Operators represented by conditional expectations and random measures
type: Item
instance of: scholarly article
title: Operators represented by conditional expectations and random measures
zbMATH Open document ID: 1117.47021
author: D. T. Rambane
author: Jacobus J. Grobler
published in: Positivity
publication date: 2006-11-03T00:00:00Z
review text: On standard measure spaces (spaces which are isomorphic to Borel subsets of complete separable metric spaces), every order bounded order continuous linear operator acting between ideals of measurable functions is generated by a random measure. The authors prove an analogue of this result for arbitrary \(\sigma\)-finite measure spaces. This then leads to the main result of the paper. Theorem. Every order continuous linear map between ideals of almost everywhere finite measurable functions on a \(\sigma\)-finite measure space is multiplication conditional expectation representable. These considerations shed further light on the structure of order continuous operators and show that multiplication operators, Riesz homomorphisms and conditional expectations constitute the building blocks of every order continuous operator.
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 5070775
zbMATH Keywords: random measure
zbMATH Keywords: multiplication conditional expectation operator
zbMATH Keywords: pseudo-integral operator
zbMATH Keywords: order continuous operator
reviewed by: Şafak Alpay
MaRDI profile type: Publication
full work available at URL: s11117-005-4399-7
OpenAlex ID: W1965783192
cites work: Q3748810
cites work: Operator algebras and invariant subspaces
cites work: Lattice Properties of Operators Defined By Random Measures
cites work: Q4188290
cites work: Q4183855
DOI: 10.1007/S11117-005-4399-7
Recommended article: Conditional expectation for operator-valued measures and functions
Recommended article: Conditional expectations and operator decompositions
Recommended article: Conditional expectations and dilations of positive operator-valued measures
Recommended article: Q5452951
Recommended article: Q4856891
Recommended article: Q4422013
Recommended article: Characterizations of conditional expectation-type operators
Recommended article: Operator means of probability measures
Recommended article: Q3781620
description: scientific article; zbMATH DE number 5070775