  We continue our study of operator algebras with contractive approximate
identities (cais). In earlier papers we have introduced and studied a new
notion of positivity in operator algebras, with an eye to extending certain
$C^*$-algebraic results and theories to more general algebras. In the first
part of this paper we do a more systematic development of this positivity and
its associated ordering, proving many foundational facts. In much of this it is
not necessary that the algebra have an approximate identity. In the second part
we study interesting examples of operator algebras with cais, which in
particular answer questions raised in previous papers in this series. Indeed
the present work solves almost all of the open questions raised in these
papers. Many of our results apply immediately to function algebras, but we will
not take the time to point these out, although most of these applications seem
new.
