  We give several applications of a recent theorem of the second author, which
solved a conjecture of the first author with Hay and Neal, concerning
contractive approximate identities; and another of Hay from the theory of
noncommutative peak sets, thereby putting the latter theory on a much firmer
foundation. From this theorem it emerges there is a surprising amount of
positivity present in any operator algebras with contractive approximate
identity. We exploit this to generalize several results previously available
only for $C^*$-algebras, and we give many other applications.
