Representations of positive projections 1 Introduction - Mathematics ...

6 Special subalgebras
As before we assume that A is a Dedekind complete f-algebra with unit
element 1 and that B is an f-subalgebra with 1 2 B. Furthermore we
assume that P : A ! A is an order continuous strictly positive projection
onto B. In Section 7 we will construct complete Boolean subalgebras E of
CA with the property that for every e 2 E there exists 2 [0 1] such that
P (e) = 1. In the present section we will collect some properties of such
special Boolean subalgebras.
First we introduce some notation. We de ne the subset S = SP of CA by
S = fp 2CA : P (p) = 1 for some 2 [0 1]g : (28)
The following simple observation will be useful.
Lemma 6.1 The subset S is closed for monotone convergence in CA.
Proof. Since p 2Simplies that 1 ; p 2S, it su ces to show that S is
closed for upwards convergence in CA. To this end suppose that p 2Sand
p 2CA such thatp " p in CA. Then P (p )= 1 for all and 0 " 1.
Let = sup . By the order continuity ofP we have P (p ) " P (p) and so
P (p) = 1. Hence p 2S.
For any non-empty subset D of CA we denote by S (D) the complete
Boolean subalgebra of CA generated by D.
Lemma 6.2 Let S CA be de ned by (28).
(i). If F is a Boolean subalgebra of CA and F S , then S (F) S.
(ii). If F is a Boolean subalgebra of CA such that F S, and if we write
P (e) = (e)1 for all e 2F, then : F ! [0 1] is a strictly positive (in
general, nitely additive) measure on F.
(iii). If E is a complete Boolean subalgebra of CA such that E S, and if
we write P (e) = (e)1 for all e 2E, then : E ![0 1] is a completely
additive and strictly positive measure.
Proof.
(i). This follows via a standard argument for Boolean algebras from Lemma
6.1.
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