Regular operators from and into a small Riesz space

Abramovich, Y. A.; Wickstead, Anthony

doi:10.1016/0019-3577(91)90014-x

Cited by 16 publications

(34 citation statements)

References 4 publications

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“…. , m} and y ∈ L + with h(y) = f j (y) (1),...,(α) we have f j (y) (α+1) = 0. Clearly we also have for all j ∈ {1, .…”

Section: (α)mentioning

confidence: 99%

“…, 0) ∈ Z the bound (3.26) yields that m i=1 h (α+1) x * i > 0. These considerations allow us to define (1),..., (α) . To see this, fix i ∈ {1, .…”

Section: Which Contradicts (324) By the Symmetry Of The Situation Wmentioning

confidence: 99%

“…, 0, h) we can assume without loss of generality that m+1 j=1 y * j = e so that y * m+1 = e − m j=1 y * j . Then we obtain that the set of functions {( f (1) i , .…”

Section: Which Contradicts (324) By the Symmetry Of The Situation Wmentioning

confidence: 99%

“…This problem has been studied more than the one on the Riesz-Kantorovich formula and a few results can be found for example in [1,2,12].…”

Section: Introductionmentioning

confidence: 99%

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The Riesz–Kantorovich formula for lexicographically ordered spaces

Schouten-Straatman

2017

Positivity

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If L and M are partially ordered vector spaces, then one can consider regular linear maps from L to M, i.e. linear maps which can be written as the difference of two positive linear maps. If the space L is directed, then the space L r (L , M) of all regular linear operators becomes a partially ordered vector space itself. We will mainly concern ourselves with the questions when the space L r (L , M) is itself a Riesz space and how, even if it is not a Riesz space, its lattice operations work. The so-called Riesz-Kantorovich theorem gives sufficient conditions for which L r (L , M) is a Riesz space and it also specifies the lattice operations by means of the Riesz-Kantorovich formula: if S, T ∈ L r (L , M) and x ∈ L with x ≥ 0 then the supremum S ∨ T in the point x is given byIt is still an open problem if whenever in a more general setting the supremum of two regular operators exists in L r (L , M), it automatically is given by the RieszKantorovich formula. Our main result concerns the special case where L is a partially ordered vector space with a strong order unit and M is a (possibly infinite) product of copies of the real line, equipped with the lexicographic ordering. It will turn out that under some mild continuity and regularity conditions the lattice operations on L r (L , M) are indeed given by the Riesz-Kantorovich formula, even though the space L r (L , M) is not necessarily a Riesz space.B W. M. Schouten

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“…. , m} and y ∈ L + with h(y) = f j (y) (1),...,(α) we have f j (y) (α+1) = 0. Clearly we also have for all j ∈ {1, .…”

Section: (α)mentioning

confidence: 99%

“…, 0) ∈ Z the bound (3.26) yields that m i=1 h (α+1) x * i > 0. These considerations allow us to define (1),..., (α) . To see this, fix i ∈ {1, .…”

Section: Which Contradicts (324) By the Symmetry Of The Situation Wmentioning

confidence: 99%

“…, 0, h) we can assume without loss of generality that m+1 j=1 y * j = e so that y * m+1 = e − m j=1 y * j . Then we obtain that the set of functions {( f (1) i , .…”

Section: Which Contradicts (324) By the Symmetry Of The Situation Wmentioning

confidence: 99%

“…This problem has been studied more than the one on the Riesz-Kantorovich formula and a few results can be found for example in [1,2,12].…”

Section: Introductionmentioning

confidence: 99%

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The Riesz–Kantorovich formula for lexicographically ordered spaces

Schouten-Straatman

2017

Positivity

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“…In [2] the space l ∞ 0 of all real sequences which are constant except for a finite number of terms is investigated. This vector lattice is not Dedekind complete.…”

Section: A Space Of Operatorsmentioning

confidence: 99%

Bands in pervasive pre-Riesz spaces

Gaans¹,

Kalauch²

2008

Operators and Matrices

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Pre-Riesz spaces are partially ordered vector spaces which are order dense subspaces of vector lattices. A band in a pre-Riesz space can be extended to a band in the ambient vector lattice. The corresponding restriction property does not hold in general. We provide sufficient conditions on the underlying space such that the restriction property for bands holds. As an application, we consider the space L r (l ∞ 0 ) of all regular operators on the space l ∞ 0 of all finally constant sequences. We establish that L r (l ∞ 0 ) is pre-Riesz and that its subspace of all order continuous operators is a band in L r (l ∞ 0 ).Subject Classification (MSC2000): 46A40, 06F20

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The centre of spaces of regular operators

Wickstead¹

2002

Math. Z.

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Regular operators from and into a small Riesz space

Cited by 16 publications

References 4 publications

The Riesz–Kantorovich formula for lexicographically ordered spaces

The Riesz–Kantorovich formula for lexicographically ordered spaces

Bands in pervasive pre-Riesz spaces

The centre of spaces of regular operators

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