The nuclear dimension of algebras associated to topological flows and orientable line foliations
Abstract.
We show that if is a locally compact metrizable space with finite covering dimension, then the crossed product algebra associated to any continuous flow on has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. This result is also analogous to the one we obtained earlier for possibly nonfree actions of on . As an application, we obtain bounds for the nuclear dimension of algebras associated to orientable line foliations. Some novel techniques in this paper include the use of conditional expectations constructed from clopen subgroupoids, as well as the introduction of what we call fiberwise groupoid coverings that help us build a link between foliation algebras and crossed products.
Contents
1. Introduction
Nuclear dimension for algebras was introduced by Winter and Zacharias in [winterzacharias], as a noncommutative generalization of covering dimension. Since then, it has come to play a crucial role in structure and classification of algebras: indeed, it is now known that simple unital separable algebras with finite nuclear dimension and which satisfy the UCT are classified via the Elliott invariant ([TWW, EGLN]). It was shown in [winterzacharias] that if is a locally compact metrizable space, then coincides with the covering dimension of , 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. There has been considerable interest in seeing to what extent finite nuclear dimension is preserved under forming crossed products, which in particular led to the development of the notion of Rokhlin dimension for various group actions; see [HWZ, hirshbergphillips, szabo, SWZ, gardella] for actions of finite groups, , , and compact group actions.
The case of flows, that is, actions of , was addressed in a joint paper [HSWW16] by the authors and Szabó and Winter. We developed a theory of Rokhlin dimension for flows which to a great extent parallels the theory for actions of (though the technicalities worked out to be rather different). This generalized Kishimoto’s Rokhlin property for flows ([kishimoto96]). In particular, we showed that if is a flow on a algebra with finite Rokhlin dimension, and has finite nuclear dimension, then the crossed product has finite nuclear dimension. We furthermore showed that if is a locally compact metrizable space with finite covering dimension, then any flow on induced from a free topological flow on has finite Rokhlin dimension. Thus, has finite nuclear dimension.
This leaves the case of nonfree flows. Those cannot have finite Rokhlin dimension. We addressed the parallel case of nonfree actions on topological spaces in [HirshbergWu16], where we showed that if is a locally compact metrizable space with finite covering dimension and is any automorphism of then has finite nuclear dimension. As in the case of actions of , nonfree actions of are quite prevalent. Indeed, if is a compact smooth manifold then any vector field on gives rise to a flow, but typically such flows may have fixed points or periodic orbits.
The purpose of this paper is to provide a parallel to [HirshbergWu16] for possibly nonfree flows. We show in Theorem LABEL:thmmain that if is a locally compact, metrizable with finite covering dimension, and is a flow on , then
In fact, this estimate for flows implies a similar estimate for actions, by applying the mapping torus construction to obtain a flow from a action (see Corollary LABEL:cor:mappingtorus). Thus we also recover the main theorem of [HirshbergWu16], albeit with a less sharp bound. It is worth pointing out that despite the similarity between these results, there is some significant difference in the technical tools used in them (see the comments after Corollary LABEL:cor:mappingtorus).
As an application, we give an estimate to the nuclear dimension of algebras associated to orientable line foliations. It was shown by Whitney ([Whitney]) that any orientable line foliation on a locally compact metrizable space arises from a flow. A crucial object in the study of index theorems for foliations ([ConnesSkandalis]) is the construction of the holonomy groupoid of a foliation and the algebra thereof (see also [mooreschochet] for a discussion of algebras associated to foliations). However, for an orientable line foliation , this foliation algebra typically differs from the crossed product by the flow that gives rise to the foliation.
To make a link between foliation algebras and crossed product algebras, we develop a general theory for a kind of groupoid homomorphisms between topological groupoids that we call fiberwise groupoid covering maps (see Definition LABEL:def:fiberwisegroupoidcovering). They have the ability to induce quotient maps between the corresponding maximal groupoid algebras (see Theorem LABEL:thm:fiberwisegroupoidcoveringlocallycompact and LABEL:thm:fiberwisegroupoidcoveringquotient). A canonical quotient map from the transformation groupoid of a flow to the holonomy groupoid of the induced foliation is shown to be a fiberwise groupoid covering map (see Proposition LABEL:prop:foliationcovering). As a consequence, can be obtained as a quotient of the crossed product associated to the flow (see Corollary LABEL:cor:foliationcovering). Thus we have
where is the underlying space of the foliation (see Corollary LABEL:cor:foliationalgebradimnuc).
We now briefly sketch the idea of our proof of the main theorem. The basic idea is similar to the case of integer actions, though the techniques are different. The case in which the flow has uniformly compact orbits, that is, when all points are fixed or periodic with period bounded by some constant , was already done in [HirshbergWu16, Section 3] (indeed, we included it in that generality for use in the present paper), except for an estimate of the covering dimension of the quotient space , which we complete in Section LABEL:section:_bounded_periods. We showed there that one can find in such a case a bound on the nuclear dimension of , which, crucially, does not depend on the maximal period length .
Now, if we fix a some , we can consider the set of points which are periodic with orbit length , which we denote by , and we let . Then one shows that is an invariant closed subset, and we have an equivariant extension
Although the restriction of to does not necessarily have finite Rokhlin dimension, we can use a quantitative version of the techniques we used in [HSWW16], based on [BarLRei081465306017991882] and [KasprowskiRueping], to construct dimensionally controlled long and thin covers of , provided “long” means “not too long compared to ”. This technique, which was used in [HSWW16] to construct Rokhlin elements, is then used to construct decomposable approximations for , for certain finite subsets and a level of precision which improves as increases. (One could also use those covers to construct Rokhlintype elements and then use them to construct decomposable approximations; however in the abelian setting, constructing suitable flowwise Lipschitz partitions of unity, which in the setting of [HSWW16] is a step towards constructing Rokhlin elements, is enough to obtain the required decomposable approximations, with an improved bound, so we do not need those Rokhlin elements for our result.) Having constructed those decomposable approximations for both and for , we patch them together to get decomposable approximations for in a manner similar to the one done in [HirshbergWu16] for the case of actions, where the required precision of the final approximation predetermines how large has to be at the beginning of this analysis.
We note that in previous proofs of finite nuclear dimension for uniform Roe algebras and, more generally, groupoid algebras, such as those in [winterzacharias] and [GWY], Arveson’s extension theorem is used in the construction of the downward completely positive map in the nuclear approximation. In our case, we give instead an explicit description of this map as a sum of compositions of compressions by a partition of unity of the unit space and conditional expectations associated to the clopen inclusion of relatively compact subgroupoids into larger ones. In the course of doing this, we prove a general result that any clopen inclusion of locally compact groupoids induces a conditional expectation between their reduced groupoid algebras (see Theorem LABEL:thm:clopensubgroupoid). Aside from simplifying the proof somewhat, the fact that it keeps track of the Cartan subalgebras could make it useful for future work.
Acknowledgements:
The authors would like to thank George Elliott, Alexander Kumjian, and Claude Schochet for some helpful discussions. Part of the research underlying this paper was carried out during the authors’ visits to the MittagLeffler Institute, University of Münster, the Penn State University, Centre de Recerca Matemàtica in Barcelona, the Fields Institute, and the Banff International Research Station.
2. Preliminaries
Throughout the paper, we use the following conventions. To simplify formulas, we may use the notations , and . If is a algebra, we denote by the positive part, and by the set of positive elements of norm at most . If is a locally compact Hausdorff group and is a algebra, we denote by an action, that is, a continuous homomorphism , where is topologized by pointwise convergence. If is a metric space, and , we denote .
We are interested here in the case in which is commutative, that is, for some locally compact Hausdorff space (namely the spectrum ). By Gel’fand’s theorem, an action is completely determined by a continuous action on the spectrum, and vice versa. They are related by the identity for any and any . Thus taking a algebraic point of view, we will denote by an action on a locally compact Hausdorff space by homeomorphisms, and save the notation for the corresponding action on .
When , the additive group of the real numbers, we often call a topological dynamical system a flow. We also make the following definitions. For any , we let be the minimal period of the flow at , or in other words, the length of the orbit of (with the convention if for any ). Thus this number is constant within each orbit. For a positive real number , we decompose as , where
This decomposition is then invariant under the flow. Intuitively, and are the shortperiod part and the longperiod part of the flow, respectively.
We also recall some basic facts about (possibly nonHausdorff) locally compact groupoids and their algebras. A topological groupoid is a small category whose morphisms are all invertible and whose set of morphisms is equipped with a topology such that the multiplication map given by and the inversion map given by are both continuous, where (the subset of consisting of all units, called the unit space) are the domain and range maps, and (called the set of composable pairs) is given the subset topology inherited from the product topology on . We also write and for .
Definition 2.1.
Let be a topological space. We say that is locally Hausdorff if any point has a closed neighborhood which is compact Hausdorff.
Notation 2.2.
Let be a topological space. We denote by the set of complexvalued functions for which there exists a Hausdorff open set so that vanishes outside of , and is continuous and compactly supported. Note that such functions may not be continuous when is not Hausdorff. We then define to be the linear span of in the linear space of all complexvalued functions on .
Observe that if is an open subset of , then we have canonical embeddings and . Also, if is locally compact and Hausdorff, then both and are simply the set of compactlysupported continuous functions on .
Definition 2.3.
([paterson, Definition 2.2.1 and 2.2.2]]) A locally compact locally Hausdorff groupoid is a topological groupoid that satisfies the following axioms:

is locally compact Hausdorff in the relative topology inherited from ;

there is a countable family of compact Hausdorff subsets of such that the family of interiors of members of is a basis for the topology of . (In particular, is locally Hausdorff;)

for any , is locally compact Hausdorff in the relative topology inherited from ;

admits a (left) Haar system , in the sense that each is a positive regular Borel measure on the locally compact Hausdorff space , such that

the support of each is the whole of ,

for any , the function given by the integral , belongs to ,

for any and , we have

Note that most of the technicality in this definition is due to our need to deal with the nonHausdorffness of certain groupoids coming from holonomies of line foliations. If a locally compact groupoid is indeed Hausdorff, then the existence of a Haar system is known to be a consequence of the first three axioms, Nevertheless, the uniqueness may still fail (a counterexample is given by the pair groupoid construction below); hence we always fix a left Haar measure as part of the data of a locally compact groupoid.
Example 2.4.
Basic examples of locally compact (Hausdorff) groupoids include:

the pair groupoid^{1}^{1}1It is also known as the trivial groupoid in [paterson]. for a locally compact metrizable space , where is the diagonal, , and , for any , and the positive regular Borel measures on with full support are in onetoone correspondence with the left Haar measures of ;

the transformation groupoid for an action of a locally compact group on a locally compact metrizable space , which is defined to be with , , and , where the left Haar measure of induces a left Haar system for ;

if is a locally compact groupoid and is an open subset of , then , the reduction of to , as defined by , is an open subset of that inherits the structure of a locally compact groupoid from , where the left Haar system is induced by the inclusion .
The main motivation for considering Haar systems lies in the representation theory for a groupoid , which is often better studied through representations of the convolution algebra , where the convolution product is defined by the formula (c.f., [paterson, (2.20) and (2.21)])
(2.4.1) 
or equivalently
(2.4.2) 
for any and any , and the operation is given by
(2.4.3) 
for any and any (c.f., [paterson, (2.22)]).
Important for us is the reduced algebra of a locally compact groupoid . To this end, we write for the right Haar measure corresponding to the left Haar measure , where for any Borel set . Now any determines a representation , the left regular representation at , of on the Hilbert space , defined by the formula (c.f., [paterson, (3.41)])
for any , any and any ; thus in terms of the inner product, we have
for any . It is known that the seminorm induced by is bounded by the norm (c.f., [paterson, (2.25)]) and thus . In fact, this estimate works for any representation.
Definition 2.5.
Let be a locally compact groupoid with a left Haar system . The maximal groupoid algebra of is the envelop of . The reduced groupoid algebra of is the completion of under the norm . Equivalently, is the completion of the image of the representation of on .
We may write and instead of and if the Haar system is clear from the context. Also useful for us is the canonical embedding
(2.5.1) 
where is the multiplier algebra of , so that for any and for any in the dense subalgebra in , the convolution product of by from the left and right are also in and are given by
for any .
Example 2.6.
For the groupoids in Example 2.4, we have:

for any locally compact Hausdorff space and a positive regular Borel measure on with full support, the reduced algebra of its pair groupoid together with the left Haar measure associated to is isomorphic to the algebra of compact operators on (c.f., for example, [paterson, Theorem 3.1.2]);

for any action of a locally compact group on a locally compact metrizable space , the reduced algebra of its transformation groupoid is isomorphic to the reduced crossed product , and in fact, when is unimodular with a Haar measure , the canonical identification between and , where a function on is identified with an iterated function on and , intertwines the convolution products and the operation. Here the convolution product and the operation on are defined by
and
for any and , so that it forms a dense subalgebra of ;

if is the reduction of a locally compact groupoid to an open subset of the unit space , then the inclusion induces an embedding , whose image coincides with the hereditary subalgebra , where we used the canonical embeddings .
Next we review some facts about order zero maps and nuclear dimension. If is a algebra, and are order zero contractions into some algebra for , we say that the map is a piecewise contractive decomposable completely positive map.
The following fact concerning order zero maps is standard and used often in the literature. It follows immediately from the fact that cones over finite dimensional algebras are projective. See [wintercoveringII, Proposition 1.2.4] and the proof of [winterzacharias, Proposition 2.9]. We record it here for further reference.
Lemma 2.7.
Let be a finite dimensional algebra, let be a algebra and let be an ideal. Then any piecewise contractive decomposable completely positive map lifts to a piecewise contractive decomposable completely positive map . ∎
We record for the reader’s convenience a few lemmas from [HirshbergWu16] which we will reuse in this paper.
Lemma 2.8 ([HirshbergWu16, Lemma 1.2]).
Let be a separable and nuclear algebra and a dense subset of the unit ball of . Then if and only if for any finite subset and for any there exists a algebra and completely positive maps