Conditional expectations on Riesz spaces

Abstract

Conditional expectations operators acting on Riesz spaces are shown to commute with a class of principal band projections. Using the above commutation property, conditional expectation operators on Riesz spaces are shown to be averaging operators. Here the theory of f -algebras is used when defining multiplication on the Riesz spaces. This leads to the extension of these conditional expectation operators to their so-called natural domains, i.e., maximal domains for which the operators are both averaging operators and conditional expectations. The natural domain is in many aspects analogous to L 1 .

An Andô-Douglas type theorem in Riesz spaces with a conditional expectation

“…We recall here that a Dedekind complete Riesz space has the Principal Projection Property. The following result, proved in [12,Theorem 3.2], is presented here for the readers convenience. The following lemma from [13, Lemma 2.2] which enables one to separate non-equal elements in a Riesz space by scalar multiples of a band projection, will be used in the last section of this paper.…”

“…We recall some definitions regarding conditional expectations on Riesz spaces from [12]. Let E be a Riesz space with a weak order unit.…”

“…For further details on bands and band projections we refer the reader to [26], and for commutation relations between band projections and conditional expectations we recommend [12] and [13] to the reader. We recall here that a Dedekind complete Riesz space has the Principal Projection Property.…”

“…exist and will be denoted by lim sup If E is a Dedekind complete Riesz space and T is a strictly positive conditional expectation operator on E, then E has a T -universal completion,Ẽ. In the terminology of [12], the T -universal completion of E is the natural domain of T in the universal completion, E u , of E, see [10,18,25] for the case of measure spaces.…”

“…Formulations of stochastic processes on Riesz spaces can be found in [11][12][13] and [22,23]. Our work here, being concerned with the existence and uniqueness of conditional expectations on Riesz spaces specified in terms of their ranges spaces, can be considered as a sequel to [12]. Generalizations of conditional expecations to other (vector valued) contexts can be found in [5,8,9,21].…”