Conditional expectations on Riesz spaces
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 \(L^1\).
scientific article; zbMATH DE number 2153760
scientific article