Hereditary subalgebras of operator algebras

In recent work of the second author, a technical result was proved establishing a bijective correspondence between certain open projections in a C*-algebra containing an operator algebra A, and certain one-sided ideals of A. Here we give several remarkable consequences of this result. These include a generalization of the theory of hereditary subalgebras of a C*-algebra, and the solution of a ten year old problem on the Morita equivalence of operator algebras. In particular, the latter gives a very clean generalization of the notion of Hilbert C*-modules to nonselfadjoint algebras. We show that an `ideal' of a general operator space X is the intersection of X with an `ideal' in any containing C*-algebra or C*-module. Finally, we discuss the noncommutative variant of the classical theory of `peak sets'.