Abstract
We compute the nuclear dimension of separable, simple, unital, nuclear, {mathcal {Z}}-stable mathrm {C}^*-algebras. This makes classification accessible from {mathcal {Z}}-stability and in particular brings large classes of mathrm {C}^*-algebras associated to free and minimal actions of amenable groups on finite dimensional spaces within the scope of the Elliott classification programme.
Highlights
Nuclear dimension is a non-commutative generalisation of topological covering dimension to C∗-algebras, introduced in [79]
A unital abelian C∗-algebra consists of continuous functions on a compact Hausdorff space X ; in this case the nuclear dimension recaptures the dimension of X
Through the work of generations of researchers ([19,27,38,48,64,75] building on numerous works going back to [18]), we have a complete classification of separable, simple, unital C∗-algebras of finite nuclear dimension satisfying Rosenberg and Schochet’s universal coefficient theorem (UCT) [56]
Summary
Nuclear dimension is a non-commutative generalisation of topological covering dimension to C∗-algebras, introduced in [79]. In this paper we show that for separable, simple, unital, and nuclear C∗algebras, finite nuclear dimension is entailed by the tensorial absorption condition of Z-stability, where Z is the Jiang–Su algebra of [34] (Z-stability will be described further below) Combining this with the main result of [74] gives the following theorem, which was predicted by the Toms–Winter conjecture. Beyond this setting is a strategy for combining behaviour at individual traces to obtain global properties uniformly over the trace space while respecting the affine structure.3 We resolve this by introducing a completely new technique— complemented partitions of unity—enabling us to perform partition of unity arguments inside a Z-stable nuclear C∗-algebra. Theorem H Let A be a separable, simple, unital, nuclear, Z-stable C∗algebra with T (A) compact and non-empty, and with complemented partitions of unity. The follow-up paper [8] further develops the theory of uniform property and complemented partitions of unity, giving other applications to the Toms–Winter conjecture
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