  This paper may be viewed as having two aims. First, we continue our study of
algebras of operators on a Hilbert space which have a contractive approximate
identity, this time from a more Banach algebraic point of view. Namely, we
mainly investigate topics concerned with the ideal structure, and hereditary
subalgebras (HSA's), which are in some sense generalization of ideals. Second,
we study properties of operator algebras which are hereditary subalgebras in
their bidual, or equivalently which are `weakly compact'. We also give several
examples answering natural questions that arise in such an investigation.
