# Nuclear dimension and $\mathcal{Z}$-stability of pure C∗-algebras

@article{Winter2012NuclearDA, title={Nuclear dimension and \$\mathcal\{Z\}\$-stability of pure C∗-algebras}, author={Wilhelm Winter}, journal={Inventiones mathematicae}, year={2012}, volume={187}, pages={259-342} }

In this article I study a number of topological and algebraic dimension type properties of simple C∗-algebras and their interplay. In particular, a simple C∗-algebra is defined to be (tracially) $(m,\bar{m})$-pure, if it has (strong tracial) m-comparison and is (tracially) $\bar{m}$-almost divisible. These notions are related to each other, and to nuclear dimension.The main result says that if a separable, simple, nonelementary, unital C∗-algebra with locally finite nuclear dimension is $(m…

## 130 Citations

### Nuclear dimension, $$\mathcal{Z }$$Z-stability, and algebraic simplicity for stably projectionless $$C^*$$C∗-algebras

- Mathematics
- 2014

The main result here is that a simple separable $$C^*$$C∗-algebra is $$\mathcal{Z }$$Z-stable (where $$\mathcal{Z }$$Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it…

### On classification of simple non-unital amenable C*-algebras, II

- Mathematics
- 2017

We present a classification theorem for amenable simple stably projectionless C*-algebras with generalized tracial rank one whose $K_0$ vanish on traces which satisfy the Universal Coefficient…

### Classifying crossed product C*-algebras

- Mathematics
- 2013

I combine recent results in the structure theory of nuclear ${\rm C}^*$-algebras and in topological dynamics to classify certain types of crossed products in terms of their Elliott invariants. In…

### Strict comparison and $$ \mathcal{Z} $$-absorption of nuclear C∗-algebras

- Mathematics
- 2012

For any unital separable simple infinite-dimensional nuclear C∗-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$-absorption, strict comparison and property (SI) are…

### Nuclear dimension and Z-stability of non-simple C*-algebras

- Mathematics
- 2013

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…

### Rokhlin Dimension for Flows

- Mathematics
- 2016

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of $${C^*}$$C∗-algebras. This generalizes Kishimoto’s Rokhlin property for flows, and is analogous to the notion of…

### $${\mathcal {Z}}$$ -Stability of Crossed Products by Strongly Outer Actions

- Mathematics
- 2012

We consider a certain class of unital simple stably finite C*-algebras which absorb the Jiang-Su algebra $${\mathcal {Z}}$$ tensorially. Under a mild assumption, we show that the crossed product of a…

### The C∗-algebra of a minimal homeomorphism of zero mean dimension

- Mathematics
- 2017

Let $X$ be an infinite compact metrizable space, and let $\sigma: X\to X$ be a minimal homeomorphism. Suppose that $(X, \sigma)$ has zero mean topological dimension. The associated C*-algebra…

### Classification of C*-algebras generated by representations of the unitriangular group $UT(4,\mathbb{Z})$

- Mathematics
- 2015

### Strict Comparison of Positive Elements in Multiplier Algebras

- MathematicsCanadian Journal of Mathematics
- 2017

Abstract Main result: If a ${{C}^{*}}$ -algebra $\mathcal{A}$ is simple, $\sigma $ -unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its…

## References

SHOWING 1-10 OF 45 REFERENCES

### THE STABLE AND THE REAL RANK OF ${\mathcal Z}$-ABSORBING C*-ALGEBRAS

- Mathematics
- 2004

Suppose that A is a C*-algebra for which $A \cong A \otimes {\mathcal Z}$, where ${\mathcal Z}$ is the Jiang–Su algebra: a unital, simple, stably finite, separable, nuclear, infinite-dimensional…

### Asymptotic unitary equivalence and classification of simple amenable C∗-algebras

- Mathematics
- 2008

AbstractLet C and A be two unital separable amenable simple C∗-algebras with tracial rank at most one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ϕ1,ϕ2:C→A are two…

### SimpleC*-algebra generated by isometries

- Mathematics
- 1977

AbstractWe consider theC*-algebra
$$\mathcal{O}_n $$
generated byn≧2 isometriesS1,...,Sn on an infinite-dimensional Hilbert space, with the property thatS1S*1+...+SnS*n=1. It turns out that…

### 𝓏-Stable ASH Algebras

- MathematicsCanadian Journal of Mathematics
- 2008

Abstract The Jiang–Su algebra $Z$ has come to prominence in the classification program for nuclear ${{C}^{*}}$ -algebras of late, due primarily to the fact that Elliott’s classification conjecture in…

### On the classification of simple inductive limit C*-algebras, II: The isomorphism theorem

- Mathematics
- 2007

In this article, it is proved that the invariant consisting of the scaled ordered K-group and the space of tracial states, together with the natural pairing between them, is a complete invariant for…

### On topologically finite-dimensional simple C*-algebras

- Mathematics
- 2003

We show that, if a simple C*-algebra A is topologically finite-dimensional in a suitable sense, then not only K0(A) has certain good properties, but A is even accessible to Elliott’s classification…

### Decomposition rank and $\mathcal{Z}$ -stability

- Mathematics
- 2009

AbstractWe show that separable, simple, nonelementary, unital C*-algebras with finite decomposition rank absorb the Jiang–Su algebra
$\mathcal{Z}$
tensorially. This has a number of consequences for…

### Dimension functions on simpleC*-algebras

- Mathematics
- 1978

In order to make available for C*-atgebras the results of Goodearl and Handelman [5] on existence and uniqueness of rank functions on regular rings, we associate in the present note with every…