Relation and radical approach to the theory of C*-algebras

TL;DR

This paper applies relation and radical theory to the study of C*-algebra ideals, revealing new insights into their structure and properties through a unified approach involving various subclasses of C*-algebras.

Contribution

It introduces a relation-radical framework for analyzing C*-algebras and explores the hierarchy and connections between different algebraic properties and radicals.

Findings

Many known results follow from the relation-radical approach.

Hierarchy and interrelations between C*-properties are established.

Links between lattice radicals and topological radicals are analyzed.

Abstract

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 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 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 in all lattices Id_A. 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 in the lattices Id_A and the topological radicals on A.