Regularity of projections revisited
  The concept of regularity in the meta-topological setting of projections in the double dual of a C*-algebra addresses the interrelations of a projection p with its closure, for instance in the form that such projections act identically, in norm, on elements of the C*-algebra. This concept has been given new actuality with the recent plan of Peligrad and Zsido to find a meaningful notion of Murray-von Neumann type equivalence among open projections.   Although automatic in the commutative case, it has been known since the late sixties that regularity fails for many projections. The original investigations, however, did not answer a question such as: "Are all open and dense projections regular in A, when A is simple?"   We report here that this and related questions have negative answers. In the other direction, we supply positive results on regularity of large open projections. 
Charles A. Akemann and Soren Eilers