An algebraic approach to the radius of comparison (Q2839948)

The radius of comparison was introduced by \textit{A. S. Toms} [Proc. Lond. Math. Soc. (3) 96, No. 1, 1--25 (2008; Zbl 1143.46037)] as one of several noncommutative versions of covering dimension for \(C^*\)-algebras. The paper under review gives an algebraic (as opposed to functional-theoretic) reformulation of this invariant. This allows to prove that the radius of comparison decreases by passage to a quotient, and is lower semicontinuous with respect to inductive limits.NEWLINENEWLINENew 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. Finally, the radius of comparison is used to study stability of \(C^*\)-algebras, in particular, to check stability of full hereditary subalgebras of stable \(C^*\)-algebras and of matrix algebras over \(C^*\)-algebras with certain properties.