Extreme points of the set of elements majorised by an integrable function: Resolution of a problem by Luxemburg and of its noncommutative counterpart

D. Dauitbek; J. Huang; F. Sukochev

doi:10.1016/j.aim.2020.107050

Extreme points of the set of elements majorised by an integrable function: Resolution of a problem by Luxemburg and of its noncommutative counterpart

D. Dauitbek
, J. Huang^*
, F. Sukochev

^*Corresponding author for this work

Abay Kazakh National Pedagogical University
Farabi University
Institute of Mathematics and Mathematical Modelling
University of New South Wales

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

Let f be an arbitrary integrable function on a finite measure space (X,Σ,ν). We characterise the extreme points of the set Ω(f) of all measurable functions on (X,Σ,ν) majorised by f, providing a complete answer to a problem raised by W.A.J. Luxemburg in 1967. Moreover, we obtain a noncommutative version of this result.

Original language	English
Article number	107050
Journal	Advances in Mathematics
Volume	365
DOIs	https://doi.org/10.1016/j.aim.2020.107050
State	Published - 13 May 2020
Externally published	Yes

Keywords

Extreme points
Finite von Neumann algebras
Majorisation
Noncommutative L-space
Probability spaces
Spectral scales

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10.1016/j.aim.2020.107050

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N2 - Let f be an arbitrary integrable function on a finite measure space (X,Σ,ν). We characterise the extreme points of the set Ω(f) of all measurable functions on (X,Σ,ν) majorised by f, providing a complete answer to a problem raised by W.A.J. Luxemburg in 1967. Moreover, we obtain a noncommutative version of this result.

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KW - Extreme points

KW - Finite von Neumann algebras

KW - Majorisation

KW - Noncommutative L-space

KW - Probability spaces

KW - Spectral scales

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Extreme points of the set of elements majorised by an integrable function: Resolution of a problem by Luxemburg and of its noncommutative counterpart

Abstract

Keywords

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