Functional calculus on Riesz spaces

Buskes, Gerard; Pagter, B. de; Rooij, Arnoud van

doi:10.1016/0019-3577(91)90028-6

Cited by 39 publications

(60 citation statements)

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“…Similar concerns where discussed by Buskes, de Pagter and van Rooij [1,7]. They proposed to avoid the use of the axiom of choice by restricting the size of the Riesz spaces [7].…”

Section: Theorem 1 [Stone-yosida] Let R Be An Archimedean Riesz Spacementioning

confidence: 93%

Constructive Pointfree Topology Eliminates Non-constructive Representation Theorems from Riesz Space Theory

Spitters

2010

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In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that Archimedean almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters. Keywords Formal topology · Axiom of choice · Riesz space · Constructive analysisThe Stone-Yosida representation theorem [25,28] for Riesz spaces shows how to embed every Riesz space [20,29] into the Riesz space of continuous functions on its spectrum.Definition 1 A Riesz space is a vector space with compatible lattice operationsi.e. f ∧ g + f ∨ g = f + g and if f 0 and a 0, then af 0. A (strong) unit 1 in a Riesz space is an element such that for all f there exists a natural number n such that | f | n · 1. A Riesz space is Archimedean if for all n, n|x| y implies that x = 0.A Riesz space with a strong unit is Archimedean.Definition 2 A representation of a Riesz space R with unit is a linear map σ : R → Ê which respects the lattice operations and the unit. We denote by the space of representations equipped with the weakest topology which makes all the maps e r (σ ) := σ (r) continuous.

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“…Similar concerns where discussed by Buskes, de Pagter and van Rooij [1,7]. They proposed to avoid the use of the axiom of choice by restricting the size of the Riesz spaces [7].…”

Section: Theorem 1 [Stone-yosida] Let R Be An Archimedean Riesz Spacementioning

confidence: 93%

Constructive Pointfree Topology Eliminates Non-constructive Representation Theorems from Riesz Space Theory

Spitters

2010

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“…Then, the set L 0 (μ, X ) consists exactly of all Bochner measurable X -valued functions defined almost everywhere in . Consequently, (1)(2)(3)(4)(5) and satisfying the condition: ρ X (c) = c for every constant function c : → X . Now, a natural question arises: When L ∞ (μ, X ) admits a lifting or, which is the same, when the measurable Banach bundle X = × {X } (with the above mentioned measurability structure) is liftable?…”

Section: Lattices Of Measurable Vector Functionsmentioning

confidence: 96%

Measurable bundles of Banach lattices

Kusraev

2010

Positivity

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“…In this paper we need homogeneous functional calculus on vector lattices, see [4,10,19,22,26,28]. Denote by H (R N ) the vector lattice of all continuous positively homogeneous functions f :…”

Section: 7mentioning

confidence: 99%