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We compute the nuclear dimension of separable, simple, unital, nuclear, Z-stable C*-algebras. This makes classification accessible from Z-stability and in particular brings large classes of C*-algebras associated to free and minimal actions of amenable groups on finite dimensional spaces within the scope of the Elliott classification programme.
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The class of simple separable KK-contractible (KK-equivalent to ${0}$) C*-algebras which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras $Aotimes mathcal W$ is classifiable, where $A$ is a simple separable C*-algebra with finite nuclear dimension and $mathcal W$ is the simple inductive limit of Razak algebras with unique trace, which is bounded.
From a suitable groupoid G, we show how to construct an amenable principal groupoid whose C*-algebra is a Kirchberg algebra which is KK-equivalent to C*(G). Using this construction, we show by example that many UCT Kirchberg algebras can be realised as the C*-algebras of amenable principal groupoids.
We define a notion of tracial $mathcal{Z}$-absorption for simple not necessarily unital C*-algebras. This extends the notion defined by Hirshberg and Orovitz for unital (simple) C*-algebras. We provide examples which show that tracially $mathcal{Z}$-absorbing C*-algebras need not be $mathcal{Z}$-absorbing. We show that tracial $mathcal{Z}$-absorption passes to hereditary C*-subalgebras, direct limits, matrix algebras, minimal tensor products with arbitrary simple C*-algebras. We find sufficient conditions for a simple, separable, tracially $mathcal{Z}$-absorbing C*-algebra to be $mathcal{Z}$-absorbing. We also study the Cuntz semigroup of a simple tracially $mathcal{Z}$-absorbing C*-algebra and prove that it is almost unperforated and weakly almost divisible.
We introduce the concept of finitely coloured equivalence for unital *-homomorphisms between C*-algebras, for which unitary equivalence is the 1-coloured case. We use this notion to classify *-homomorphisms from separable, unital, nuclear C*-algebras into ultrapowers of simple, unital, nuclear, Z-stable C*-algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classification theorem for certain order zero maps, also in terms of tracial data. As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z-stable C*-algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a homotopy equivalence implies isomorphism result for large classes of C*-algebras with finite nuclear dimension.
Let $C$ and $A$ be two unital separable amenable simple C*-algebras with tracial rank no more than one. Suppose that $C$ satisfies the Universal Coefficient Theorem and suppose that $phi_1, phi_2: Cto A$ are two unital monomorphisms. We show that there is a continuous path of unitaries ${u_t: tin [0, infty)}$ of $A$ such that $$ lim_{ttoinfty}u_t^*phi_1(c)u_t=phi_2(c)tforal cin C $$ if and only if $[phi_1]=[phi_2]$ in $KK(C,A),$ $phi_1^{ddag}=phi_2^{ddag},$ $(phi_1)_T=(phi_2)_T$ and a rotation related map $bar{R}_{phi_1,phi_2}$ associated with $phi_1$ and $phi_2$ is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class ${cal A}$ of unital separable simple amenable CA s which is strictly larger than the class of separable CA s whose tracial rank are zero or one. The class contains all unital simple ASH-algebras whose state spaces of $K_0$ are the same as the tracial state spaces as well as the simple inductive limits of dimension drop circle algebras. Moreover it contains some unital simple ASH-algebras whose $K_0$-groups are not Riesz. One consequence of the main result is that all unital simple AH-algebras which are ${cal Z}$-stable are isomorphic to ones with no dimension growth.
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