Positivity 13 (2009), 543–558
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/030543-16, published online October 4, 2008
An Andˆo-Douglas type theorem in Riesz
spaces with a conditional expectation
Bruce A. Watson
Abstract. In this paper we formulate and prove analogues of the Hahn-Jordan
decomposition and an Andˆo-Douglas-Radon-Nikod´ym theorem in Dedekind
complete Riesz spaces with a weak order unit, in the presence of a Riesz space
conditional expectation operator. As a consequence we can characterize those
subspaces of the Riesz space which are ranges of conditional expectation ope-
rators commuting with the given conditional expectation operators and which
have a larger range space. This provides the ﬁrst step towards a formulation
of Markov processes on Riesz spaces.
Mathematics Subject Classiﬁcation (2000). 47B60, 60G40, 60G48, 60G42.
Keywords. Riesz spaces, Andˆo-Douglas theorem, Radon-Nikod´ym theorem,
R.G. Douglas in [7, Theorem 3] characterized the range spaces of contractive pro-
jections on L
and hence characterized the subspaces of L
which are range spaces
of conditional expectations in L
. The results of Douglas were extended to the L
space context by Andˆo, . A survey of these and related results can be found
in [19, pages 392-401]. Using Banach lattice techniques, Y.A. Abramovich, C.D.
Aliprantis and O. Burkinshaw in , for p = 1, and, Y.A. Abramovich and C.D.
Aliprantis in [1, Sections 5.3, 5.4], for ∞ >p≥ 1, give extremely elegant proofs of
the results of Douglas and Andˆo. In , S.J. Bernau and H.E. Lacey consider the
closely related problem of characterizing the range spaces of contractive projections
spaces. W.A.J. Luxemburg and B. de Pagter in [15, Proposition 4.2] prove
an Andˆo-Douglas type theorem in the context of Dedekind complete f-algebras
This work was supported in part by the Centre for Applicable Analysis and Number Theory and
by South African National Research Foundation grant FA2007041200006.