Topological radicals, V. From algebra to spectral theory

TL;DR

This paper explores the spectral theory of joint spectral radius, introducing the scattered radical, and establishes continuity conditions for spectral radii in GCR C*-algebras, linking algebraic and spectral properties.

Contribution

It develops the theory of the scattered radical and provides new continuity results for spectral radii in GCR C*-algebras, connecting algebraic structures to spectral analysis.

Findings

Continuity of spectrum and spectral radii under certain conditions

Joint spectral radius is continuous on precompact subsets in GCR C*-algebras

Joint spectral radius coincides with Berger-Wang radius in GCR C*-algebras

Abstract

We introduce and study procedures and constructions in the theory of the joint spectral radius that are related to the spectral theory. In particular we devlop the theory of the scattered radical. Among applications we find some sufficient conditions of continuity of the spectrum and spectral radii of various types, and prove that in GCR C*-algebras the joint spectral radius is continuous on precompact subsets and coincides with the Berger-Wang radius.