Haar systems on equivalent groupoids

By Dana P. Williams

Abstract

For second countable locally compact Hausdorff groupoids, the property of possessing a Haar system is preserved by equivalence.

1. Introduction

Beginning with the publication of Renault’s seminal paper Reference 11, locally compact groupoids have been an especially important way to construct operator algebras. Just as with the time honored group -algebra construction, this is done by turning into a convolution algebra and then completing. In the group case, there is always a (left) Haar measure on which allows us to form the convolution product. In the groupoid case, the natural convolution formula requires a family of measures with support for each . We want the family to be left-invariant in that where . In order that the convolution formula return a continuous function, we need the continuity condition that

is continuous for all (the necessity is the main result in Reference 15). Such a family is called a (continuous) Haar system for . An annoying gap in the theory is that there is no theorem guaranteeing that Haar systems exist. The only significant positive existence result we are aware of is that if is open in and the range map is open (and hence the source map as well), then the family consisting of counting measures is always a Haar system. Groupoids with open and for which the range map is open are called étale. (It is also true that Lie groupoids necessarily have Haar systems Reference 9, Theorem 2.3.1, but this result is crucially dependent on the manifold structure and hence is not in the spirit of this note.) It is also well known that if is any locally compact groupoid with a Haar system, then its range and source maps must be open. (This is a consequence of Remark 2.2 and Lemma 2.3.) Thus if a locally compact groupoid has a range map which is not open, then it can’t possess a Haar system. Such groupoids do exist; for example, see Reference 15, §3. However, to the best of our knowledge, there is no example of a locally compact groupoid with open range and source maps which does not possess a Haar system. We have yet to find an expert willing to conjecture (even off the record) that all such groupoids need have Haar systems, but the question remains open.

The purpose of this paper is to provide some additional examples where Haar systems must exist. The main result being that if and are equivalent second countable locally compact groupoids (as defined in Reference 7 for example), and if has a Haar system, then so does . Since equivalence is such a powerful tool, this result gives the existence of Haar systems on a great number of interesting groupoids. For example, every transitive groupoid with open range and source maps has a Haar system (Proposition 3.4).

The proof given here depends on several significant results from the literature. The first is that if is a continuous, open surjection with second countable, then there is a family of Radon measures on such that and

is continuous for all . (This result is due to Blanchard, who makes use of a theorem of Michael Reference 6.) The second is the characterization in Reference 5, Proposition 5.2 of when the imprimitivity groupoid of a free and proper -space has a Haar system. The third is the concept of a Bruhat section or cut-off function. These are used in Reference 3, Chapter 7 to construct invariant measures. They also appear prominently in Reference 13, Lemma 25 and Reference 16, §6.

Since Blanchard’s result requires separability, we can only consider second countable groupoids here.

2. The theorem

Theorem 2.1.

Suppose that is a second countable, locally compact Hausdorff groupoid with a Haar system . If is a second countable, locally compact groupoid which is equivalent to , then has a Haar system.

As in Reference 12, p. 69 or Reference 1, Definition 1.1.1, if is a continuous map between locally compact spaces and , then a -system is a family of (positive) Radon measures on such that and for every , the function

is continuous. We say that is full if for all .

If and are both (left) -spaces and is equivariant, then we say is equivariant if , where for all . Alternatively,

for all and .

Remark 2.2.

It is useful to keep in mind that a Haar system on is a full, equivariant -system on for the range map .

In many cases, such as Reference 5, §5, -systems are reserved for continuous and open maps . In the case of full systems, the next lemma implies that there is no loss in generality. (This part of the result does not require second countability.) Conversely, if is second countable and is an open surjection, then Blanchard has proved that full systems must exist. Blanchard’s result will be crucial in the proof of the main result.

Lemma 2.3 (Blanchard).

Suppose that is a continuous surjection between second countable locally compact Hausdorff spaces. Then is open if and only if it admits a full system.

Proof.

Suppose that is a full system. Let be a nonempty open subset. Fix and such that . Choose open such that and is compact. Let such that and . Since is full, . But is continuous and . Hence is an interior point of . Since was arbitrary, is open.

The converse is much more subtle and is due to Blanchard. The proof can be extracted from Reference 2, §3.1 as follows. Note that the openness of implies that is a continuous field of -algebras over Reference 18, Theorem C.26. Now we can apply Reference 2, Proposition 3.9 thanks to Reference 2, Proposition 3.5.⁠Footnote1

1

There is a slight gap in the proof of Reference 2, Lemme 3.8: it is asserted there that if is a separable -algebra, then its dual , equipped with the topology of uniform convergence on compact subsets of , is a Fréchet space. However, if is infinite dimensional, then is not first countable in this topology. Unfortunately, this invalidates the application of Reference 2, Lemme 3.6 in the proof of Reference 2, Proposition 3.9. Fortunately, Reference 2, Lemme 3.6 can be modified (using the notation of Reference 2, Lemme 3.6) to the case where the topological vector space contains a complete convex subset containing the domain . Thus the main results of Reference 2, §3 remain true as stated.

We also need what is sometimes called a Bruhat section or cut-off function for . The construction is modeled after Lemma 1 in Appendix I of Reference 3, Chapter 7. Recall that a subset is called -compact if is compact whenever is compact in . We write for the set of continuous functions on with -compact support.

Lemma 2.4.

Let be a continuous open surjection between second countable locally compact Hausdorff spaces. Then there is a such that .

Proof.

Let be a countable, locally finite cover of by pre-compact open sets . Let be a partition of unity on subordinate to . Let be such that . Then we can define

Since is locally finite, the above sum is finite in a neighborhood of any . Hence is well defined and continuous. Local finiteness also implies that every compact subset of meets at most finitely many . Since is subordinate to , it follows that has -compact support. If , then there is an such that . Then there is a such that and . Hence the result.

Proof of Theorem 2.1.

Let be a -equivalence. Then the opposite module, , is an -equivalence. Therefore, in view of Reference 5, Proposition 5.2, it will suffice to produce a full -equivariant -system for the structure map . Equivalently, we need a full equivariant -system for the map .⁠Footnote2 Hence the main theorem is a consequence of Proposition 2.5 below.

2

The gymnastics with the opposite space is just to accommodate a preference for left-actions. This has the advantage of making closer contact with the literature on -systems.

The following proposition is even more than what is called for in the proof of Theorem 2.1: it shows that every proper -space has a full equivariant -system for the moment map whether the action is free or not.⁠Footnote3 It should be noted that pairs where is a proper -space and and equivariant -system play an important role in the constructions in Reference 12 and Reference 4. This makes even more interesting the assertion that such ’s always exist.

3

The literature is inconsistent as to whether the moment map for a groupoid action need be open. In this paper, it is critical that we use the definition of groupoid action that requires moment maps to be open.

Proposition 2.5.

Let be a second countable locally compact Hausdorff groupoid with a Haar system . Suppose that is a proper -space with open momment map . Then there is a full equivariant -system .

Blanchard’s Lemma 2.3 implies that there is a full -system . The idea of the proof of Proposition 2.5 is to use the Haar system on to average this system to create an equivariant system. The technicalities are provided by the next lemma. Notice that since acts properly, the orbit map is a continuous open surjection between locally compact Hausdorff spaces.

Lemma 2.6.

Let , , and be as above.

(a)

If , then

defines an element of .

(b)

If and , then

defines an element of .

Proof.

(a) This is straightforward if with and . But we can approximate in the inductive limit topology with sums of such functions.

(b) Let . By assumption on , is compact. Since acts properly on , the set

is compact. It follows that

defines an element of . Then

The assertion follows.

Proof of Proposition 2.5.

Using Lemma 2.4, we fix such that . Then we define a Radon measure on by

Since is a Haar system and , we see immediately that

is continuous.

Clearly, . Suppose and is such that . Then there is a such that . Hence there is a such that . Note that and . Since has full support and since everything in sight is continuous and nonnegative,

Hence and we have

Hence to complete the proof of the theorem, we just need to establish equivariance. But

which, since is a Haar system on , is

which, since , is

This completes the proof.

Proposition 2.5 is interesting even for a group action. The result itself is no doubt known to experts, but is amusing nonetheless.

Corollary 2.7.

Suppose that is a locally compact group acting properly on a space . Then has at least one invariant measure with full support.

3. Examples and comments

Recall that a topological groupoid is étale if and only if the range map is a local homeomorphism. (Since inversion is a homeomorphism, is a local homeomorphism if and only if is as well.) As pointed out in the introduction, any étale groupoid has a Haar system. As a consequence of Theorem 2.1, any second countable groupoid equivalent to has a Haar system (provided is second countable). In this section, we want to look at some additional examples. In some cases it is possible and enlightening to describe the Haar system in finer detail.

3.1. Proper principal groupoids

Recall that is called principal if the natural action of on given by is free. We call proper if this action is proper in that is proper from . If is a proper principal groupoid with open range and source maps, then the orbit space is locally compact Hausdorff, and it is straightforward to check that implements an equivalence between and the orbit space . Since the orbit space clearly has a Haar system, the following is a simple corollary of Theorem 2.1.

Proposition 3.1 (Blanchard).

Every second countable proper principal groupoid with open range and source maps has a Haar system.

Remark 3.2.

If is a second countable proper principal groupoid with open range and source maps, then the orbit map sending to is continuous and open. Hence Blanchard’s Lemma 2.3 implies there is a full -system . It is not hard to check that is a Haar system for

Since is a groupoid isomorphism of and , we get an elementary description for a Haar system on .

While there certainly exist groupoids that fail to have open range and source maps — and hence cannot have Haar systems — most of these examples are far from proper and principal. In fact the examples we’ve seen are all group bundles which are as a far from principal as possible. This poses an interesting question.

Question 3.3.

Must a second countable, locally compact, proper principal groupoid have open range and source maps?

3.2. Transitive groupoids

Recall that a groupoid is called transitive if the natural action of on given by is transitive. If is transitive and has open range and source maps, then is equivalent to any of its stability groups for ; the equivalence is given by with the obvious left -action and right -action. Second countability is required to see that the restriction of the range map to onto is open.⁠Footnote4 Since locally compact groups always have a Haar measure, the following is an immediate consequence of Theorem 2.1. (Similar assertions can be found in Reference 14.)

4

Proving the openness of is nontrivial. It follows from Reference 10, Theorem 2.1 or Theorems 2.2A and 2.2B in Reference 7. The assertion and equivalence fail without the second countability assumption as observed in Reference 7, Example 2.2.

It should be noted that in both Reference 10 and Reference 7 openness of the range and source maps on a topological groupoid is a standing assumption.

Proposition 3.4 (Seda).

If is a second countable, locally compact transitive groupoid with open range and source maps, then has a Haar system.

As before, we don’t know the answer to the following.

Question 3.5.

Must a second countable, locally compact, transitive groupoid have open range and source maps?

3.3. Blowing up the unit space

While there are myriad ways that groupoid equivalences arise in applications, one standard technique deserves special mention (see Reference 17 for example). Suppose that is a second countable locally compact groupoid with a Haar system (or at least open range and source maps). Let be a continuous and open map. Then we can form the groupoid

(The operations are as expected: and .) The idea is that we use to “blow-up” the unit space of to all of . If is the homormorphism , then we get a -equivalence given by “the graph of (see Reference 5, §6):

The left -action is given by and the right -action by . The openness of the range map for is required to see that the structure map is open, while the openness of is required to see that is open. Assuming has a Haar system and is second countable, Theorem 2.1 implies that has a Haar system. However in this case we can do a bit better and write a tidy formula for the Haar system on the blow-up. We still require Blanchard’s Lemma 2.3 where there is a full -system for any continuous open map (provided that is second countable).

Proposition 3.6.

Suppose that is a locally compact Hausdorff groupoid with a Haar system , and that is second countable. Let be a continuous open map, and let