Topological radicals, IX: relations in ideals of C*-algebras

In this paper, we pursue three aims. The first one is to apply Amitsur’s relations and radicals theory to the study of the lattices \(\hbox {Id}_{{A}}\) of closed two-sided ideals of C*-algebras A. We show that many new and many well-known results about C*-algebras follow naturally from this approach. To use “relation-radical” approach, we consider various subclasses of the class \({\mathfrak {A}}\) of all C*-algebras, which we call C*-properties, as they often linked to some properties of C*-algebras. We consider C*-properties P consisting of CCR- and of GCR-algebras; of C*-algebras with continuous trace; of real rank zero, AF, nuclear C*-algebras, etc. Each P defines reflexive relations \(\ll _{{P}}\) in all lattices \(\hbox {Id}_{A}.\) Our second aim is to determine the hierarchy and interconnection between properties in \({\mathfrak {A}}.\) Our third aim is to study the link between the radicals of relations \(\ll _{{P}}\) in the lattices \(\hbox {Id}_{{A}}\) and the topological radicals on \({\mathfrak {A}}.\)