Operator algebras with contractive approximate identities II

We make several contributions to our recent program investigating structural properties of algebras of operators on a Hilbert space. For example, we make substantial contributions to the noncommutative peak interpolation program begun by Hay and the first author, Hay and Neal. Another sample result: an operator algebra has a contractive approximate identity iff the linear span of the elements with positive real part is dense. We also extend the theory of compact projections to the most general case. Despite the title, our algebras are often allowed to have no approximate identity.