Classification of essential commutants of abelian von Neumann algebras (Q583559)

The main purpose of this paper is to classify the \(C^*\)-algebras of the form \({\mathfrak A}'+{\mathfrak K}\), where \({\mathfrak A}'\) denotes the commutant of an abelian von Neumann algebra \({\mathfrak A}\), and \({\mathfrak K}\) is the set of compact operators. By the famous result of Johnson and Parrott, \({\mathfrak A}'+{\mathfrak K}\) is the same as the essential commutant of \({\mathfrak A}\). These algebras were studied by Plastiras in the special case in which \({\mathfrak A}\) is generated by its minimal projections and in addition all of these projections are finite dimensional. Using a theorem of Andersen, we are able to generalize Plastiras' main results to general abelian von Neumann algebras. We also study the automorphism groups and derivations of these algebras.