Nuclear dimension, Z-stability, and algebraic simplicity for stably projectionless C*-algebras

Abstract

The main result here is that a simple separable C*-algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of [Toms, "K-theoretic rigidity and slow dimension growth"; Winter, "Nuclear dimension and Z-stability of pure C*-algebras"] to the nonunital setting. As a consequence, finite nuclear dimension implies Z-stability even in the case of a separable C*-algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of being simple and unital, and the proof of the main result makes heavy use of this concept.

Fixed typos, etc., and added corollary regarding finitely many ideals. 37 pages. To appear in Mathematische Annalen. The published version will differ

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Cited by in corpus (22)

• Quasidiagonality of nuclear C*-algebras
• Nuclear dimension and Z-stability
• Nuclear dimension of simple stably projectionless C*-algebras
• Nuclear dimension and Z-stability of non-simple C*-algebras
• Equivariant property (SI) revisited
• Rokhlin dimension for flows
• On the structure of the Cuntz semigroup in (possibly) nonunital C*-algebras
• On classification of non-unital amenable simple C*-algebras, III, the range and the reduction
• The classification of Rokhlin flows on C*-algebras
• Uniform property Gamma
• Simple stably projectionless C*-algebras with generalized tracial rank one
• Dynamical Complexity and $K$-Theory of $L^p$ Operator Crossed Products
• Unitary groups and augmented Cuntz semigroups of separable simple Z-stable C*-algebras
• Unitary orbits of self-adjoint operators in simple Z-stable C*-algebras
• On tracial $\mathcal{Z}$-stability of simple non-unital C*-algebras
• Extensions of C*-algebas by a small ideal
• Tracial oscillation zero and stable rank one
• Finite group actions on certain stably projectionless C*-algebras with the Rohlin property
• Group actions on Smale space C*-algebras
• C*-algebras and their nuclear dimension
• On $C^{\ast }$-algebras associated to actions of discrete subgroups of $SL(2,\mathbb{R})$ on the punctured plane
• A Criterion for $\mathcal{Z}$-Stability with Applications to Crossed Products