  The closed one-sided ideals of a C*-algebra are exactly the closed subspaces
supported by the orthogonal complement of a closed projection. Let A be a (not
necessarily selfadjoint) subalgebra of a unital C*-algebra B which contains the
unit of B. Here we characterize the right ideals of A with left contractive
approximate identity as those subspaces of A supported by the orthogonal
complement of a closed projection in B** which also lies in the weak* closure
of A. Although this seems quite natural, the proof requires a set of new
techniques which may may be viewed as a noncommutative version of the subject
of peak interpolation from the theory of function spaces. Thus, the right
ideals with left approximate identity are closely related to a type of peaking
phenomena in the algebra. In this direction we introduce a class of closed
projections which generalizes the notion of a peak set in the theory of uniform
algebras to the world of operator algebras and operator spaces.
