Nuclear dimension and Z-stability

Abstract

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

23 Pages. New section added. Invent. Math., to appear

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Cited by in corpus (39)

• Quasidiagonality of nuclear C*-algebras
• Nuclear dimension of simple C*-algebras
• Tensor products and regularity properties of Cuntz semigroups
• Logic and operator algebras
• Nuclear dimension and Z-stability of non-simple C*-algebras
• Rokhlin dimension for actions of residually finite groups
• Equivariant property (SI) revisited
• Fraïssé limits of C*-algebras
• Rokhlin dimension for flows
• Regularity properties and Rokhlin dimension for compact group actions
• Decomposition rank of approximately subhomogeneous C*-algebras
• Actions of amenable groups and crossed products of Z-absorbing C*-algebras
• Uniform property Gamma
• Rokhlin dimension: absorption of model actions
• The Cuntz-Toeplitz algebras have nuclear dimension one
• Sofic boundaries of groups and coarse geometry of sofic approximations
• Crossed products of nuclear C*-algebras by free groups and their traces
• Decomposable approximations revisited
• Almost elementary étale groupoids
• The nuclear dimension of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• The Toeplitz algebra has nuclear dimension one
• Unitary orbits of self-adjoint operators in simple Z-stable C*-algebras
• Cartan subalgebras in C*-algebras. Existence and uniqueness
• Locally Trivial W*-Bundles
• K-theoretic characterization of C*-algebras with approximately inner flip
• Quasidiagonal traces on exact $C^{\ast }$-algebras
• Decomposable approximations and approximately finite dimensional C*-algebras
• Classification of tiling $C^{\ast }$-algebras
• Approaching the UCT problem via crossed products of the Razak-Jacelon algebra
• Tracially sequentially-split ${}^{\ast }$-homomorphisms between $C^{\ast }$-algebras II
• Decomposing nuclear maps
• Non-amenable simple C*-algebras with tracial approximation
• Invariant ergodic measures and the classification of crossed product $C^{\ast }$-algebras
• Sturmian subshifts and their C*-algebras
• Equivariant property Gamma and the tracial local-to-global principle for C*-dynamics
• Some properties for certain generalized tracial approximated ${\mathrm{C}}^{\ast }$-algebras
• 2-positive almost order zero maps and decomposition rank
• Classifiable ${\mathrm{C}}^{\ast }$-algebras from minimal $\mathbb{Z}$-actions and their orbit-breaking subalgebras
• Constructions in minimal amenable dynamics and applications to the classification of ${\mathrm{C}}^{\ast }$-algebras