Related papers: C*-algebras and their nuclear dimension

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…

Kazhdan's notion of property T has recently been imported to the C$^*$-world by Bekka. Our objective is to extend a well known fact to this realm; we show that a nuclear C$^*$-algebra with property T is finite dimensional (for all intents…

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…

We introduce the decomposition rank, a notion of covering dimension for nuclear C^*-algebras. The decomposition rank generalizes ordinary covering dimension and has nice permanence properties; in particular, it behaves well with respect to…

We provide some background on the category of classifiable $\mathrm{C}^*$-algebras, whose objects are infinite-dimensional, simple, separable, unital $\mathrm{C}^*$-algebras that have finite nuclear dimension and satisfy the universal…

We show that the homoclinic C*-algebras of mixing Smale spaces are classifiable by the Elliott invariant. To obtain this result, we prove that the stable, unstable, and homoclinic C*-algebras associated to such Smale spaces have finite…

We characterise subhomogeneity for twisted \'etale groupoid C*-algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear…

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 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 growth rank of a C*-algebra, a (N \cup {\infty})-valued invariant which measures how far an algebra is from absorbing the Jiang-Su algebra Z tensorially. We prove that its range is exhausted by simple nuclear C*-algebras,…

We show that for any locally compact Hausdorff space $Y$ with finite covering dimension and for any continuous flow $\mathbb{R} \curvearrowright Y$, the resulting crossed product $C^*$-algebra $C_0(Y) \rtimes \mathbb{R}$ has finite nuclear…

We give a definition of hypergraph C*-algebras. These generalize the well-known graph C*-algebras as well as ultragraph C*-algebras. In contrast to those objects, hypergraph C*-algebras are not always nuclear. We provide a number of…

I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and amenability for classification, and on the…

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…

Let $G$ be a finitely generated virtually abelian group and $[\sigma]\in H^2(G;\mathbb{T})$ such that $\sigma(x,y)$ is always a root of unity. We show that the nuclear dimension of the twisted group $C^*$-algebra $C^*(G,\sigma)$ is equal to…

The completely positive rank is an analogue of topological covering dimension, defined for nuclear C*-algebras via completely positive approximations. These may be thought of as simplicial approximations of the algebra, which leads to the…

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 construct a simple C*-algebra with nuclear dimension zero that is not isomorphic to its tensor product with the Jiang-Su algebra Z, and a hyperfinite II_1 factor not isomorphic to its tensor product with the separable hyperfinite II_1…

We exhibit examples of simple separable nuclear C*-algebras, along with actions of the circle group and outer actions of the integers, which are not equivariantly isomorphic to their opposite algebras. In fact, the fixed point subalgebras…

The subject of quasidiagonality is of much interest in many places - among other things, in the classification program for simple unital separable nuclear C*-algebras. In this note, we give two characterizations of nuclearity and…