An algebraic approach to the radius of comparison

Blackadar, Bruce; Robert, Leonel; Tikuisis, Aaron; Toms, Andrew; Winter, Wilhelm

doi:10.1090/S0002-9947-2012-05538-3

An algebraic approach to the radius of comparison
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by Bruce Blackadar, Leonel Robert, Aaron P. Tikuisis, Andrew S. Toms and Wilhelm Winter PDF

Trans. Amer. Math. Soc. 364 (2012), 3657-3674 Request permission

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 $\mathrm {M}_n(D)$ is stable.

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