Representations of positive projections 1 Introduction - Mathematics ...

De nition 2.5
S(E) =
( nX
j=1
jej : j 2 Rej 2Ej =1:::nn2 N
)
: (2)
It is clear that S(E) is a linear subspace of L. Furthermore, every s 2 S(E)
can be written as
s =
nX
j=1
jej, with ei ^ ej =0(i 6= j),
nX
j=1
ej = w: (3)
From this observation it follows easily that S(E) is actually a Riesz subspace
of L, in which w is a strong order unit. Moreover, since E L (E), it follows
that S(E) L (E). For 0 u 2 L we will denote by Pu the band projection
in L onto fug dd .
Proposition 2.6 If L is a Dedekind -complete Riesz space with weak order
unit 0 w 2 L and E is a Boolean -subalgebra of C (w), then the following
statements hold.
(i). If 0 f 2 L, then f 2 L (E) if and only if there exists a sequence
fsng 1
n=1 in S(E) such that 0 sn " f.
(ii). All the band projections in L (E) are all of the form (Pe) jL(E) and all
the bands in L (E) are given by feg dd \ L (E) with e 2E.
(iii). L (E) is a complete Riesz subspace of L if and only if E is a complete
Boolean subalgebra of C (w).
In particular, if L is Dedekind complete and if E is a complete Boolean subalgebra
ofC (w), then L (E) is a complete Riesz subspace ofL and hence, L (E)
is Dedekind complete and regularly embedded in L.
Proof.
(i). This follows immediately from the Freudenthal spectral theorem ( [9],
Theorem 40.3) and from the fact that L (E) isa -complete Riesz subspace
of L.
(ii). If e 2 E and 0 f 2 L (E) then f ^ (ne) 2 L (E) for all n = 1 2:::
and f ^ (ne) " Pef in L. Since L (E) is a -complete Riesz subspace
of L, it follows that Pef 2 L (E). This shows that Pe [L(E)] L(E)
and now it is clear that (Pe) jL(E) is a band projection in L (E). Since
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