  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.
