Representations of positive projections 1 Introduction - Mathematics ...

for all n. As we have observed before, since we assume that M (N ) bL,
this implies that Rf 2 bL. Hence Rf 2 bL \ M (X ) and it is clear that
bL\M (X ) E. Consequently, wehavea positive linear operator R : E !
E with Ran (R) =E \ M (X
all f 2 E.
). It is easy to see that P ( f) = (Rf) for
De ning 1 = jE, it is clear that 1 : E ! L is a -order continuous
Riesz homomorphism. We will show nowthat1 is surjective. Let 0 g 2 L
be given and take 0 fh 2 bL such that f = g and h = Pg. De ne
F = x 2 X :
Letting fn = f ^ (n1) we have
Z
Y
m (x y) f (x y) d (y) =1 .
(Rfn) =P ( fn) Pg = h,
which implies that (Rfn ; h) + =0. Therefore, the sets
Fn = f(x y) 2 :Rfn (x y) >h(x y)g
satisfy Fn 2 N for all n. Furthermore, it follows from 0 fn " f that
Rfn (x y) "
Z
Y
m (x z) f (x z) d (z)
for all (x y) 2 and so F Y 2 S 1
n=1 Fn. This shows that F Y 2
N . De ning ~ f = f1 (F Y ) c, it is clear that ~ f 2 E and ~ f = g, so we
may conclude that 1 is surjective. Therefore, 1 : E ! L is the desired
representation for P on L.
The above lemma shows that for the construction of representations of
positive projections we may restrict our attention to spaces with a strong
order unit. All such Dedekind complete Riesz spaces have the structure
of an f-algebra and complete Riesz subspaces containing the unit element
are f-subalgebras. Therefore, in the next sections we will study in detail
the properties and structure of positive projections in f-algebras onto fsubalgebras.
4 Some properties of f-subalgebras
For the basic theory of f-algebras we refer the reader to the books [1], [13]
and [16]. Let A be an Archimedean f-algebra with a unit element 1, which
is weak order unit in A. We denote the Boolean algebra of all components
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