arbitrary Dedekind complete vector lattices. The special attention that is paid to vector sublattices of the orthomorphisms is motivated by explaining their relevance for representation theory on vector lattices.

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“…Remark 6.11. For comparison, we recall that, for a regular vector sublattice F of a vector lattice E, it is always the case that a o (F) = a uo (F) in E, and that these coinciding subsets are order closed subsets of E; see [13,Theorem 2.13].…”

Section: Equality Of Adherences Of Vector Sublatticesmentioning

confidence: 99%

Convergence structures and Hausdorff uo-Lebesgue topologies on vector lattice algebras of operators

Yang¹,

Jeu²

2020

Preprint

0

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A vector sublattice of the order bounded operators on a Dedekind complete vector lattice can be supplied with the convergence structures of order convergence, strong order convergence, unbounded order convergence, strong unbounded order convergence, and, when applicable, convergence with respect to a Hausdorff uo-Lebesgue topology and strong convergence with respect to such a topology. We determine the general validity of the implications between these six convergences on the order bounded operator and on the orthomorphisms. Furthermore, the continuity of left and right multiplications with respect to these convergence structures on the order bounded operators, on the order continuous operators, and on the orthomorphisms is investigated, as is their simultaneous continuity. A number of results are included on the equality of adherences of vector sublattices of the order bounded operators and of the orthomorphisms with respect to these convergence structures. These are consequences of more general results for vector sublattices of arbitrary Dedekind complete vector lattices. The special attention that is paid to vector sublattices of the orthomorphisms is motivated by explaining their relevance for representation theory on vector lattices.

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“…It is inspired by[6, Definition 1.3.1] 5. This definition is consistent with that in[13] 6. There is no notation for the closure operation in the o-topology in[13].…”

mentioning

confidence: 98%

“…In[19, p. 82], our σ-o-adherence is called the pseudo order closure. In[13], our o-adherence of a subset A is called the order closure of A, and it is denoted by A o . These two terminologies, as well as the notation A o , could suggest that taking the (pseudo) order closure is a (sequential) closure operation for a topology.…”

mentioning

confidence: 99%

Vector lattices with a Hausdorff uo-Lebesgue topology

Deng,

de Jeu

2020

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0

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We investigate the construction of a Hausdorff uo-Lebesgue topology on a vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal, and what the properties of the topologies thus obtained are. When the vector lattice has an order dense ideal with a separating order continuous dual, it is always possible to supply it with such a topology in this fashion, and the restriction of this topology to a regular sublattice is then also a Hausdorff uo-Lebesgue topology. A regular vector sublattice of L 0 (X , Σ, µ) for a semi-finite measure µ falls into this category, and the convergence of nets in its Hausdorff uo-Lebesgue topology is then the convergence in measure on subsets of finite measure. When a vector lattice not only has an order dense ideal with a separating order continuous dual, but also has the countable sup property, we show that every net in a regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology always contains a sequence that is uoconvergent to the same limit. This enables us to give satisfactory answers to various topological questions about uo-convergence in this context.

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Smallest order closed sublattices and option spanning

Cited by 10 publications

References 24 publications

Convergence structures and Hausdorff uo-Lebesgue topologies on vector lattice algebras of operators

Convergence structures and Hausdorff uo-Lebesgue topologies on vector lattice algebras of operators

Convergence structures and Hausdorff uo-Lebesgue topologies on vector lattice algebras of operators

Vector lattices with a Hausdorff uo-Lebesgue topology

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