For second countable locally compact Hausdorff groupoids, the property of possessing a Haar system is preserved by equivalence.
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 construction, this is done by turning -algebra 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 has a Haar system. The third is the concept of a Bruhat section or cut-off function. These are used in -spaceReference 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
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 , is a family of (positive) Radon measures -system on such that and for every the function ,
is continuous. We say that is full if for all .
If and are both (left) and -spaces is equivariant, then we say is equivariant if where , for all Alternatively, .
for all and .
In many cases, such as Reference 5, §5, are reserved for continuous and open maps -systems 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.
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 if -compact is compact whenever is compact in We write . for the set of continuous functions on with support. -compact
The following proposition is even more than what is called for in the proof of Theorem 2.1: it shows that every proper has a full equivariant -space for the moment map whether the action is free or not. -systemFootnote3 It should be noted that pairs where is a proper and -space and equivariant play an important role in the constructions in -systemReference 12 and Reference 4. This makes even more interesting the assertion that such always exist. ’s
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.
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.
Proposition 2.5 is interesting even for a group action. The result itself is no doubt known to experts, but is amusing nonetheless.
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.
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.
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 and right -action Second countability is required to see that the restriction of the range map to -action. 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.)
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.
As before, we don’t know the answer to the following.
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 , given by “the graph of -equivalence (see ”Reference 5, §6):
The left is given by -action and the right by -action 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 for any continuous open map -system (provided that is second countable).