Open projections in operator algebras. II: Compact projections (Q2891265)

This paper continues the authors' investigations into the generalization of the properties of \(C^*\)-algebras to the larger class of approximately unital (not necessarily self-adjoint) operator algebras. The crucial property of such an algebra \(A\) is that the intersection of the closed unit ball in \(A\) with the set of elements in the closed unit ball centered at the unit in \(A\) (adjoined if necessary) shares many of the properties of the positive part of the unit ball in a \(C^*\)-algebra.NEWLINENEWLINE In the first part of this paper [ibid. 208, No. 2, 117--150 (2012; Zbl 1259.46045)], the authors studied various generalizations of the `openness' of projections in the biduals of \(C^*\)-algebras and their consequences. In this second part, they concern themselves with the generalization of `compactness', of projections in the second duals of \(C^*\)-algebras. The fact that many of their results are predictable should not detract from the interesting nature of their generalization to a wide class of non-self-adjoint operator algebras.