A note on Riesz spaces with property-b

Alpay, Şafak; Altin, Birol; Tonyali, C.

doi:10.1007/s10587-006-0054-0

Cited by 15 publications

(16 citation statements)

References 6 publications

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“…Indeed Proposition 7 in [7] shows that if I is an ideal with the b-property in a Dedekind complete Riesz space, then I is a band if and only if I has the b-property in E.…”

Section: Definitionmentioning

confidence: 99%

On Riesz spaces with b-property and strongly order bounded operators

Alpay

Altin

2011

Rend. Circ. Mat. Palermo

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A Riesz space E is said to have the b-property if each subset that is order bounded in the bidual remains to be order bounded in E. Properties of a Riesz space with the bproperty, the relationship between the b-property and various classes of operators are studied.Keywords b-property · b-weakly compact operator · Strongly order bounded operator · Preregular operator Mathematics Subject Classification (2000) 46A40 · 46B40 · 47B60 · 47B65

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“…Indeed Proposition 7 in [7] shows that if I is an ideal with the b-property in a Dedekind complete Riesz space, then I is a band if and only if I has the b-property in E.…”

Section: Definitionmentioning

confidence: 99%

On Riesz spaces with b-property and strongly order bounded operators

Alpay

Altin

2011

Rend. Circ. Mat. Palermo

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“…In [4] it is proved that a Banach lattice E is a KB-space if and only it has order continuous norm and has b-property. This can be generalized as follows: A Riesz space E is a band in E ∼∼ if and only if it is an ideal in E ∼∼ and has b-property.…”

Section: ş Alpay and Z Ercan Positivitymentioning

confidence: 99%

“…A Banach lattice E is a KB-space iff E has order continuous norm and has b-property in E (see [4]). On the other hand it is well-known that a Banach lattice is a KB-space if and only if E is weakly sequentially complete.…”

Section: ş Alpay and Z Ercan Positivitymentioning

confidence: 99%

Characterizations of Riesz spaces with b-property

Alpay

Ercan

2008

Positivity

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A Riesz space E is said to have b-property if each subset which is order bounded in E ∼∼ is order bounded in E. The relationship between b-property and completeness, being a retract and the absolute weak topology |σ| (E ∼ , E) is studied. Perfect Riesz spaces are characterized in terms of b-property. It is shown that b-property coincides with the Levi property in Dedekind complete Frechet lattices. Mathematics Subject Classification (2000). Primary 46A40.

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“…In [5], it is seen that L b (E, F ) has property (b) if and only if F has property (b), where both E and F are two Banach lattices and F is Dedekind complete. The following proposition generalizes the result to the case of Riesz spaces.…”

Section: Proposition 3 Let E Be a Dedekind Complete Banach Lattice mentioning

confidence: 99%

On Banach lattices with Levi norms

Altin

2007

Proc. Amer. Math. Soc.

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Abstract. Schmidt proved that an operator T from a Banach lattice E into a Banach lattice G with property (P ) is order bounded if and only if its adjoint is order bounded, and in this case T satisfies |T | = |T | . In the present paper the result is generalized to Banach lattices with Levi-Fatou norm serving as range, and some characterizations of Banach lattices with a Levi norm are given. Moreover, some characterizations of Riesz spaces having property (b) are also obtained.

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A note on Riesz spaces with property-b

Cited by 15 publications

References 6 publications

On Riesz spaces with b-property and strongly order bounded operators

On Riesz spaces with b-property and strongly order bounded operators

Characterizations of Riesz spaces with b-property

On Banach lattices with Levi norms

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