Representations of positive projections 1 Introduction - Mathematics ...

(ii). for every 2 Z(B 0) there exists 2 Z(K) such that jB0 =
(iii). [0Q 0u] L = Q 0 [0u] K for all 0 u 2 K (where the subscripts L and
K indicate that the order intervals are taken in L and K respectively)
(iv). Q 0(K) is an ideal in L.
Proof. We will only show that (iv) implies (i), as the remaining implications
follow by standard arguments. So we assume that Q 0(K) isan ideal
in L. De ne Q 0 2P(K) by
Q 0 =inffR 2P(K) :Q 0
and de ne : Q 0(K) ! Q 0(K) by (f) =Q 0f for all f 2 Q 0(K). It is clear
that is a surjective Riesz homomorphism. We claim that is injective.
Suppose that 0 f 2 Q 0(K) and Q 0f =0. Since
and ; I ; Q 0 f =0,it follows that
Rg
I ; Q 0 =supfR 2P(K) :RQ 0 =0g
sup fRf : R 2P(K)RQ 0 =0g =0
i.e., Rf = 0 for all R 2P(K) withRQ 0 =0. Let Rf be the band projection
in L onto ffg dd . Since f 2 K, it is easy to see that Rf 2P(K), and Q 0f =0
implies that RfQ 0 =0. Hence f =0,which proves the claim.
Now let Q 2P(L) be given such that Q Q 0. Let Q 1 be the restriction
of Q to Q 0(K). Then Q 1 is a band projection in Q 0(K), as Q 0(K) isanideal
in L. By the rst part of the proof, there exists a band projection R in Q 0(K)
such that R = Q 1 , i.e., Q 0Rf = Q 1Q 0f = Qf for all f 2 Q 0(K).
Considering R as an element ofP(K), this shows that Q 0R = Q and we may
conclude that Q 0 satis es (27), i.e., Q 0 is Z(K)-full.
We will say that L is nowhere full with respect to K if 0 is the only
band projection in L which is full with respect to Z (K). By de nition,
this is equivalent to saying that Z(L) is nowhere full with respect to Z(K).
Therefore the following result is now an immediate consequence of Theorem
5.4.
Theorem 5.7 Let L be a Dedekind complete Riesz space and K an order
dense Dedekind complete regular Riesz subspace ofL, such that L is nowhere
full with respect to K. Suppose that 0 < '0 2 Ls n is strictly positive. If
2 Ks is such that 0 ('0) jK , then there exists a band projection
Q 2P(L) such that =('0 Q) jK , i.e., is the restriction of acomponent
of '0. 40