The regular completion of a C*-algebra with large projections

Abstract

Abstract When a unital C*-algebra A is prime and has very large projections, it is shown that the regular completion A^ of the algebra A is a simple, wild type III AW*-factor that has no non-zero σ-finite projections. For example, the Weaver algebra, the Crabb algebra and the Katsura algebra are prime C*-algebras that have very large projections. As a corollary, such algebras have no non-zero abelian elements, that is, they are not of type I.