An algebraic approach to the radius of comparison

TL;DR

This paper extends the concept of the radius of comparison to general C*-algebras, providing algebraic reformulations, new properties, and applications including stability criteria for subalgebras and examples of finite radius of comparison.

Contribution

It introduces an algebraic reformulation of the radius of comparison for C*-algebras and explores its implications for stability and classification.

Findings

New algebraic reformulation of the radius of comparison

Characterization of stability for hereditary subalgebras

Examples of C*-algebras with finite radius of comparison

Abstract

The radius of comparison is an invariant for unital C*-algebras which extends the theory of covering dimension to noncommutative spaces. We extend its definition to general C*-algebras, and give an algebraic (as opposed to functional-theoretic) reformulation. This yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces. We then give several applications of these results. New examples of C*-algebras with finite radius of comparison are given, and the question of when the Cuntz classes of finitely generated Hilbert modules form a hereditary subset of the Cuntz semigroup is addressed. Most interestingly, perhaps, we treat the question of when a full hereditary subalgebra B of a stable C*-algebra A is itself stable, giving a characterization in terms of the radius of comparison. We also use the radius of comparison to quantify the least n for which a C*-algebra D without bounded 2-quasitraces or unital quotients has the property that M_n(D) is stable.