Representations of positive projections 1 Introduction - Mathematics ...

Proof. If f
(o)
(o)
;! f, then it is clear that f ; f 0
(o)
;! 0. Now suppose
that f ; f 0 ;! 0, i.e., there exist a net fu g in L satisfying u # 0such
2B
that for every 2 B there exists ; 0 2 A A such that jf ; f 0j u
; 0 0 whenever ( ) . Replacing and 0 00 0 by some , we
may assume without loss of generality that = 0 . Fix some 0 2 B . It is
clear that ff :
0
0g is order bounded. Moreover,
_ f ; ^
f 0 = _
jf ; f 0j u
for all 0. Since u # 0, it follows that
_
0
0
0
^ f = ^
0
_ f ,
and this implies that (4) holds with 0 = 0, so ff g is order convergent.
Next we will prove a simple extension result for positive operators which
will be used in a later section of the paper. Given a Riesz subspace K of the
Riesz space L we will denote by K0 the set of all f 2 L for which there exists
(o)
anetff g in K such that f ;! f in L. It is easy to see that K0 is a Riesz
subspace of L.
Lemma 2.14 Let K be a Riesz subspace of the Archimedean Riesz space L
and let M be aDedekind complete Riesz space. Suppose that 0 T : K ! M
is a positive linear operator with the property that Tf
(o)
;! 0 in M whenever
ff g is a net in K satisfying f
(o)
;! 0 in L. Then there exists a unique
positive linear operator 0 T 0 : K 0 ! M such that for every f 2 K 0 and any
net ff g K with f
(o)
;! f we have Tf
(o)
;! T 0f (in particular, T 0
jK = T ).
Proof. Take f 2 K 0 and let ff g be a net in K such that f
(o)
;! f in
L. Then f ; f 0
(o)
;! 0inL and so, by hypothesis, Tf ; Tf 0
(o)
;! 0inM.
Now it follows from Lemma 2.13 that fTf g is order convergent in M, i.e.,
(o)
Tf ;! g for some g 2 M. We claim that g does not depend on the choice
of the particular net ff g. Indeed, suppose that fh g is any other net in K
such that h
(o)
;! f in L. Then f ; h
on T it follows that Tf ; Th
(o)
(o)
;! 0 in L and by the assumption
;! 0 in M. This implies that Th
(o)
;! g.
De ning T 0 f = g, it is easy to verify that T 0 : K 0 ! M is a positive linear
operator which has, by de nition, the desired property. The uniqueness is
obvious.
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