Nuclear dimension and 𝒵-stability of non-simple 𝒞*-algebras

Robert, Leonel; Tikuisis, Aaron

doi:10.1090/tran/6842

Nuclear dimension and $\mathcal {Z}$-stability of non-simple $\mathrm {C}^*$-algebras
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Trans. Amer. Math. Soc. 369 (2017), 4631-4670 Request permission

Abstract:

We investigate the interplay of the following regularity properties for non-simple $\mathrm C^*$-algebras: finite nuclear dimension, $\mathcal Z$-stability, and algebraic regularity in the Cuntz semigroup. We show that finite nuclear dimension implies algebraic regularity in the Cuntz semigroup, provided that known type I obstructions are avoided. We demonstrate how finite nuclear dimension can be used to study the structure of the central sequence algebra, by factorizing the identity map on the central sequence algebra, in a manner resembling the factorization arising in the definition of nuclear dimension.

Results about the central sequence algebra are used to attack the conjecture that finite nuclear dimension implies $\mathcal {Z}$-stability, for sufficiently non-type I, separable $\mathrm C^*$-algebras. We prove this conjecture in the following cases: (i) the $\mathrm C^*$-algebra has no simple purely infinite ideals of quotients and its primitive ideal space has a basis of compact open sets, (ii) the $\mathrm C^*$-algebra has no purely infinite quotients and its primitive ideal space is Hausdorff. In particular, this covers $\mathrm C^*$-algebras with finite decomposition rank and real rank zero. Our results hold more generally for $\mathrm C^*$-algebras with locally finite nuclear dimension which are $(M,N)$-pure (a regularity condition of the Cuntz semigroup).

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