The nuclear dimension of C⁎-algebras associated to homeomorphisms

Abstract

We show that if X is a finite dimensional locally compact Hausdorff space, then the crossed product of C0(X) by any automorphism has finite nuclear dimension. This generalizes previous results, in which the automorphism was required to be free. As an application, we show that group C * -algebras of certain non-nilpotent groups have finite nuclear dimension.Nuclear dimension for C * -algebras was introduced by Winter and Zacharias in [WZ10], as a noncommutative generalization of covering dimension. This is a variant of the previous notion of decomposition rank ([KW04]), and is also applicable to non-quasidiagonal C * -algebras. Since then, it has come to play a major role in structure and classification of C * -algebras. It was shown in [WZ10] that if X is a locally compact metrizable space, then dim nuc (C 0 (X)) coincides with the covering dimension of X, and the property of having finite nuclear dimension is preserved under various constructions: forming direct sums and tensor products, passing to quotients and hereditary subalgebras, and forming extensions. An important problem which was left open in [WZ10] is to understand the behavior of finite nuclear dimension under forming crossed products. It was shown in [TW13] that finite nuclear dimension passes to crossed products by minimal homeomorphisms: if X is a compact metric space with finite covering dimension and h : X → X is a minimal homeomorphism, then denoting α(f ) = f • h, we have dim nuc (C(X) ⋊ α Z) < ∞. This was re-proved in a different way in [HWZ15]. The paper [HWZ15] develops a notion of Rokhlin dimension for an automorphism of a C * -algebra (extended in [HP15] to the non-unital setting). It was shown there that in general, if A has finite nuclear dimension and α ∈ Aut(A) has finite Rokhlin dimension, then A ⋊ α Z has finite nuclear dimension as well, and furthermore, for a minimal homeomorphism as above, the induced automorphism on C(X) always has finite Rokhlin dimension. Szabó ([Sza15b]) then showed that the minimality condition can be weakened to freeness: if X is as above and h : X → X has no periodic points, then α has finite Rokhlin dimension (and therefore, by [HWZ15, Theorem 4.1], the crossed product has finite nuclear dimension). In fact, Szabó's result works for actions of Z m as well. This uses the marker property, introduced by Gutman in [Gut15b]. Those results were further extended to free actions of finitely generated nilpotent groups in [SWZ14].For the case of integer actions arising from homeomorphisms, this leaves the case of actions which also have periodic points. Those include important examples. For instance, suppose G is a countable abelian group and G has finite covering dimension, and suppose α is an automorphism of G. The group C * -algebra C * (G⋊ α Z) is isomorphic to a crossed product C( G) ⋊ Z, and such actions are never free: an

Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and $$C^*$$ C ∗ -algebras

“…We should note that there are many other results in the literature that give estimates on nuclear dimension or other C˚-algebraic regularity properties based on conditions on actions: see for example [36,19,37,20,11]. Many of these results go further than ours in at least some ways: for example, [36] proves fairly general finite dimensionality results for Z n -actions, [20] treats non-free Z-actions, [11] treats some Z-actions on non-finite dimensional spaces, and several works deal with some actions on noncommutative C˚-algebras. It would be interesting to clarify the relationships holding between the various conditions involved in these results and ours.…”

Dynamic asymptotic dimension for actions of virtually cyclic groups

“…As already mentioned in the introduction, the notion of dynamic asymptotic dimension provides a way to compute an upper bound on the nuclear dimension of C * -algebras arising from topological dynamical systems. The following corollary generalizes [10,Corollary 8.23] to the setting considered here and additionally removes the metrisability assumption following the ideas of [13,Corollary 5.4].…”

“…In the case of actions of Z, the main result of [13] removes the freeness and metrisability assumption from [19] (at the price of replacing the linear bound from [19] and our Corollary 2.5 with a quadratic upper bound).…”

“…By Lemma 3.3, there is a countable set S such that if Y is a factor of X satisfying S ⊆ C(Y ), then Γ Y is free. In the special case of commutative C * -algebras, Lemma 1.3 of [13] asserts that for such a set S ⊆ C(X), there is a metrisable factor Y of X such that S ⊆ C(Y ) and dim Y ≤ dim X. By [17,Theorem 3.8], freeness and finite-dimensionality imply that Γ Y satisfies the bounded topological small boundary property with respect to d = dim Y , which in turn (together with freeness) implies the marker property for Γ Y by [17,Theorem 4.6].…”

Finite decomposition rank for virtually nilpotent groups

“…Combined with [23,Theorem 5.3] and our Theorem 6.4 below, this implies that among groups of the form Z d ⋊ Z, the virtually nilpotent ones are precisely those whose C*-algebras have finite decomposition rank. In a similar vein, Hirshberg and Wu [19] recently proved that all groups of the form Z d ⋊ Z have finite nuclear dimension; thus, there exist (non virtually nilpotent) groups of this form which have infinite decomposition rank but finite nuclear dimension. Finally, in [15], Giol and Kerr provided examples of topological dynamical systems such that C(X) ⋊ Z has arbitrarily large radius of comparison.…”