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Kissin, Edward, Shulman, Victor and Turovskii, Yuri (2024) Topological radicals, IX. Relations in ideals of C*-algebras. Annals of Functional Analysis, 16 (1). pp. 1-49. ISSN 2639-7390
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 IdA 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 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≪P in all lattices IdA.
Our second aim is to determine the hierarchy and interconnection between properties in A.
Our third aim is to study the link between the radicals of relations ≪P in the lattices IdA and the topological radicals on A.
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