POSITIVE OPERATORS

1.3. Order Projections 33
(2) If x ⊥ A ⊕ A d , then x ⊥ A and x ⊥ A d both hold. Therefore,
x ∈ A d ∩ A dd = {0}. This shows that (A ⊕ A d ) d = {0}. By part (1) the
ideal A ⊕ A d is order dense in E.
(3) This follows immediately from part (1).
Anet{xα} of a Riesz space is said to be order convergent to a vector x
(in symbols xα−→ o x) whenever there exists another net {yα} with the same
index set satisfying yα ↓ 0and|xα − x| ≤yα for all indices α (abbreviated as
|xα − x| ≤yα ↓ 0). A subset A of a Riesz space is said to be order closed
whenever {xα} ⊆A and xα−→ o x imply x ∈ A.
Lemma 1.37. A solid subset A of a Riesz space is order closed if and only
if {xα} ⊆A and 0 ≤ xα ↑ x imply x ∈ A.
Proof. Assume that a solid set A of a Riesz space has the stated property
and let a net {xα} ⊆A satisfy xα−→ o x. Pick a net {yα} with the same
index net satisfying yα ↓ 0and|xα − x| ≤yα for each α. Now note that we
have (|x|−yα) + ≤|xα| for each α and 0 ≤ (|x|−yα) + ↑|x|, and from this
it follows that x ∈ A. That is, A is order closed.
An order closed ideal is referred to as a band. Thus, according to
Lemma 1.37 an ideal A is a band if and only if {xα} ⊆A and 0 ≤ xα ↑ x
imply x ∈ A (or, equivalently, if and only if D ⊆ A + and D ↑ x imply
x ∈ A). In the early developments of Riesz spaces a band was called a
normal subspace (G. Birkhoff [36], S. Bochner and R. S. Phillips [39]),
while F. Riesz was calling a band a famille complète.
Let A be a nonempty subset of a Riesz space E. Then the ideal generated
by A is the smallest (with respect to inclusion) ideal that includes
A. A moment’s thought reveals that this ideal is
EA =
x ∈ E : ∃ x1,...,xn ∈ A and λ ∈ R + with |x| ≤λ
n
i=1
|xi| .
The ideal generated by a vector x ∈ E will be denoted by Ex. By the
preceding discussion we have
Ex = y ∈ E : ∃ λ>0with|y| ≤λ|x| .
Every ideal of the form Ex is referred to as a principal ideal.
Similarly, the band generated by a set A is the smallest band that
includes the set A. Such a band always exists (since it is the intersection
of the family of all bands that include A, andE is one of them.) Clearly,
the band generated by A coincides with the band generated by the ideal
generated by A. The band generated by a vector x is called the principal
band generated by x and is denoted by Bx.