Relations and radicals in the lattice of ideals of $C^*$-algebras

In the paper we pursued three aims (in this talk I will mainly dwell on the 1st aim). The first one was to apply Amitsur’s relations and radicals theory for abstract lattices to the study of the lattices $\mathrm{Id}_A$ of closed two-sided ideals of $C^*$-algebras $A$. We showed that many new and well-known results about $C^*$-algebras follow naturally from this approach. To use “relation-radical” approach, we considered various subclasses of $C^*$-algebras, which we call $C^*$-properties, as they often linked to some properties of $C^*$-algebras. We considered $C^*$-properties $P$ consisting of CCR- and of GCR-algebras; of algebras with continuous trace; of real rank zero, AF, nuclear algebras, etc. Each property $P$ defines a reflexive relation $\ll_P$ in lattices $\mathrm{Id}_A$.
Our 2nd aim was to determine the hierarchy and interconnection between various properties. Our 3d aim was to study the link between the radicals of relations $\ll_P$ in the lattices $\mathrm{Id}_A$ and the topological radicals.
(Joint work with Victor S. Shulman and Yurii V. Turovskii)