# Nuclear dimension and $$\mathcal Z$$Z-stability

@article{Sato2015NuclearDA, title={Nuclear dimension and \$\$\mathcal Z\$\$Z-stability}, author={Yasuhiko Sato and Stuart White and Wilhelm Winter}, journal={Inventiones mathematicae}, year={2015}, volume={202}, pages={893-921} }

Simple, separable, unital, monotracial and nuclear $$\mathrm {C}^{*}$$C∗-algebras are shown to have finite nuclear dimension whenever they absorb the Jiang–Su algebra $$\mathcal {Z}$$Z tensorially. This completes the proof of the Toms–Winter conjecture in the unique trace case.

## 54 Citations

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### Ranks of Operators in Simple $$C^*$$-algebras with Stable Rank One

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Let $A$ be a separable, unital, simple C*-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of $A$ is…

### Quasidiagonality of nuclear C*-algebras

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### Approaching the UCT problem via crossed products of the Razak–Jacelon algebra

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We show that the UCT problem for separable, nuclear $\mathrm C^*$-algebras relies only on whether the UCT holds for crossed products of certain finite cyclic group actions on the Razak-Jacelon…

## References

SHOWING 1-10 OF 70 REFERENCES

### 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…

### 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…

### $\mathcal Z$-stability and finite dimensional tracial boundaries

- Mathematics
- 2012

We show that a simple separable unital nuclear nonelementary $C^*$-algebra whose tracial state space has a compact extreme boundary with finite covering dimension admits uniformly tracially large…

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

- Mathematics
- 2012

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)…

### 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…

### Decomposition rank and Z-stability

- Mathematics
- 2008

We show that separable, simple, unital C*-algebras with finite decomposition rank absorb the Jiang-Su algebra Z tensorially. This has a number of consequences for Elliott's program to classify…

### Strict comparison and Z-absorption of nuclear C*-algebras

- Mathematics
- 2011

For any unital separable simple infinite-dimensional nuclear C*-algebra with finitely many extremal traces, we prove that Z-absorption, strict comparison, and property (SI) are equivalent. We also…

### 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…

### 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…

### Discrete amenable group actions on von Neumann algebras and invariant nuclear C*-subalgebras

- Mathematics
- 2011

Let $G$ be a countable discrete amenable group, ${\cal M}$ a McDuff factor von Neumann algebra, and $A$ a separable nuclear weakly dense C$^*$-subalgebra of ${\cal M}$. We show that if two centrally…