相关论文: On the nuclear dimension of strongly purely infini…
We introduce the nuclear dimension of a C*-algebra; this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank. Our notion behaves well with respect to inductive…
We prove that Z-stable, simple, separable, nuclear, non-unital C*-algebras have nuclear dimension at most 1. This completes the equivalence between finite nuclear dimension and Z-stability for simple, separable, nuclear, non-elementary…
We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra $A$ generated by an irreducible representation of such a group has…
We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we…
We show that every nuclear $\mathcal O_\infty$-stable *-homomorphism with a separable exact domain has nuclear dimension at most 1. In particular separable, nuclear, $\mathcal O_\infty$-stable C*-algebras have nuclear dimension 1. We also…
We write arbitrary separable nuclear C*-algebras as limits of inductive systems of finite-dimensional C*-algebras with completely positive connecting maps. The characteristic feature of such CPC*-systems is that the maps become more and…
Consider a graph C*-algebra C*(E) with a purely infinite ideal I (possibly all of C*(E)) such that I has only finitely many ideals and C*(E)/I is approximately finite dimensional. We prove that the nuclear dimension of C*(E) is 1. If I has…
Nuclear $C^*$-algebras having a system of completely positive approximations formed with convex combinations of a uniformly bounded number of order zero summands are shown to be approximately finite dimensional.
We show that if X is a finite dimensional locally compact Hausdorff space, then the crossed product of C_0(X) by any automorphism has finite nuclear dimension. This generalizes previous results, in which the automorphism was required to be…
Simple, separable, unital, monotracial and nuclear C$^*$-algebras are shown to have finite nuclear dimension whenever they absorb the Jiang-Su algebra $\mathcal{Z}$ tensorially. This completes the proof of the Toms-Winter conjecture in the…
We compute the nuclear dimension of extensions of C*-algebras involving commutative unital quotients and stable Kirchberg ideals. We identify the finite directed graphs whose C*-algebras are covered by this theorem.
The main result here is that a simple separable C*-algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the…
It is shown that if a C*-algebra has nuclear dimension $n$ then its Cuntz semigroup has the property of $n$-comparison. It then follows from results by Ortega, Perera, and Rordam that $\sigma$-unital C*-algebras of finite nuclear dimension…
We show that if A is a separable, nuclear, O_infty-absorbing (or strongly purely infinite) C*-algebra, which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form M_k(C_0(G,v)),…
The problem of expressing a selfadjoint element that is zero on every bounded trace as a finite sum (or a limit of sums) of commutators is investigated in the setting of C*-algebras of finite nuclear dimension. Upper bounds -- in terms of…
We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear…
We introduce the completely positive rank, a notion of covering dimension for nuclear $C^*$-algebras and analyze some of its properties. The completely positive rank behaves nicely with respect to direct sums, quotients, ideals and…
We investigate the interplay of the following regularity properties for non-simple C*-algebras: finite nuclear dimension, Z-stability, and algebraic regularity in the Cuntz semigroup. We show that finite nuclear dimension implies algebraic…
We show that a $C^*$-algebra $A$ is nuclear iff there is a constant $K$ and $\alpha<3$ such that, for any bounded homomorphism $u\colon A \to B(H)$, there is an isomorphism $\xi\colon H\to H$ satisfying $\|\xi^{-1}\|\|\xi\| \le…
We obtain an improved upper bound for the nuclear dimension of extensions of $\mathcal{O}_\infty$-stable $\rm{C}^*$-algebras. In particular, we prove that the nuclear dimension of a full extension of an $\mathcal{O}_\infty$-stable…