Open projections in operator algebras II: Compact projections
Volume 209 / 2012
Studia Mathematica 209 (2012), 203-224
MSC: Primary 46L07, 46L85, 47L30, 46L52; Secondary 32T40, 46H10, 46L05, 46L30.
DOI: 10.4064/sm209-3-1
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 result 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 associated face in the state space of the algebra is weak$^*$ closed.
