Conditional expectations on Riesz spaces
type: Item
instance of: scholarly article
title: Conditional expectations on Riesz spaces
zbMATH Open document ID: 1075.46002
author: Wen-Chi Kuo
author: Coenraad C. A. Labuschagne
author: Bruce A. Watson
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-07T00:00:00Z
review text: A positive order continuous projection \(T\) on \(E\) with range \(R (T)\), a Dedekind complete Riesz subspace of \(E\), is called a conditional expectation if \(Te\) is a weak order unit of \(E\) whenever \(e\) is a weak order unit in \(E\). The authors show that if \(T\) is a conditional expectation on a Dedekind complete Riesz space with weak order unit, then every band projection onto a principal band generated by an element in the range of \(T\) commutes with \(T\). Expressed in terms of orthomorphisms, this takes the form of the following Theorem: Let \(E\) be a Dedekind complete Riesz space with weak order unit and \(T\) be a conditional expectation of \(E\). Then \(TJ = JT\) for all \(J \in \) Orth \(R (T)\). Applied to \(L^1 (\mu)\), this is the well-known result that conditional expectations are averaging. If \(E\) is a Dedekind complete Riesz space with order unit \(e \in E_+\), then \(E\) can be uniquely endowed with a multiplicative structure which turns \(E\) into an \(f\)-algebra. Thus, if \(E\) is a Dedekind complete Riesz space with order unit \(e\), then each conditional expectation operator \(T\) on \(E\) with \(Te =e\) satisfies \(T (gf) = g Tf\) for \(f \in E\), \(g \in R (T)\). These ideas lead to the extension of conditional expectation operators to their natural domains, that is to say maximal domains, for which the operators are both averaging operators and conditional expectations. In the final part of this very interesting paper, the authors show that the natural domain is in many ways analogous to \(L^1\).
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 47B60
zbMATH DE Number: 2153760
zbMATH Keywords: averaging operators
zbMATH Keywords: \(f\)-algebras
zbMATH Keywords: conditional expectations
zbMATH Keywords: Riesz spaces
reviewed by: Şafak Alpay
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.08.050
OpenAlex ID: W1969011584
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DOI: 10.1016/J.JMAA.2004.08.050
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description: scientific article; zbMATH DE number 2153760