simplicity and representations of topological full groups of groupoids
Abstract.
Given an ample groupoid with compact unit space, we study the canonical representation of the topological full group in the full groupoid algebra . In particular, we show that the image of this representation generates if and only if admits no tracial state. The techniques that we use include the notion of groups covering groupoids.
As an application, we provide sufficient conditions for simplicity of certain topological full groups, including those associated with topologically free and minimal actions of nonamenable and countable groups on the Cantor set.
1. Introduction
Topological full groups associated to group actions on the Cantor set have given rise to examples of groups with interesting new properties ([7], [13]). In the context of groupoids, the topological full group was introduced by H. Matui in [10], who investigated their relation with homology groups of groupoids.
Following a slightly different approach, V. Nekrashevych ([12]) defined the topological full group of an ample groupoid with compact unit space to consist of the bisections such that . In this paper, we study the unitary representation given by , for every . Let denote the algebra generated by in .
Our main result is as follows:
Theorem (Theorems 4.3 and 4.6).
Let be an ample groupoid with compact unit space such that the orbit of each has at least three points. Then is a hereditary subalgebra of . Moreover, admits no tracial state if and only if .
This generalizes part of [6, Proposition 5.3] (see Remark 4.7). If, in addition, is a second countable, essentially principal and minimal groupoid on the Cantor set, then is stably isomorphic to (Corollary 4.4).
The paper is organized as follows. In Section 2, we collect basic definitions about groupoids, establish notation and present some relevant examples.
In Section 3, we study groups covering groupoids. Given an ample groupoid with compact unit space, a subgroup is said to cover if . We investigate under which conditions covers and show that admits a character if and only if admits a invariant probability measure (Corollary 3.7).
In Section 4, we analyze the representation of the topological full group in the full and the reduced groupoid algebras to reach the main theorem above.
In Section 5, we apply the results of Sections 3 and 4 in order to study simplicity of the topological full group. Given an ample groupoid with compact unit space, let denote the canonical representation of in . Recall that a group is said to be simple if its reduced algebra is simple. In [8], A. Le Boudec and N. Matte Bon showed that a countable group of homeomorphisms on a Hausdorff space is simple if the rigid stabilizers are nonamenable. By using this result, we show the following:
Theorem (Theorem 5.2).
Let be a second countable, essentially principal, minimal and ample groupoid with compact unit space. If

is not amenable, or

does not weakly contain the trivial representation,
then is simple.
2. Preliminaries
In this section we introduce relevant concepts and establish notation. Throughout the paper, we let denote the nonnegative integers.
2.1. Ample groupoids
A topological groupoid is ample if is locally compact, Hausdorff, étale (in the sense that the range and source maps are local homeomorphisms onto ) and the unit space is totally disconnected. The orbit of a point is the set , and is said to be minimal if , for every .
A bisection is a subset such that and are injective. Note that, if is open, then and are homeomorphisms onto their images. We will denote by the inverse semigroup of open bisections of , and by the subinverse semigroup of compact open bisections. There is a homomorphism from to the inverse semigroup of homeomorphisms between open subsets of , given by . As observed in [17], is injective if and only if is essentially principal (that is, ).
In the following we let be the algebra of complex valued, continuous and compactly supported functions on , and we let and denote the reduced and full groupoid algebras, respectively. For an introduction to (étale) groupoids and their algebras, the reader is refered to, e.g., [15] or [19].
If is minimal and essentially principal, then is simple (see, e.g., [19, Proposition 4.3.7]).
A regular Borel measure on is invariant if , for each . Clearly, is invariant if and only if , for each . The following proposition is wellknown.
Proposition 2.1.
Let be an ample groupoid with compact unit space. The following conditions are equivalent:

admits a invariant probability measure;

admits a tracial state;

admits a tracial state.
Proof.
The proof of the implications (i)(ii)(iii) can be found in [15, Theorem 3.4.4].
(iii) (i): Let be a tracial state on . Given , we have
Thus, the probability measure on induced by is invariant. ∎
Suppose admits a invariant measure . Then there is a representation given by
(1) 
for , and . Note that if is a compact open bisection, then , and is the representation by multiplication operators.
2.2. Topological full groups
Given an ample groupoid with compact unit space, the topological full group of is
This definition coincides with the one from [12]. In [10], however, H. Matui defines the topological full group of as . Therefore, if is essentially principal then is injective and the two definitions coincide.
Two examples to have in mind are as follows.
Example 2.2.
Let be an action of a group on a compact Hausdorff space . As a space, the transformation groupoid associated with is equipped with the product topology. The product of two elements is defined if and only if in which case . Inversion is given by . The unit space is naturally identified with and is ample if is totally disconnected.
The topological full group consists of sets of the form , where and are clopen sets such that . In particular, there is a canonical injective homomorphism sending .
Example 2.3.
Let be the full onesided shift and consider the DeaconuRenault groupoid
The product of is welldefined if and only if in which case . Inversion is given by .
Let be the set of finite words (including the empty word) on the alphabet . Given , let denote its length and let be the cylinder set of . The topology on is generated by sets of the form
for . This topology is strictly finer than the one inherited from the product topology and is ample with compact unit space. Note as well that is minimal.
The topological full group consists of sets of the form
(2) 
with .
We would now like to recall the isomorphism between Thompson’s group and , observed in [11] (see also [14]).
Thompson’s group consists of piecewise linear, right continuous bijections on which have finitely many points of nondifferentiability, all being dyadic rationals, and have a derivative which is a power of at each point of differentiability.
Given , let and . The isomorphism from to takes as in (2) and sends it to the bijection on which, restricted to , is linear, increasing and onto , for every .
The next example shows that the quotient does not always split. Since we are interested in studying the canonical representation of in , this illustrates why we have chosen to treat the topological full group as bisections, rather than homeomorphisms on the unit space.
Example 2.4.
Let be the onepoint compactification of and define an action by
for . Note that is a compact open bisection in the transformation groupoid and that the homeomorphism
for , has order .
Moreover, for any satisfying , there is an odd integer such that . In particular, has infinite order. Therefore, the quotient does not split.
2.3. Unitary representations
Let be an an ample groupoid with compact unit space. There is a unitary representation
and the algebra generated by the image of is denoted . We will denote the analogous representation of in by and by the algebra generated by the image of .
If and are two unitary representations of a group on a unital algebra, then is said to weakly contain if
for every .
Recall that the trivial representation satisfies for every .
Proposition 2.5.
Let be a unitary representation of a group on a unital algebra . Then weakly contains the trivial representation if and only if .
Proof.
The forward implication is evident, so we only prove the backward one.
Let . If , then, since is a algebra, . Hence, for every and , we have that
thus showing that weakly contains the trivial representation. ∎
3. Groups covering groupoids
It is wellknown that an ample groupoid can always be covered by compact open bisections. We investigate to which degree can be covered by compact open bisections which satisfy . We show that if covers and is a invariant probability measure on , then is also invariant.
Definition 3.1.
Given an ample groupoid with compact unit space, we say that a subgroup covers if .
The idea of covering a groupoid by compact open bisections such that has already appeared in H. Matui’s study of automorphisms of , cf. [10, Proposition 5.7].
If is essentially principal, then a subgroup covers if and only if, for each open bisection and , there are and a neighborhood of such that .
Example 3.2.
If is an action of a group on a compact Hausdorff and totally disconnected space, then the copy of in covers .
Example 3.3.
Recall that Thompson’s group consists of the elements of Thompson’s group (see Example 2.3) which have at most one point of discontinuity.
Let be the groupoid of Example 2.3. Under the identification of with , covers . This follows from the fact that if are leftclosed and rightopen intervals with endpoints in , then there exists a piecewise linear homeomorphism with a derivative which is a power of at each point of differentiability and with finitely many points of nondifferentiability, all of which belong to .
Lemma 3.4.
Let be an ample groupoid with compact unit space. If for every , then covers .
Proof.
Let . If , then there is a compact open bisection containing and such that . Let . Then .
If , then there is such that since . As before, there are such that and . Hence, . ∎
The purpose of the next example is to show that the above result may fail if one does not make any assumption on the orbits.
Example 3.5.
Consider equipped with the order topology and let be the action given by , for and . The transformation groupoid is ample with compact unit space.
Given we put . Then
is an ample subgroupoid of . Incidentally, this is the groupoid of the partial action obtained by restricting to (see [5] and [9] for more details). Observe that .
We claim that if , then . Otherwise, there is such that and . But then and contradicting the fact that and are injective on . Hence, and does not cover .
Recall that a probability measure on is invariant if for every . Moreover, if , then we say is invariant if it is invariant with respect to the action .
Proposition 3.6.
Let be an ample groupoid with compact unit space and a subgroup of . Consider the following conditions:

admits a invariant probability measure;

weakly contains the trivial representation;

admits a character;

admits a invariant probability measure.
Then (i)(ii)(iii)(iv). If covers , then (iv) (i) and all conditions are equivalent.
Proof.
(i)(ii): Suppose is a invariant measure on and let be the representation of (1). The vector is invariant for the representation . Hence, weakly contains the trivial representation.
The implication (ii)(iii) is evident.
(iii)(iv): Let be a character on and a state on which is a lift of . Then is in the multiplicative domain of . Clearly, induces a invariant probability measure on .
Now, suppose covers and let us show that (iv) (i). Let be a invariant probability measure on . We claim that is also invariant. Indeed, since covers , given , we have that . As is compact, there are and such that and for . In particular, for every . It follows that
Therefore, is a invariant probability measure on . ∎
Corollary 3.7.
Let be an ample groupoid with compact unit space. The following conditions are equivalent:

admits a invariant probability measure;

weakly contains the trivial representation;

admits a character;

admits a invariant probability measure.
4. Representations of topological full groups
In this section, we prove the main results of the article. We start with two technical lemmas.
Lemma 4.1.
Let be an ample groupoid with compact unit space. If and is a clopen subset such that are mutually disjoint, then .
Proof.
We have
The sets and are mutually disjoint and their union is in . This is also the case for the sets and and so the result follows. ∎
In order to employ Lemma 4.1, the following result will be useful.
Lemma 4.2.
Let be an ample groupoid with compact unit space. If and , then
(3)  
(4) 
Proof.
Let
and take . We will show that .
If , we take such that . Then .
On the other hand, if , we take such that and . Then so by the above. Hence proving (3).
The next result generalizes [18, Theorem 3.7], which was obtained in the setting of Cantor minimal systems.
Theorem 4.3.
Let be an ample groupoid with compact unit space. If for every , then is a hereditary subalgebra of .
Proof.
Let . We will first show that
(5) 
It suffices to prove that, given , there is a basis for consisting of compact open sets satisfying , for each . Take and let and be distinct elements in . By Lemma 4.2, there are and ,, and in and , such that
with and for every . By Lemma 4.1, we see that for every sufficiently small compact open neighborhood of . This proves (5).
Next we show that . It suffices to prove that , for every in a basis for consisting of compact open sets. Given , take such that . Then, for sufficiently small compact open neighborhood of , we have that . Let
Since , we have and, finally,
by (5). ∎
Corollary 4.4.
Let be an ample groupoid with compact unit space. If for every , then is a hereditary subalgebra of . If, in addition, is second countable, essentially principal and minimal, then is stably isomorphic to .
Proof.
The next example shows that Theorem 4.3 does not hold without the hypothesis on orbits.
Example 4.5.
Let with the order topology and let be the action of the infinite dihedral group on given by , for and . Then for every .
By arguing as in Example 3.5, one concludes that, given , there is such that .
Let be the canonical conditional expectation and let and be the two states on given by pointevaluations at and , respectively. Then and are two distinct states on whose restrictions to agree. Hence, is not a hereditary subalgebra of .
By combining Theorem 4.3 with the results of the previous section, we obtain the following:
Theorem 4.6.
Let be an ample groupoid with compact unit space. Assume that for every . The following conditions are equivalent.

admits no tracial state;

admits no character;

does not weakly contain the trivial representation;

.
Proof.
(iii)(iv): By Proposition 2.5 and Theorem 4.3, is a hereditary subalgebra of and . Hence, . Since the result follows.
(iv)(i): If has a tracial state, then admits an invariant probability measure , cf. Proposition 2.1. Since for each , cannot be a pointevaluation. Let be the representation of in as in (1). Then extends to a representation of and of . Note that the vector is invariant under and thus under . Now, if , then but is not invariant under . Indeed, if is any proper, nonempty subset which is compact and open, then . Therefore . ∎
Remark 4.7.
In [6, Proposition 5.3], U. Haagerup and K. Olesen considered a certain representation of Thompson’s group in the Cuntz algebra and showed that . Under the identifications of with (see Example 2.3) and with , one can check that and coincide. Hence, Theorem 4.6 recovers part of U. Haagerup and K. Olesen’s result.
We now state and prove a version of Theorem 4.6 regarding .
Theorem 4.8.
Let be an ample groupoid with compact unit space. Assume that for each and consider the following conditions:

admits no tracial state;

admits no character;

does not weakly contain the trivial representation;

.
Then (i) (ii) (iii) (iv).
Proof.
The implications (i)(ii)(iii)(iv) are done as in the full case.
(iv)(ii). If admits a character , then is a point evaluation at some . As is a tracial state, it follows that for each compact and open bisection with , we have . This contradicts the hypothesis that . ∎
The next example shows that the implication from (ii) to (i) in the above theorem fails in general, even in the case when is a principal, minimal and ample groupoid with unit space homeomorphic to the Cantor set.
Example 4.9.
Let be a nonamenable, countable and residually finite group. There is a descending sequence of finite index and normal subgroups of such that the canonical map is injective. Then is a topological group homeomorphic to the Cantor set. Furthermore, the action by multiplication of on is free, minimal and the Haar measure on is invariant (actions of this sort were studied in detail in [3]).
Then admits a tracial state, whereas does not admit a character, since embeds unitally in it and is nonamenable.
5. simplicity of topological full groups
As an application of the above results, we provide conditions which ensure that the topological full group of an ample groupoid is simple.
Recall than an ample groupoid is amenable if there exists a net in of nonnegative functions such that
(6) 
for , uniformly on compact subsets of . Amenability of is equivalent to nuclearity of , and it implies that and are canonically isomorphic. A counterexample of R. Willett [20] has proven that the converse is not true in general.
Lemma 5.1.
Let be an ample groupoid with compact unit space. If is an amenable subgroup which covers , then is amenable.
Proof.
We are going to construct functions satisfying (6). Let be a compact subset and let . As covers and is amenable, there are such that and a finite subset such that
for . Let . For we have . Given , take such that . Then, for ,