# 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}
}
• Published 4 March 2014
• Mathematics
• Inventiones mathematicae
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.
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## 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

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### $\mathcal Z$-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

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

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

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

• 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

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