Open projections in operator algebras II: Compact projections

Abstract

We generalize some aspects of the theory of compact projections relative to a C � -algebra, to the setting of more general algebras. Our main re- sult is that compact projections are the decreasing limits of 'peak projections', and in the separable case compact projections are just the peak projections. We also establish new forms of the noncommutative Urysohn lemma relative to an operator algebra, and we show that a projection is compact iff the asso- ciated face in the state space of the algebra is weak* closed.