Positivity (2005) 9:293–325 © Springer 2005
Representations of Positive Projections I
and B. DE PAGTER
Mathematics, California Institute of Technology, Pasadena, CA, USA;
Department of Mathematics, Delft University of Technology, Delft, The Netherlands
Abstract. In this paper we start the development of a general theory of Maharam-type
representation theorems for positive projections on Dedekind complete vector lattices. In
the approach to these results the theory off -algebras plays a crucial role.
AMS classiﬁcation: 47B65, 46A40, 06F25
About half a century ago Dorothy Maharam published an important series
of papers dealing with the characterization of measure algebras and their
decomposition properties and applied them to prove a number of deep rep-
resentation theorems for positive linear transformations between spaces of
measurable functions as kernel operators by means of what is now called
Maharam’s slice integral method. For a discussion of these decomposition
theorems for measure algebras and their subalgebras we refer the reader
also to Fremlin’s contribution in .
In , the ﬁrst author presented a brief summary and discussion of some
of Maharam’s groundbreaking work. By using the language of the theory
of Riesz spaces (vector lattices) it was suggested that an algebraization of
her work would be a worthwhile undertaking. This led to the paper 
dealing with Maharam extensions of positive linear operators in the set-
ting of the theory of f -modules (i.e., vector lattices which are modules over
f -algebras). For an alternative exposition of results on Maharam exten-
sions we refer the reader also to the book .
In the present paper, we start the development of a general theory
of Maharam-type representation theorems mentioned above for positive
projections on unital f -algebras as well as on Dedekind complete vec-
tor lattices which are what we call locally order separable. The main
results concerning representations of positive projections are collected in
the theorems in Section 7 of the forthcoming paper .
To Dorothy Maharam in admiration.