##
**Nuclear dimension, \(\mathcal Z\)-stability, and algebraic simplicity for stably projectionless \(C^\ast\)-algebras.**
*(English)*
Zbl 1319.46043

This paper generalizes deep theorems in the structure theory of simple, nuclear \(C^\ast\)-algebras by showing that the usual assumption of being unital is not necessary. Besides the importance of removing an unnecessary assumption, this paper also allows one to extend some known results to non-simple \(C^*\)-algebras with finitely many ideals.

Elliott initiated the program of classifying nuclear, separable, simple \(C^\ast\)-algebras by \(K\)-theoretic and tracial data. The Elliott program can be thought of as an attempt to transfer Connes’ celebrated result of uniqueness of the hyperfinite \(\mathrm{II}_1\)-factor from von Neumann algebras to \(C^\ast\)-algebras. It has become clear that, in order to achieve a classification, one has to assume certain additional regularity properties.

The Toms-Winter conjecture predicts that three properties of very different nature agree (for a nuclear, separable, simple, non-elementary \(C^\ast\)-algebra \(A\)):

(1) \(A\) has finite nuclear dimension. (2) \(A\) is \(\mathcal{Z}\)-stable, which means \(A\cong A\otimes\mathcal{Z}\), where \(\mathcal{Z}\) is the Jiang-Su algebra. (3) The Cuntz semigroup \(Cu(A)\) is almost unperforated.

The first condition is of topological nature. The second condition can be thought of as the \(C^\ast\)-algebraic analogue of being a McDuff factor. Finally, the last condition roughly means that positive elements in \(A\) can be compared by dimension functions.

It was shown by Rørdam that (2) implies (3); see Theorem 4.5 in [M. Rørdam, Int. J. Math. 15, No. 10, 1065–1084 (2004; Zbl 1077.46054)]. For unital algebras, Winter showed that (1) implies (2), and that (3) implies (2) under the addition assumption that \(A\) has locally finite nuclear dimension (for example, if \(A\) is approximately subhomogeneous) and \(Cu(A)\) is almost divisible; see Theorem 7.1 and Corollary 7.3 in [W. Winter, Invent. Math. 187, No. 2, 259–342 (2012; Zbl 1280.46041)].

The main result of the paper, Theorem 8.5, generalizes Winter’s result to not necessarily unital algebras.

One of the main notions, developed in Section 2, is algebraic simplicity of a \(C^\ast\)-algebra. Every unital, simple \(C^\ast\)-algebra is algebraically simple. In Corollary 2.2 it is shown that every \(\sigma\)-unital, simple \(C^\ast\)-algebra is stably isomorphic to an algebraically simple \(C^\ast\)-algebra. (One can take the hereditary sub-\(C^\ast\)-algebra generated by any element in the Pedersen ideal of the original \(C^\ast\)-algebra.) Since the properties appearing in the Toms-Winter conjecture are invariant under stable isomorphism, it is therefore enough to consider algebraically simple algebras.

In Sections 3 to 8, which contain most of the work, the author develops techniques that are used to show that the considered \(C^\ast\)-algebras are \(\mathcal{Z}\)-stable. Section 9 contains an application to approximately subhomogeneous (ASH) \(C^\ast\)-algebras. It is shown that a simple ASH algebra has slow dimension growth if and only if it is \(\mathcal{Z}\)-stable; see Corollary 9.2. Under the additional assumption that the algebra is unital, this was shown in [A. Toms, Invent. Math. 183, No. 2, 225–244 (2011; Zbl 1237.19009)].

Elliott initiated the program of classifying nuclear, separable, simple \(C^\ast\)-algebras by \(K\)-theoretic and tracial data. The Elliott program can be thought of as an attempt to transfer Connes’ celebrated result of uniqueness of the hyperfinite \(\mathrm{II}_1\)-factor from von Neumann algebras to \(C^\ast\)-algebras. It has become clear that, in order to achieve a classification, one has to assume certain additional regularity properties.

The Toms-Winter conjecture predicts that three properties of very different nature agree (for a nuclear, separable, simple, non-elementary \(C^\ast\)-algebra \(A\)):

(1) \(A\) has finite nuclear dimension. (2) \(A\) is \(\mathcal{Z}\)-stable, which means \(A\cong A\otimes\mathcal{Z}\), where \(\mathcal{Z}\) is the Jiang-Su algebra. (3) The Cuntz semigroup \(Cu(A)\) is almost unperforated.

The first condition is of topological nature. The second condition can be thought of as the \(C^\ast\)-algebraic analogue of being a McDuff factor. Finally, the last condition roughly means that positive elements in \(A\) can be compared by dimension functions.

It was shown by Rørdam that (2) implies (3); see Theorem 4.5 in [M. Rørdam, Int. J. Math. 15, No. 10, 1065–1084 (2004; Zbl 1077.46054)]. For unital algebras, Winter showed that (1) implies (2), and that (3) implies (2) under the addition assumption that \(A\) has locally finite nuclear dimension (for example, if \(A\) is approximately subhomogeneous) and \(Cu(A)\) is almost divisible; see Theorem 7.1 and Corollary 7.3 in [W. Winter, Invent. Math. 187, No. 2, 259–342 (2012; Zbl 1280.46041)].

The main result of the paper, Theorem 8.5, generalizes Winter’s result to not necessarily unital algebras.

One of the main notions, developed in Section 2, is algebraic simplicity of a \(C^\ast\)-algebra. Every unital, simple \(C^\ast\)-algebra is algebraically simple. In Corollary 2.2 it is shown that every \(\sigma\)-unital, simple \(C^\ast\)-algebra is stably isomorphic to an algebraically simple \(C^\ast\)-algebra. (One can take the hereditary sub-\(C^\ast\)-algebra generated by any element in the Pedersen ideal of the original \(C^\ast\)-algebra.) Since the properties appearing in the Toms-Winter conjecture are invariant under stable isomorphism, it is therefore enough to consider algebraically simple algebras.

In Sections 3 to 8, which contain most of the work, the author develops techniques that are used to show that the considered \(C^\ast\)-algebras are \(\mathcal{Z}\)-stable. Section 9 contains an application to approximately subhomogeneous (ASH) \(C^\ast\)-algebras. It is shown that a simple ASH algebra has slow dimension growth if and only if it is \(\mathcal{Z}\)-stable; see Corollary 9.2. Under the additional assumption that the algebra is unital, this was shown in [A. Toms, Invent. Math. 183, No. 2, 225–244 (2011; Zbl 1237.19009)].

Reviewer: Hannes Thiel (Münster)

### MSC:

46L35 | Classifications of \(C^*\)-algebras |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

46L05 | General theory of \(C^*\)-algebras |

47L40 | Limit algebras, subalgebras of \(C^*\)-algebras |

46L85 | Noncommutative topology |

### Keywords:

\(C^\ast\)-algebras; nuclear dimension; Jiang-Su algebra; \(\mathcal Z\)-stability; algebraic simplicity; Toms-Winter conjecture; Elliott classification program; Cuntz semigroup### References:

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