An algebraic approach to the radius of comparison

By B. Blackadar, Leonel Robert, Aaron Tikuisis, Andrew S. Toms, W. Winter. 2010

57 citations recorded by the source corpus.

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 Mn(D) is stable.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/abd3b140b2d8b340a6aa4421ec9c29a3c6169624

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Pullbacks, $C(X)$-algebras, and their Cuntz semigroup (2011, background): “ ﬁbres are stable, we may assume that A and its ﬁbres are stable at the outset. Let 0 < ǫ < 1 be ﬁxed, and let us suppose that a,b ∈A are positive contractions such that a(t) - b(t), for all t ∈[0,1]. We need to show that a - b. Let B := A+C[0,1]·1M(A). Then B is a C*-algebra that contains A a”
The cone of functionals on the Cuntz semigroup (2011, background): “Such a study was done partly in [6] and [2] and is continued here.”
Nuclear dimension and Z-stability of non-simple C*-algebras (2013, methodology): “t to an element of M 2(A) +. Proof. Let < 1 M+1 be such that [b] < s [a]. We follow the proof of [4, Theorem 4.4.1], with modications as follows. Note that Cuntz classes are denoted using hiin [4], though we will stick with our convention here and use []. In the proof of [4, Theorem 4.4.1], we”
Large subalgebras (2014, background): “4 of [7], does not address the following difficulty.”
Nuclear dimension and -stability of pure C ∗ -algebras (2010, unknown): “[7] or [5]; it follows from [3] that in fact x ∈ W(A) ⊂ W(A ⊗ K).”