(Non)exotic completions of the group algebras of isotropy groups
aa r X i v : . [ m a t h . OA ] D ec (NON)EXOTIC COMPLETIONS OF THE GROUP ALGEBRAS OF ISOTROPYGROUPS JOHANNES CHRISTENSEN AND SERGEY NESHVEYEV Abstract.
Motivated by the problem of characterizing KMS states on the reduced C ∗ -algebrasof étale groupoids, we show that the reduced norm on these algebras induces a C ∗ -norm on thegroup algebras of the isotropy groups. This C ∗ -norm coincides with the reduced norm for thetransformation groupoids, but, as follows from examples of Higson–Lafforgue–Skandalis, it can beexotic already for groupoids of germs associated with group actions. We show that the norm is stillthe reduced one for some classes of graded groupoids, in particular, for the groupoids associatedwith partial actions of groups and the semidirect products of exact groups and groupoids withamenable isotropy groups. Introduction
Over the last fifty years there has been a significant interest in describing KMS states for differentC ∗ -dynamical systems. By now there are several strategies how to approach this problem in concreteexamples. One such strategy, which has proven to be extremely useful, is to represent a givensystem as a C ∗ -algebra of a locally compact étale groupoid G with the time evolution defined by acontinuous real-valued -cocycle c on the groupoid. By a result of Renault [Ren80], if the groupoidis Hausdorff and principal, then all KMS β states on C ∗ ( G ) arise by integration with respect toquasi-invariant probability measures µ on G (0) with Radon-Nikodym cocycle e − βc , and they allfactor through the reduced groupoid C ∗ -algebra C ∗ r ( G ) . The situation for non-principal groupoidsis more complicated. As was shown by the second author [Nes13], in this case the KMS states areclassified by pairs ( µ, { ϕ x } x ) consisting of a quasi-invariant measure µ and a measurable field oftracial states ϕ x on the C ∗ -algebras C ∗ ( G xx ) of the isotropy groups satisfying certain conditions.The present paper is motivated by the natural question of how to describe the KMS stateson the reduced groupoid C ∗ -algebras C ∗ r ( G ) . In other words, the question is under which condi-tions on ( µ, { ϕ x } x ) the corresponding KMS state on C ∗ ( G ) factors through C ∗ r ( G ) . A sufficientcondition, which follows immediately from the construction, is that µ -almost all traces ϕ x fac-tor through C ∗ r ( G xx ) . But this condition is not in general necessary, since by an example of Wil-lett [Wil15] based on the HLS groupoids [HLS02] there exist groupoids G such that C ∗ ( G ) = C ∗ r ( G ) ,yet G has a nonamenable isotropy group; see the related Example 3.2 below. In order to obtain anecessary and sufficient condition we introduce a new, in general exotic, C ∗ -norm k·k e on the groupalgebras of the isotropy groups. Denoting by C ∗ e ( G xx ) the corresponding completions, we prove thenthat the state defined by ( µ, { ϕ x } x ) factors through C ∗ r ( G ) if and only if µ -almost all traces ϕ x factor through C ∗ e ( G xx ) , see Proposition 3.1.Although our initial motivation was to characterize KMS states on C ∗ r ( G ) , in the end the mainfocus of the paper is the norm k·k e itself. In particular, we provide several sufficient conditions onthe groupoid that ensure that this norm is equal to the reduced norm. It is easy to show that this Date : December 21, 2020. KU Leuven, Department of Mathematics (Belgium). E-mail: [email protected]. University of Oslo, Mathematics institute (Norway). E-mail: [email protected]. is supported by a DFF-International Postdoctoral Grant from the Independent Research Fund Denmark.S.N. is partially supported by the NFR funded project 300837 “Quantum Symmetry”. s always the case for the transformation groupoids, see Proposition 2.10. We show that this is alsooften the case for graded groupoids, see Theorems 4.5 and 4.11 for the precise statements.The paper consists of four sections. In Section 1 we recall basic properties of locally compactétale groupoids and their associated C ∗ -algebras and we prove a few auxiliary results. In Section 2we define the C ∗ -norm k·k e on the group algebras C G xx of the isotropy groups of a locally compactétale groupoid G . We prove that the norm k·k e agrees with the reduced norm for transformationgroupoids and, using [HLS02, Wil15], we provide an example of a groupoid of germs where it isa genuine exotic norm. In Section 3 we prove that this norm completely governs which stateson C ∗ ( G ) with C ( G (0) ) in their centralizers factor through C ∗ r ( G ) , and we give a partial extensionof this result to weights. This covers the KMS states on the reduced groupoid C ∗ -algebras for thetime evolutions defined by continuous real-valued -cocycles. We end the paper by proving tworesults in Section 4 that give sufficient conditions for the exotic norm to agree with the reducednorm. A common assumption in both cases is the existence of a grading of the groupoid, so toillustrate the relevance of our results, we begin Section 4 by discussing several examples of gradedgroupoids.As a last remark, let us comment on our assumptions on the groupoids. Since the groupoidsof germs and the groupoids associated with semigroups are increasingly popular and can easily benon-Hausdorff, we work with not necessarily Hausdorff locally compact étale groupoids throughoutthe paper, see Section 1 for the precise setup. This requires some extra care at a few places, butno fundamental changes compared to the Hausdorff case. Apart from Section 3, we do not assumethat our groupoids are second countable either.1. Preliminaries
For a groupoid G , we denote by G (0) ⊂ G its unit space and by r, s : G → G (0) the range mapand the source map, respectively, so that r ( g ) = gg − and s ( g ) = g − g . For x, y ∈ G (0) , we set G x := s − ( x ) , G x := r − ( x ) and G yx := G y ∩ G x . In particular, G xx denotes the isotropy group at x .We will be working with locally compact, but not necessarily Hausdorff, étale groupoids, by whichwe mean groupoids G endowed with a topology such that:- the groupoid operations are continuous;- the unit space G (0) is a locally compact Hausdorff space in the relative topology;- the map r is a local homeomorphism.These assumptions imply that the map s is a local homeomorphism as well, the sets G x and G x arediscrete (in particular, Hausdorff), and every point of G has a compact Hausdorff neighbourhood.It is known and not difficult to see that they also imply that G (0) is open in G .If W ⊂ G is open and r | W : W → r ( W ) and s | W : W → s ( W ) are homeomorphisms onto theopen sets r ( W ) and s ( W ) , then W is called an (open) bisection. Given two subsets U and V of G ,we denote by U V the set of pairwise products. If U and V are bisections, then so is U V .Recall how to construct the full and reduced groupoid C ∗ -algebras of G . For an open Hausdorffsubset U ⊂ G , consider the space C c ( U ) of continuous compactly supported functions on U . Wecan consider every f ∈ C c ( U ) as a function on G by continuing f by zero. Define the functionspace C c ( G ) as the sum of the spaces C c ( U ) for all U . Note that the functions in C c ( G ) are notnecessarily continuous. Later we will need the following simple lemma, proved using a partition ofunity argument. Lemma 1.1 ([Exe08, Proposition 3.10]) . If ( U i ) i ∈ I is a covering of G by open Hausdorff sets, then C c ( G ) is the sum of the spaces C c ( U i ) , i ∈ I . e make C c ( G ) into a ∗ -algebra by defining the convolution product f ∗ f of two functions f , f ∈ C c ( G ) via the formula ( f ∗ f )( g ) := X h ∈G r ( g ) f ( h ) f ( h − g ) for g ∈ G , and the involution by f ∗ ( g ) := f ( g − ) . We define a norm k·k on C c ( G ) by setting k f k := sup ρ k ρ ( f ) k , where the supremum is taken over all representations of C c ( G ) by bounded operators on Hilbertspaces. Completing C c ( G ) with this norm we obtain the full groupoid C ∗ -algebra C ∗ ( G ) of G .To introduce the reduced norm on C c ( G ) , for every point x ∈ G (0) define a representation ρ x : C c ( G ) → B ( ℓ ( G x )) by ρ x ( f ) δ g := X h ∈G r ( g ) f ( h ) δ hg , (1.1)where δ g is the Dirac delta-function. The reduced norm k·k r on C c ( G ) is defined by k f k r := sup x ∈G (0) k ρ x ( f ) k , and the reduced groupoid C ∗ -algebra C ∗ r ( G ) is then the completion of C c ( G ) with respect to thisnorm. The identity map C c ( G ) → C c ( G ) extends to a surjective ∗ -homomorphism C ∗ ( G ) → C ∗ r ( G ) .For all f ∈ C c ( G ) , we have the inequalities k f k ∞ ≤ k f k r ≤ k f k , where k·k ∞ denotes thesupremum-norm. If f ∈ C c ( U ) for a bisection U , then k f k = k f k r = k f k ∞ . (1.2)It follows from this and the definition of the product (1) that the space C ( G (0) ) with its usualC ∗ -algebra structure is embedded into C ∗ r ( G ) and C ∗ ( G ) .Next, recall the definition of induced representations. Take x ∈ G (0) and consider the full groupC ∗ -algebra C ∗ ( G xx ) of the isotropy group G xx . Denote by u g , g ∈ G xx , the canonical unitary generatorsof C ∗ ( G xx ) . Assume that ρ : C ∗ ( G xx ) → B ( H ) is a representation. Let L be the space of functions ξ : G x → H satisfying ξ ( gh ) = ρ ( u ∗ h ) ξ ( g ) for all g ∈ G x and h ∈ G xx and such that X g ∈G x / G xx k ξ ( g ) k < ∞ . We define a representation
Ind ρ : C ∗ ( G ) → B ( L ) by ((Ind ρ )( f ) ξ )( g ) := X h ∈G r ( g ) f ( h ) ξ ( h − g ) for f ∈ C c ( G ) . If ρ = λ G xx , the left regular representation of G xx , then Ind λ G xx is unitarily equivalent to therepresentation ρ x : C ∗ ( G ) → B ( ℓ ( G x )) defined by (1.1). Explicitly, in this case we have L = ℓ ( G x × G xx G xx ) , where G x × G xx G xx is the quotient of G x ×G xx by the equivalence relation ( gh ′ , h ) ∼ ( g, h ′ h ) ( g ∈ G x , h, h ′ ∈ G xx ), and the canonical bijection G x × G xx G xx → G x , ( g, h ) gh , gives rise to a unitaryintertwiner between Ind λ G xx and ρ x .For a general representation ρ of C ∗ ( G xx ) , we have a coisometric map v : L → H, vξ = ξ ( x ) , ith the adjoint given by ( v ∗ ζ )( g ) = ( ρ ( u g ) ∗ ζ, if g ∈ G xx , , otherwise . (1.3)Denote by η x the restriction map C c ( G ) ∋ f f | G xx ∈ C c ( G xx ) . We then have v (Ind ρ )( f ) v ∗ = ρ ( η x ( f )) (1.4)for all f ∈ C c ( G ) . Taking a faithful representation and then the regular representation of C ∗ ( G xx ) as ρ , we get the following result. Lemma 1.2.
The restriction map η x : C c ( G ) → C c ( G xx ) extends to a completely positive contraction ϑ x : C ∗ ( G ) → C ∗ ( G xx ) , as well as to a completely positive contraction ϑ x,r : C ∗ r ( G ) → C ∗ r ( G xx ) . The states ϕ ◦ ϑ x are of particular interest to us. Their GNS-representations are described asfollows. Lemma 1.3.
Assume ϕ is a state on C ∗ ( G xx ) and ( H ϕ , π ϕ , ξ ϕ ) is the associated GNS-triple. Considerthe induced representation Ind π ϕ : C ∗ ( G ) → B ( L ) . Then ( L, Ind π ϕ , v ∗ ξ ϕ ) is a GNS-triple associatedwith ϕ ◦ ϑ x , where v ∗ is the isometry (1.3) .Proof. Identity (1.4) for ρ = π ϕ implies that ϕ ◦ ϑ x = ((Ind π ϕ )( · ) v ∗ ξ ϕ , v ∗ ξ ϕ ) . Therefore we only need to check that the vector v ∗ ξ ϕ is cyclic.It suffices to show that if ξ ∈ L is nonzero, then there exists f ∈ C c ( G ) such that ( ξ, (Ind π ϕ )( f ) v ∗ ξ ϕ ) = 0 . Let g ∈ G x be such that ξ ( g ) = 0 . There exists h ∈ G xx such that ( ξ ( g ) , π ϕ ( u h ) ξ ϕ ) = 0 . Choosea bisection U containing gh and pick f ∈ C c ( U ) with f ( gh ) = 1 . Since U ∩ G x = { gh } we have f ( gh ) = 1 and f = 0 on G x \ { gh } . Then ((Ind π ϕ )( f ) v ∗ ξ ϕ )( g ) = X g ′ ∈G x f ( gg ′ )( v ∗ ξ ϕ )( g ′− ) = f ( gh )( v ∗ ξ ϕ )( h − ) = π ϕ ( u h ) ξ ϕ , and similarly (Ind π ϕ )( f ) v ∗ ξ ϕ = 0 on G x \ g G xx . It follows that ( ξ, (Ind π ϕ )( f ) v ∗ ξ ϕ ) = ( ξ ( g ) , π ϕ ( u h ) ξ ϕ ) = 0 , as needed. (cid:3) The maps ϑ x and ϑ x,r have large multiplicative domains. Specifically, we have the followingelementary result, which we will use repeatedly throughout the paper. Lemma 1.4.
Assume that g , g , . . . , g n are distinct points in G xx and { U i } ni =1 is a family of bisectionswith g i ∈ U i for each i . Assume f ∈ C c ( G ) is zero outside S ni =1 U i . Then f lies in the multiplicativedomains of ϑ x and ϑ x,r .Proof. Take f ′ ∈ C c ( G ) . For g ∈ G xx , we have that ( f ∗ f ′ )( g ) = X h ∈G x f ( h ) f ′ ( h − g ) . If h ∈ G x and f ( h ) = 0 , we have h ∈ U j for some j . Since g j ∈ U j and r ( g j ) = x we conclude that h = g j ∈ G xx . This implies that ( f ∗ f ′ )( g ) = X h ∈G xx f ( h ) f ′ ( h − g ) = ( η x ( f ) ∗ η x ( f ′ ))( g ) , ence η x ( f ∗ f ′ ) = η x ( f ) ∗ η x ( f ′ ) . In a similar way we check that η x ( f ′ ∗ f ) = η x ( f ′ ) ∗ η x ( f ) . (cid:3) A possibly exotic norm on the group algebras of isotropy groups
Assume that we are given a locally compact étale groupoid G and a point x ∈ G (0) . We define aC ∗ -norm on the group algebra C G xx as follows. Definition 2.1.
For h ∈ C c ( G xx ) , let k h k e := sup ρ ∈S x k ρ ( h ) k , (2.1)where S x is the collection of representations ρ of C ∗ ( G xx ) such that Ind ρ factors through C ∗ r ( G ) .Since S x contains the regular representation λ G xx , this is indeed a C ∗ -norm and k·k e ≥ k·k r .Denote by C ∗ e ( G xx ) the completion of C G xx with respect to this norm. Proposition 2.2.
For any representation ρ of C ∗ ( G xx ) , the representation Ind ρ of C ∗ ( G ) factorsthrough C ∗ r ( G ) if and only if ρ factors through C ∗ e ( G xx ) .Proof. If Ind ρ factors through C ∗ r ( G ) , then by definition ρ ∈ S x and hence ρ factors through C ∗ e ( G xx ) .Conversely, assume ρ factors through C ∗ e ( G xx ) . Since the collection S x is closed under direct sums,we can find π ∈ S x such that k h k e = k π ( h ) k for all h ∈ C G xx . Then ρ is weakly contained in π .Since weak containment is preserved under induction, we conclude that ρ ∈ S x . (cid:3) The norm k·k e can also be defined in terms of states as follows. Proposition 2.3.
A state ϕ on C ∗ ( G xx ) factors through C ∗ e ( G xx ) if and only if the state ϕ ◦ ϑ x on C ∗ ( G ) factors through C ∗ r ( G ) . Hence, for every h ∈ C c ( G xx ) , we have k h k e = sup ϕ ( h ∗ ∗ h ) / , where the supremum is taken over all states ϕ on C G xx such that ϕ ◦ η x is bounded with respect tothe reduced norm on C c ( G ) .Proof. Since a state on a C ∗ -algebra vanishes on a closed ideal if and only if the associated GNS-representation vanishes on the same ideal, the result follows from Lemma 1.3 and Proposition 2.2. (cid:3) Similarly to the existence of contractions ϑ x : C ∗ ( G ) → C ∗ ( G xx ) and ϑ x,r : C ∗ r ( G ) → C ∗ r ( G xx ) ,identity (1.4) for any faithful representation ρ of C ∗ e ( G xx ) shows that η x extends to a completelypositive contraction ϑ x,e : C ∗ r ( G ) → C ∗ e ( G xx ) . This contraction is surjective and then as a Banach space C ∗ e ( G xx ) is isometrically isomorphic tothe quotient space C ∗ r ( G ) / ker ϑ x,e . Indeed, if A ⊂ C ∗ r ( G ) is the multiplicative domain of ϑ x,e , thenby Lemma 1.4 the image of the C ∗ -algebra A in C ∗ e ( G xx ) under ϑ x,e is dense, hence it coincideswith C ∗ e ( G xx ) and the norm on C ∗ e ( G xx ) ∼ = A/ ker( ϑ x,e | A ) is the quotient norm. Since ϑ x,e is acontraction, we then conclude that the norm k·k e can also be described as the quotient norm on theBanach space C ∗ r ( G ) / ker ϑ x,e , as claimed.From the practical point of view this is still not very useful, as it is not clear what the kernelof ϑ x,e is. The following theorem sharpens the above observation and will allow us to describe thekernel. Theorem 2.4.
For any locally compact étale groupoid G , x ∈ G (0) and h ∈ C c ( G xx ) , we have k h k e = inf {k f k r : f ∈ C c ( G ) , η x ( f ) = h } . (2.2) urthermore, let V be a neighbourhood base at x partially ordered by containment. For each V ∈ V ,choose a function q V ∈ C c ( G (0) ) such that q V ( x ) = 1 , ≤ q V ≤ and supp q V ⊂ V . For h ∈ C c ( G xx ) ,choose any function f ∈ C c ( G ) such that η x ( f ) = h . Then k h k e = lim V ∈V k q V ∗ f ∗ q V k r . (2.3)We divide the proof of the theorem into several lemmas. Denote the right hand side of (2.2)by k h k ′ e . Lemma 2.5. If η x ( f ) = h , then the limit in (2.3) exists and equals k h k ′ e .Proof. Let us show first that if f ∈ C c ( G ) satisfies η x ( f ) = 0 , then lim V ∈V k q V ∗ f ∗ q V k r = 0 . (2.4)By Lemma 1.1 we can write f = P mi =1 f i , with f i ∈ C c ( U i ) for a bisection U i . Assume G xx ∩ (cid:16) m [ i =1 U i (cid:17) = { g , . . . , g n } . Take an index i . Assume first that G xx ∩ U i = ∅ . In this case we eventually have q V ∗ f i ∗ q V = 0 .Indeed, let K i ⊂ U i be the support of f i | U i . For every g ∈ K i we must have that either r ( g ) = x or s ( g ) = x , so we can find a neighbourhood W g of g in U i with either x / ∈ r ( W g ) or x / ∈ s ( W g ) .By compactness of K i there is a neighbourhood W of x in G (0) such that for each g ∈ K i we haveeither r ( g ) / ∈ W or s ( g ) / ∈ W . This implies that if x ∈ V ⊂ W , then q V ∗ f i ∗ q V = 0 , proving ourclaim.Consider now indices i such that G xx ∩ U i = ∅ . For every such i there is a unique index k i suchthat G xx ∩ U i = { g k i } . Put W k = \ i : k i = k U i . For every k , the set W k is a bisection containing g k and we have X i : k i = k f i ( g k ) = f ( g k ) = 0 . (2.5)Since ( r − ( V ) ∩ U i ) V ∈V is a neighbourhood base for g k i in U i and f i | U i is continuous at g k i , wehave lim V k q V ∗ f i ∗ q V − f i ( g k i ) q V ∗ W ki ∗ q V k ∞ = 0 . For x ∈ V ⊂ V ⊂ r ( W k i ) , we have q V ∗ f i ∗ q V − f i ( g k i ) q V ∗ W ki ∗ q V ∈ C c ( U i ) . Hence, by (1.2),we may conclude that lim V k q V ∗ f i ∗ q V − f i ( g k i ) q V ∗ W ki ∗ q V k r = 0 , which together with (2.5) gives lim V (cid:13)(cid:13)(cid:13) X i : k i = k q V ∗ f i ∗ q V (cid:13)(cid:13)(cid:13) r = 0 for all k = 1 , . . . , n . This, combined with the fact that we eventually have q V ∗ f i ∗ q V = 0 for i suchthat G xx ∩ U i = ∅ , proves (2.4).Assume now that η x ( f ) = h . For every W ∈ V , we have lim V k q V ∗ f ∗ q V − ( q V q W ) ∗ f ∗ ( q W q V ) k r = 0 y (2.4) applied to f − q W ∗ f ∗ q W or simply because k q V − q W q V k ∞ → . As k ( q V q W ) ∗ f ∗ ( q W q V ) k r ≤k q W ∗ f ∗ q W k r , this implies that lim sup V k q V ∗ f ∗ q V k r ≤ k q W ∗ f ∗ q W k r . Since this is true for all W ∈ V , we conclude that the limit in (2.3) indeed exists. Furthermore,since η x ( q V ∗ f ∗ q V ) = η x ( f ) = h and k q V ∗ f ∗ q V k r ≤ k f k r , we have k h k ′ e ≤ lim V k q V ∗ f ∗ q V k r ≤ k f k r . (2.6)Now, take another function f ′ ∈ C c ( G ) such that η x ( f ′ ) = h . By (2.4) applied to f − f ′ weconclude that the limit of k q V ∗ f ∗ q V k r is independent of f such that η x ( f ) = h . Together withthe inequalities (2.6) and the definition of k h k ′ e , this proves that this limit is k h k ′ e . (cid:3) It is clear that k·k ′ e is a seminorm on C c ( G xx ) , but we can now say much more. Lemma 2.6.
The seminorm k·k ′ e is a C ∗ -norm on the group algebra C G xx , and k·k ′ e ≥ k·k e .Proof. Since ϑ x,e : C ∗ r ( G ) → C ∗ e ( G xx ) is a contraction, if h ∈ C c ( G xx ) and f ∈ C c ( G ) satisfy η x ( f ) = h ,we get k h k e = k η x ( f ) k e ≤ k f k r . By the definition of k·k ′ e it follows that k h k e ≤ k h k ′ e . In particular, k·k ′ e is a norm.Next, let us check that k h ∗ h k ′ e ≤ k h k ′ e k h k ′ e . Fix ε > and choose f ∈ C c ( G ) suchthat η x ( f ) = h and k f k r < k h k ′ e + ε . Suppose h is supported on { g , . . . , g n } ⊂ G xx . Let U , . . . , U n be bisections such that G xx ∩ U i = { g i } . Choose ϕ i ∈ C c ( U i ) such that ϕ i ( g i ) = 1 and let f = P i h ( g i ) ϕ i . Then η x ( f ) = h and by Lemma 1.4 we have η x ( f ∗ f ) = η x ( f ) ∗ η x ( f ) = h ∗ h .By Lemma 2.5, by replacing ϕ i with q V ∗ ϕ i ∗ q V (where q V is as in Theorem 2.4), we may assumethat k f k r < k h k ′ e + ε . Hence k h ∗ h k ′ e ≤ k f ∗ f k r ≤ ( k h k ′ e + ε )( k h k ′ e + ε ) , which implies that k h ∗ h k ′ e ≤ k h k ′ e k h k ′ e , as ε > was arbitrary.In order to show that k·k ′ e is a C ∗ -norm, it remains to check that k h k ′ e ≤ k h ∗ ∗ h k ′ e . Similarly to theprevious paragraph, by Lemma 1.4 we can find f ∈ C c ( G ) such that η x ( f ) = h and η x ( f ∗ ∗ f ) = h ∗ ∗ h .Then we have k h k ′ e = lim V k q V ∗ f ∗ q V k r = lim V k q V ∗ f ∗ ∗ q V ∗ f ∗ q V k r ≤ lim V k q V ∗ f ∗ ∗ f ∗ q V k r = k h ∗ ∗ h k ′ e , which completes the proof of the lemma. (cid:3) Proof of Theorem 2.4.
We already know that k·k e ≤ k·k ′ e . In order to prove the opposite inequalityit suffices to show that if ϕ is a state on C G xx bounded with respect to k·k ′ e , then it is also boundedwith respect to k·k e . By Proposition 2.3 this means that we have to show that ϕ ◦ η x is boundedwith respect to the reduced norm on C c ( G ) . But this is obvious, since by definition k·k ′ e is thequotient reduced norm on C c ( G ) / ker η x . (cid:3) Corollary 2.7.
The space ker( η x : C c ( G ) → C c ( G xx )) is dense in ker( ϑ x,e : C ∗ r ( G ) → C ∗ e ( G xx )) .Proof. Take a ∈ ker ϑ x,e and ε > . We can find f ∈ C c ( G ) such that k a − f k r < ε . Then k η x ( f ) k e = k ϑ x,e ( f − a ) k e < ε. By Theorem 2.4 it follows that we can find f ∈ C c ( G ) such that η x ( f ) = η x ( f ) and k f k r < ε .Then for f := f − f ∈ C c ( G ) we have η x ( f ) = 0 and k a − f k r ≤ k a − f k r + k f k r < ε . (cid:3) orollary 2.8. If { x } ⊂ G (0) is an invariant subset, so that G yx = ∅ for all y = x , then G \ G xx is alocally compact étale groupoid and we have a short exact sequence → C ∗ r ( G \ G xx ) → C ∗ r ( G ) → C ∗ e ( G xx ) → . (2.7) Proof.
It is easy to see that
G \ G xx is an open subgroupoid of G . As ϑ x,e : C ∗ r ( G ) → C ∗ e ( G xx ) is a ∗ -homomorphism in the present case and ker η x is dense in ker ϑ x,e , we then only need to show that C c ( G \ G xx ) is dense in ker η x with respect to the reduced norm on C c ( G ) . But this follows from (2.4),since f − q V ∗ f ∗ q V ∈ C c ( G \ G xx ) for any f ∈ C c ( G ) as long as q V = 1 in a neighbourhood of x . (cid:3) Remark . If one is interested only in the setting of Corollary 2.8, then it is actually easier tofirst realize that there is a C ∗ -norm k·k e on C G xx making (2.7) exact, and then that this norm mustsatisfy (2.1) and (2.2), cf. [Ren80, Proposition II.4.5] (but note that this proposition erroneouslyclaims that we get the reduced norm) and [Wil15, Lemma 2.7]. Similar statements can be provedfor any closed invariant subset of G (0) in place of { x } . ♦ As a first example consider the transformation groupoids. Assume X is a locally compact Haus-dorff space and a discrete group Γ acts on X by homeomorphisms. The transformation groupoid G := Γ ⋉ X corresponding to such an action is the set Γ × X with product ( g, hx )( h, x ) = ( gh, x ) , so that r ( g, x ) = gx, s ( g, x ) = x and ( g, x ) − = ( g − , gx ) . Endowed with the product topology, it becomes a locally compact Hausdorff étale groupoid, andthen C ∗ r (Γ) ∼ = C ( X ) ⋊ r Γ . For the isotropy groups we have G xx = Γ x × { x } ∼ = Γ x , where Γ x is thestabilizer of x . Proposition 2.10.
For any transformation groupoid G = Γ ⋉ X , the norm k·k e on C G xx ∼ = C Γ x coincides with the reduced norm.Proof. We have an isometric embedding of C ∗ r (Γ x ) into M ( C ( X ) ⋊ r Γ) . Using this embedding,for any a ∈ C Γ x and f ∈ C c ( X ) such that k f k ∞ = 1 and f ( x ) = 1 , we have af ∈ C c ( G ) and η x ( af ) = a . It follows that k a k e ≤ k af k r ≤ k a k r . Hence k a k e = k a k r . (cid:3) Examples where the norm k·k e is exotic (that is, it is neither reduced nor maximal [KS12]) are,however, just one step away from the transformation groupoids. Namely, they can be found amongthe associated groupoids of germs, which are defined as follows.Given an action of a discrete group Γ on a locally compact Hausdorff space X , define an equiva-lence relation on Γ × X by saying ( g, x ) ∼ ( h, x ) if gy = hy for all y in a neighbourhood of x . Then G = (Γ × X ) / ∼ is a groupoid with the unit space X and the composition [ g, hx ][ h, x ] = [ gh, x ] ,where [ g, x ] denotes the equivalence class of ( g, x ) . Equipped with the quotient topology, G becomesa locally compact étale groupoid. In general, such groupoids are non-Hausdorff.It follows from examples of Higson–Lafforgue–Skandalis [HLS02, Section 2], see also [Wil15], thatthe norm k·k e can be exotic for groupoids that are group bundles. In the next example we make aminimal modification of their construction to demonstrate the same phenomenon for groupoids ofgerms. Example . Let Γ be a discrete group and (Γ n ) ∞ n =1 be a decreasing sequence of finite index normalsubgroups of Γ such that T ∞ n =1 Γ n = { e } . Let X n := Γ / Γ n and view X n as a finite discrete set. Wehave an action of Γ on X n by translations. Let X be the one-point compactification of F ∞ n =1 X n .The actions of Γ on X n together with the trivial action on the point ∞ ∈ X define an action of Γ on X .Let G be the corresponding groupoid of germs. Explicitly, we have G = ∞ G n =1 ((Γ / Γ n ) × X n ) ⊔ (Γ × {∞} ) , ach finite set (Γ / Γ n ) × X n is open and has the discrete topology, while the sets ∪ n ≥ N ( { π n ( g ) } × X n ) ∪{ ( g, ∞ ) } for N ∈ N form a neighbourhood base at ( g, ∞ ) , where π n : Γ → Γ / Γ n is the quotientmap. Note that G is Hausdorff.Consider the point x = ∞ ∈ X = G (0) , which is the only point of G (0) with nontrivial isotropy.Then G xx = Γ × {∞} ∼ = Γ . Using (2.3) it is easy to see that, for all a ∈ C Γ , k a k e = lim N →∞ max (cid:8) sup n ≥ N k λ n ( a ) k , k a k r (cid:9) = lim n →∞ k λ n ( a ) k , (2.8)where λ n denotes the quasi-regular representation of Γ on ℓ (Γ / Γ n ) and the second equality abovefollows because the sequence {k λ n ( a ) k} n is nondecreasing and the regular representation of Γ isweakly contained in L n λ n , so that k a k r ≤ sup n k λ n ( a ) k , cf. [Wil15, Lemma 2.7].Following [HLS02, Wil15], an explicit example with k·k e = k·k r , k·k max can be obtained as follows.Take Γ = SL ( Z ) and Γ n = ker( SL ( Z ) → SL ( Z / n Z )) . Since Γ is nonamenable and the trivialrepresentation of Γ is contained in λ n for every n , we cannot have k·k e = k·k r . On the other hand,the trivial representation is isolated in the irreducible unitary representations weakly containedin L ∞ n =1 λ n by Selberg’s theorem, yet Γ does not have property (T). Hence we cannot have k·k e = k·k max either.3. Application to states and weights with diagonal centralizers
Throughout this section we assume that G is a second countable locally compact étale groupoid.(But we still do not require G to be Hausdorff.)Consider a Radon measure µ on G (0) . A µ -measurable field of states on the C ∗ -algebras of theisotropy groups is a collection { ϕ x } x ∈G (0) , where ϕ x is a state on C ∗ ( G xx ) for all x ∈ G (0) , such thatthe function G (0) ∋ x X g ∈G xx f ( g ) ϕ x ( u g ) is µ -measurable for all f ∈ C c ( G ) .By [Nes13, Theorem 1.1] , there is a bijective correspondence between the states on C ∗ ( G ) con-taining C ( G (0) ) in their centralizers and the pairs ( µ, { ϕ x } x ∈G (0) ) consisting of a (regular, Borel)probability measure µ on G (0) and a µ -measurable field of states { ϕ x } x ∈G (0) . The state ϕ corre-sponding to ( µ, { ϕ x } x ∈G (0) ) is given by ϕ ( f ) = Z G (0) X g ∈G xx f ( g ) ϕ x ( u g ) dµ ( x ) for all f ∈ C c ( G ) . In other words, ϕ = Z G (0) ϕ x ◦ ϑ x dµ ( x ) , (3.1)with the integral understood in the weak ∗ sense. To be precise, the formulation in [Nes13] requires G to be Hausdorff, but as has already been observed in [NS19], the result is true in the non-Hausdorffcase as well, with essentially the same proof; see also Remark 3.5 below. Proposition 3.1.
For any second countable locally compact étale groupoid G , a state ϕ on C ∗ ( G ) with centralizer containing C ( G (0) ) factors through C ∗ r ( G ) if and only if the state ϕ x factors through C ∗ e ( G xx ) for µ -almost all x , where ( µ, { ϕ x } x ∈G (0) ) is the pair associated with ϕ .Proof. Let I be the kernel of the canonical map C ∗ ( G ) → C ∗ r ( G ) . Since G is second countable, theC ∗ -algebra C ∗ ( G ) is separable. (Indeed, we can cover G by countably many bisections U n , choosecountable subsets of C c ( U n ) that are dense in the supremum-norm, then the sums of finitely manyelements in the union of all these subsets will be dense in C ∗ ( G ) by (1.2) and Lemma 1.1.) Let F be countable dense subset of I + . The state ϕ factors through C ∗ r ( G ) if and only if it vanishes on F .One the other hand, by Proposition 2.3, the state ϕ x factors through C ∗ e ( G xx ) if and only if ϕ x ◦ ϑ x vanishes on F . Since ϕ ( a ) = Z G (0) ( ϕ x ◦ ϑ x )( a ) dµ ( x ) for all a ∈ F , we get the result. (cid:3) Example . As discussed in the introduction, our main motivation for writing this paper was tocharacterize the KMS states on C ∗ r ( G ) for the time evolutions defined by real-valued -cocycles on G .A KMS state for such a time evolution on C ∗ ( G ) is always given by a pair ( µ, { ϕ x } x ∈G (0) ) satisfyingsome extra conditions [Nes13, Theorem 1.3], and the question was when it factors through C ∗ r ( G ) .Consider, for instance, the groupoid G from Example 2.11 defined by a group Γ and a decreasingsequence (Γ n ) ∞ n =1 of finite index normal subgroups. Take the zero cocycle, so we want to describethe tracial states on C ∗ r ( G ) . Then Proposition 3.1 and [Nes13, Theorem 1.3] imply that suchtraces are classified by pairs ( µ, τ ) , where µ is a probability measure on X such that µ | X n isa scalar multiple of the counting measure for all n and, if µ ( ∞ ) > , τ is a tracial state on C ∗ e (Γ) = C ∗ e ( G ∞∞ ) , otherwise τ is irrelevant. (This description also easily follows from Corollary 2.8,since C ∗ r ( G \ G ∞∞ ) ∼ = c - L ∞ n =1 B ( ℓ (Γ / Γ n )) .)Note that there can be many more tracial states on C ∗ e (Γ) than those that factor through C ∗ r (Γ) ,which are the only obvious ones that give rise to tracial states on C ∗ r ( G ) . Consider, for example, thefree group Γ = F on two generators, and define the subgroups (Γ n ) ∞ n =1 as in [Wil15, Lemma 2.8].It then follows from [Wil15, Lemma 2.8] and (2.8) that k·k e agrees with the full norm on C Γ = C F .Therefore there are plenty of tracial states on C ∗ e (Γ) , but only one tracial state on C ∗ r (Γ) , namely,the canonical trace. ♦ For completeness, let us describe the GNS-representation defined by the state ϕ given by a pair ( µ, { ϕ x } x ∈G (0) ) . Let ( H x , π x , ξ x ) be the GNS-triple associated with ϕ x ◦ ϑ x . Recall that by Lemma 1.3the representation π x is induced from the GNS-representation defined by ϕ x . Lemma 3.3.
There is a unique structure of a µ -measurable field of Hilbert spaces on ( H x ) x ∈G (0) such that the sections ( π x ( f ) ξ x ) x ∈G (0) are measurable for all f ∈ C c ( G ) . Namely, a section ( ζ x ) x ∈G (0) is measurable if and only if the function x ( ζ x , π x ( f ) ξ x ) is µ -measurable for all f ∈ C c ( G ) .Proof. Choose a sequence { f n } n of elements of C c ( G ) that is dense in C ∗ ( G ) . Then { π x ( f n ) ξ x } n isdense in H x for every x ∈ G (0) . By [Tak02, Lemma IV.8.10] it follows that there is a unique structureof a µ -measurable field of Hilbert spaces on ( H x ) x ∈G (0) such that the sections ( π x ( f n ) ξ x ) x ∈G (0) are measurable for all n , namely, a section ( ζ x ) x ∈G (0) is measurable if and only if the function x ( ζ x , π x ( f n ) ξ x ) is µ -measurable for all n ∈ N . To finish the proof of the lemma it is then enoughto show that the section ( π x ( f ) ξ x ) x ∈G (0) is measurable for every f ∈ C c ( G ) . But this is clearly true,since we can find a subsequence { f n k } k converging to f in C ∗ ( G ) , and then π x ( f n k ) ξ x → π x ( f ) ξ x for all x ∈ G (0) . (cid:3) We can therefore consider the direct integral π of the GNS-representations π x on H := Z ⊕G (0) H x dµ ( x ) . Consider also the vector ξ := ( ξ x ) x ∈G (0) ∈ H . Proposition 3.4.
With the above notation, ( H, π, ξ ) is a GNS-triple associated with ϕ .Proof. We only need to show that the vector ξ is cyclic. Consider the representation m of L ∞ ( G (0) , µ ) on H = R ⊕G (0) H x dµ ( x ) by diagonal operators. As π x ( f ) ξ x = f ( x ) ξ x for all f ∈ C ( G (0) ) and x ∈ G (0) , e then have π ( f ) ξ = m ( f ) ξ for all f ∈ C ( G (0) ) . It follows that for all a ∈ C ∗ ( G ) and f ∈ C ( G (0) ) we have m ( f ) π ( a ) ξ = π ( a ) m ( f ) ξ = π ( af ) ξ. Therefore the subspace π ( C ∗ ( G )) ξ is invariant under the diagonal operators, hence the projection P onto this subspace is a decomposable operator. Thus, P = ( P x ) x ∈G (0) for a measurable family ofprojections P x . For µ -almost all x , we also have that the projection P x commutes with π x ( C ∗ ( G )) and P x ξ x = ξ x . Since the vector ξ x is cyclic, it follows that P x = 1 a.e., hence P = 1 . (cid:3) Remark . The above proposition can be approached from a different angle by trying to disinte-grate the GNS-representation of a given state ϕ . This leads to a proof of decomposition (3.1) thatrelies only on disintegration of representations rather than on Renault’s disintegration theorem forgroupoids [Ren87], which is a more delicate result, especially in the non-Hausdorff setting. Let ussketch this argument, which elaborates on the discussion in [Nes13, Section 2].So, assume we have a state ϕ on C ∗ ( G ) with C ( G (0) ) in its centralizer. Let ( H, π, ξ ) bethe corresponding GNS-triple. By [Nes13, Lemma 2.2] and its proof, there is a representation ρ : C ( G (0) ) → π ( C ∗ ( G )) ′ ⊂ B ( H ) such that ρ ( h ) π ( a ) ξ = π ( ah ) ξ for all h ∈ C ( G (0) ) and a ∈ C ∗ ( G ) .We can then disintegrate π with respect to ρ ( C ( G (0) )) , so that π = R L G (0) π x dν ( x ) for representa-tions π x : C ∗ ( G ) → B ( H x ) and ρ becomes the representation by diagonal operators. If ξ = ( ξ x ) x ,we define a probability measure µ on G (0) by dµ ( x ) = k ξ x k dν ( x ) and, for µ -almost all x ∈ G (0) ,states ψ x on C ∗ ( G ) by ψ x = k ξ x k − ( π x ( · ) ξ x , ξ x ) . Then ϕ = R G (0) ψ x dµ ( x ) .Since π ( h ) ξ = ρ ( h ) ξ , we have π x ( h ) ξ x = h ( x ) ξ x , and hence ψ x ( ah ) = ψ x ( ha ) = h ( x ) ψ x ( a ) , for µ -a.e. x and all h ∈ C ( G (0) ) and a ∈ C ∗ ( G ) . By the analogue of (2.4) for the full norm, which isproved using the same arguments, we conclude that, for µ -a.e. x and all f ∈ C c ( G ) , the value ψ x ( f ) depends only on η x ( f ) , so that ψ x = ϕ x ◦ η x on C c ( G ) for a linear functional ϕ x on C c ( G xx ) . Thislinear functional is positive, hence it extends to a state on C ∗ ( G xx ) , since by Lemma 1.4, for every h ∈ C c ( G xx ) , we can find f ∈ C c ( G ) such that h ∗ ∗ h = η x ( f ∗ ∗ f ) . ♦ Proposition 3.1 can be extended to a class of weights on C ∗ ( G ) . Let us first recall some termi-nology. A weight ϕ on a C ∗ -algebra A is a map ϕ : A + → [0 , + ∞ ] satisfying ϕ ( a + b ) = ϕ ( a ) + ϕ ( b ) and ϕ ( λa ) = λϕ ( a ) for all a, b ∈ A + and λ ≥ . A weight ϕ is called densely defined when { a ∈ A + | ϕ ( a ) < ∞} is dense in A + , and it is called lower semi-continuous if { a ∈ A + | ϕ ( a ) ≤ λ } is closed for all λ ≥ . We call ϕ proper if ϕ is nonzero, densely defined and lower semi-continuous.We use the standard notation N ϕ := { a ∈ A | ϕ ( a ∗ a ) < ∞} and M + ϕ := { a ∈ A + | ϕ ( a ) < ∞} . Any weight ϕ extends uniquely to a linear functional on the subspace M ϕ := N ∗ ϕ N ϕ = span M + ϕ ,which is a dense ∗ -subalgebra of A if ϕ is densely defined.Let { ϕ x } x ∈G (0) be a µ -measurable field of states for a non-zero Radon measure µ on G (0) . Thenthe map x ( ϕ x ◦ ϑ x )( a ) is µ -measurable for all a ∈ C ∗ ( G ) + , and hence we can define a map ϕ : C ∗ ( G ) + → [0 , + ∞ ] by ϕ ( a ) := Z G (0) ( ϕ x ◦ ϑ x )( a ) dµ ( x ) for a ∈ C ∗ ( G ) + . (3.2)Since G can be covered by bisections and µ is Radon, it follows from Lemma 1.1 that ϕ is finiteon C c ( G ) + . By Fatou’s lemma it is also lower semi-continuous. It follows that ϕ is a proper weighton C ∗ ( G ) , with C c ( G ) ⊂ M ϕ . In conclusion we can associate a proper weight ϕ on C ∗ ( G ) to anypair ( µ, { ϕ x } x ∈G (0) ) consisting of a Radon measure µ on G (0) and a µ -measurable field of states { ϕ x } x ∈G (0) . We say that the weight ϕ factors through C ∗ r ( G ) if there exists a proper weight ˜ ϕ n C ∗ r ( G ) such that ϕ ( a ) = ˜ ϕ ( π ( a )) for all a ∈ C ∗ ( G ) + , where π : C ∗ ( G ) → C ∗ r ( G ) is the canonicalsurjection. Proposition 3.6.
Let G be a second countable locally compact étale groupoid and ϕ be the properweight on C ∗ ( G ) associated to a pair ( µ, { ϕ x } x ∈G (0) ) consisting of a Radon measure µ and a µ -measurable field of states { ϕ x } x ∈G (0) . Then the following conditions are equivalent: (1) ϕ factors through C ∗ r ( G ) ; (2) ϕ ((ker π ) + ) = 0 ; (3) ϕ x factors through C ∗ e ( G xx ) for µ -almost all x .Proof. That (1) implies (2) is clear. Assume now that (2) is true. Let { V n } ∞ n =1 be an increasingsequence of open sets in G (0) with compact closures such that G (0) = ∪ ∞ n =1 V n . Since µ is a Radonmeasure, the measures µ n := µ | V n are all finite regular Borel measures on G (0) . Let ψ n be thepositive linear functional on C ∗ ( G ) associated to ( µ n , { ϕ x } x ∈G (0) ) . It then follows from (3.2) thatfor any a ∈ C ∗ ( G ) + we have ϕ ( a ) = Z G (0) ( ϕ x ◦ ϑ x )( a ) dµ ( x ) ≥ Z V n ( ϕ x ◦ ϑ x )( a ) dµ ( x ) = ψ n ( a ) . By assumption this must imply that ψ n ((ker π ) + ) = 0 for all n ∈ N , and hence (3) is true byProposition 3.1.Assume now that (3) is true. Then similarly to ϕ we can define a proper weight ˜ ϕ on C ∗ r ( G ) by ˜ ϕ ( a ) := Z G (0) ( ϕ x ◦ ϑ x,e )( a ) dµ ( x ) for a ∈ C ∗ r ( G ) + , where we view ϕ x as a state on C ∗ e ( G xx ) whenever it makes sense, which is the case for µ -a.e. x byassumption. Since ϕ x ◦ ϑ x = ϕ x ◦ ϑ x,e ◦ π , we obviously have ϕ ( a ) = ˜ ϕ ( π ( a )) for all a ∈ C ∗ ( G ) + .This completes the proof. (cid:3) Remark . There is no known analogue of decomposition (3.1) for arbitrary proper weights withcentralizer containing C ( G (0) ) . If one assumes that G is Hausdorff and restricts attention to KMSweights with respect to the time evolution defined by a real-valued -cocycle on G , then it followsfrom Theorem 6.3 in [Chr20] that every KMS weight is given by a pair ( µ, { ϕ x } x ∈G (0) ) (satisfyingsome extra conditions), and hence Proposition 3.6 can be used to describe all KMS weights on C ∗ r ( G ) in this setting. 4. Graded groupoids
Given a locally compact étale groupoid G and a discrete group Γ , by a Γ -valued -cocycle on G we mean a continuous homomorphism Φ :
G → Γ . We then say that Φ defines a Γ -grading on G .In this section we prove several extensions of Proposition 2.10 of the form that if G is a gradedgroupoid satisfying some extra assumptions, then the norm k·k e coincides with the reduced norm.Let us first give a few examples of graded groupoids. Example . Assume a discrete group Γ acts by automorphisms on a locallycompact étale groupoid G . Then the semidirect product Γ ⋉ G is the space Γ × G with product ( γ , g )( γ , g ) = ( γ γ , γ − ( g ) g ) . Therefore the unit space of Γ × G is { e } × G (0) , and if we identify it with G (0) , then the range andsource maps are r ( γ, g ) = r ( γ ( g )) , s ( γ, g ) = s ( g ) . Equipped with the product topology, Γ ⋉ G becomes a locally compact étale groupoid. The map Φ : Γ ⋉ G → Γ , Φ( γ, g ) = γ , defines a grading on Γ ⋉ G . ote that in general the isotropy groups Γ ⋉ G are not determined by those of G and the stabilizersof the Γ -action, but for every x ∈ G (0) we at least have (Γ ⋉ G ) xx ∩ ker Φ = { e } × G xx ∼ = G xx , so that (Γ ⋉ G ) xx is an extension of a subgroup of Γ by G xx . ♦ Example . Given a discrete group Γ and a locally compact Hausdorffspace X , a partial action of Γ on X is given by the following data. For every γ ∈ Γ , we are givenopen subsets X γ − and X γ of X and a homeomorphism X γ − → X γ , which we denote simply by γ ,such that X e = X and, for all γ , γ ∈ Γ , we have ( γ γ ) x = γ ( γ x ) whenever the right hand sideis well-defined (in other words, the homeomorphism γ γ is an extension of the composition of γ and γ ).Every partial action of Γ defines a transformation-type groupoid G [Aba04]: G = { ( y, γ, x ) ∈ X × Γ × X : y = γx } , with the obvious product ( z, γ , y )( y, γ , x ) = ( z, γ γ , x ) . (4.1)Equipped with the topology inherited from the product topology on X × Γ × X , the groupoid G becomes a locally compact Hausdorff étale groupoid. The map Φ :
G → Γ , Φ( y, γ, x ) = γ , defines a Γ -grading on G . This grading is injective in the sense that ker(Φ | G xx ) = { x } for all x ∈ G (0) .The simplest way of getting a partial action of Γ is to start with a genuine action on a locallycompact Hausdorff space Y containing X as an open subset. We then get a partial action of Γ on X by restriction, so that X γ = X ∩ γ ( X ) . In this case the groupoid G of the partial action issimply the reduction of the transformation groupoid Γ × Y by X . But not all partial actions canbe obtained this way, at least if we require Y to be Hausdorff. A necessary condition for a partialaction to arise as a reduction of a genuine action is that the graphs { ( x, γx ) : x ∈ X γ − } must beclosed in X × X for all γ ∈ Γ . This condition is actually also sufficient, see [Aba03, Proposition 2.10and Remark 2.11].As an example, assume we are given partial homeomorphisms S and S on X , so that we aregiven open subsets A i and B i of X ( i = 1 , ) and homeomorphisms S i : A i → B i . From this wecan construct a partial action of the free group F with two generators s and s . Namely, given areduced nonempty word γ = s a i . . . s a n i n ∈ F , the action of γ is given by S a i . . . S a n i n . If the graphof S or S is not closed in X × X , this partial action is not a reduction of a genuine action. ♦ Example . We recall the construction of Exel and Re-nault [ER07] generalizing the Deaconu–Renault groupoids of local homeomorphisms [Ren80, Dea95].Let Γ be a discrete group and S ⊂ Γ be a submonoid such that SS − ⊂ S − S . Assume S actsby local homeomorphisms on a locally compact Hausdorff space X . We define a groupoid by G := { ( y, γ, x ) ∈ X × Γ × X : ∃ s, t ∈ S such that γ = s − t and sy = tx } . (4.2)The product is defined by (4.1). The sets { ( y, s − t, x ) : x ∈ A, y ∈ B, sy = tx } , where A, B ⊂ X are open and s, t ∈ S , form a basis of topology on G . This topology is strongerthan, and in general different from, the one induced by the product topology on X × Γ × X . Themap Φ :
G → Γ , Φ( y, γ, x ) = γ , defines an injective Γ -grading on G .In most concrete examples of this construction studied in the literature the group Γ is abelian,which is not particularly interesting for our purposes, so let us give a class of examples with poten-tially more complicated isotropy groups.Assume G is a discrete group, θ ∈ Aut( G ) , and consider the group Γ :=
Z ⋉ θ G . Consider thesubmonoid S := Z + ⋉ θ G ⊂ Γ . Then SS − = S − S = Γ . Assume we have an action of S on X by ocal homeomorphisms. Then the corresponding groupoid is G = { ( y, k − l, g, x ) ∈ X × ( Z ⋉ θ G ) × X : k, l ≥ , σ l ( y ) = σ k ( gx ) } , (4.3)where σ is the local homeomorphism defined by the action of ∈ Z + .Note that to have an action of S by local homeomorphisms is the same as having a local home-omorphism σ : X → X and an action of G on X by homeomorphisms such that σ ( gx ) = θ ( g ) σ ( x ) for all g ∈ G and x ∈ X . A simple, but possibly not the most exciting, way of producing suchexamples is to consider a local homeomorphism σ : X → X and an action of Γ on X , and thenconsider the local homeomorphism σ = ( σ , on X := X × X together with the action of G onthe second factor of X . ♦ In some cases a graded groupoid can be transformed into a new injectively graded one.
Example . Let G be a locally compact étale groupoid with a grading Ψ :
G → Γ . Assume thatthe following condition is satisfied: if Ψ( g ) = e for some g ∈ G xx and x ∈ G (0) , then r = s in aneighbourhood of g . We then construct a new groupoid G Ψ as follows.Put X := G (0) and consider the set G Ψ := { ( y, γ, x ) ∈ X × Γ × X : ∃ g ∈ G yx such that Ψ( g ) = γ } . The product is again defined by (4.1).To define a topology on G Ψ , take a bisection U of G and consider the set U Ψ := { ( r ( g ) , Ψ( g ) , s ( g )) : g ∈ U } . We claim that these sets form a basis of a topology on G Ψ . It is clear that the sets U Ψ cover G Ψ .Assume now that U and V are two bisections of G and ( y, γ, x ) ∈ U Ψ ∩ V Ψ . Let g ∈ U and h ∈ V bethe unique elements such that g, h ∈ G yx . Then g − h ∈ G xx and Ψ( g − h ) = e . By our assumption,there is a bisection O containing g − h such that r = s on O and Ψ( O ) = { e } . Note that U O (theset of pairwise products) is a bisection and gO = h . Hence we can find an open neighbourhood W of g such that W ⊂ U , s ( W ) ⊂ r ( O ) and h ∈ W O ⊂ V . Since r ( kO ) = r ( k ) , s ( kO ) = s ( k ) and Ψ( kO ) = Ψ( k ) for all k ∈ W , we then have ( y, γ, x ) ∈ W Ψ ⊂ U Ψ ∩ V Ψ .Observe next that the open sets U Ψ are bisections. It is then easy to see that G Ψ is a locallycompact Hausdorff étale groupoid. The map Φ : G Ψ → Γ , Φ( y, γ, x ) = γ , defines an injective Γ -grading on G Ψ . For the isotropy groups we have ( G Ψ ) xx ∼ = Ψ( G xx ) .The above construction can be viewed as a generalization of Example 4.3. Indeed, consider anaction of S ⊂ Γ on X by local homeomorphisms as in that example. For every s ∈ S , s = e , wecan cover X by open sets U s,i such that s | U s,i is a homeomorphism onto sU s,i . Consider the freegroup F with generators g s,i for all s and i . We can define a partial action of F on X similarly toExample 4.2, with g s,i acting by s | U s,i . Let G be the corresponding groupoid. It is F -graded, butwe rather view it as Γ -graded using the homomorphism φ : F → Γ such that φ ( g s,i ) = s for all s and i .To check that our assumption on the corresponding cocycle Ψ :
G → Γ is satisfied, assume g a s ,i . . . g a n s n ,i n x = x for some x ∈ X and g a s ,i . . . g a n s n ,i n ∈ F ( a j = ± ) such that s a . . . s a n n = e . Wehave to show that g a s ,i . . . g a n s n ,i n y = y for all y close to x . By the assumption SS − ⊂ S − S , wecan write st − for any given s, t ∈ S as p − q , which implies that for any indices i, j and any point y ∈ X we can find indices k, l such that g s,i g − t,j z = g − p,k g q,l z for all z close to y . Using this propertyrepeatedly, we can move all the negative powers to the left, that is, without loss of generality wemay assume that a = · · · = a m = − and a m +1 = · · · = a n = 1 for some m . But then for s := s m +1 . . . s n = s m . . . s e have sg a s ,i . . . g a n s n ,i n y = sy , and since s acts by a local homeomorphism, we conclude that g a s ,i . . . g a n s n ,i n y = y for all y close to x . It is now straightforward to check that the groupoid G Ψ ,with its topology, is exactly the groupoid (4.2).Consider a related example inspired by [HL18, Example 3.3]. Let G be the groupoid (4.3) definedby an action of Z + ⋉ θ G on X by local homeomorphisms. Take any discrete group Γ together witha homomorphism φ : G → Γ such that Ω := { γ ∈ Γ : φ ( g ) γ = γφ ( θ ( g )) for all g ∈ G } 6 = ∅ . For any continuous G -invariant function d : X → Ω , we can then define a -cocycle Ψ :
G → Γ by Ψ( y, k − l, g, x ) := d ( y ) d ( σ ( y )) . . . d ( σ l − ( y )) d ( σ k − ( x )) − . . . d ( σ ( x )) − d ( x ) − φ ( g ) , if σ l ( y ) = σ k ( gx ) . Our condition on Ψ reads as follows: if σ l ( x ) = σ k ( gx ) and Ψ( x, k − l, g, x ) = e ,then σ l ( y ) = σ k ( gy ) for all y close to x . If it is satisfied, we get an injectively Γ -graded locallycompact Hausdorff étale groupoid G Ψ . For G = { e } this groupoid coincides with the one defined in[HL18, Example 3.3]. ♦ Theorem 4.5.
Assume Γ is a discrete group and G is a Γ -graded locally compact étale groupoid,with grading Φ :
G → Γ . For a fixed x ∈ G (0) , assume the group ker(Φ | G xx ) is amenable and Φ( G xx ) isexact. Then k·k e = k·k r on C G xx . We need some preparation to prove this theorem. The -cocycle Φ defines a Γ -grading on C c ( G ) , orequivalently, a coaction δ : C c ( G ) → C c ( G ) ⊙ C Γ of the Hopf algebra ( C Γ , ∆) , where ∆( λ g ) = λ g ⊗ λ g and ⊙ denotes the algebraic tensor product. Namely, if we view the elements of C c ( G ) ⊙ C Γ as C Γ -valued functions on G , then δ ( f )( g ) = f ( g ) λ Φ( g ) . The following lemma and its proof are standard.
Lemma 4.6.
The map δ extends to an injective ∗ -homomorphism C ∗ r ( G ) → C ∗ r ( G ) ⊗ C ∗ r (Γ) , whichdefines a coaction of ( C ∗ r (Γ) , ∆) on C ∗ r ( G ) .Proof. For every y ∈ G (0) , define a unitary operator W y : ℓ ( G y ) ⊗ ℓ (Γ) → ℓ ( G y ) ⊗ ℓ (Γ) by ( W y ξ )( g, γ ) := ξ ( g, Φ( g ) − γ ) . A simple computation shows then that for every f ∈ C c ( G ) we have W y ( ρ y ( f ) ⊗
1) = ( ρ y ⊗ ι ) (cid:0) δ ( f ) (cid:1) W y . It follows that we have an injective ∗ -homomorphism δ y : ρ y ( C ∗ r ( G )) → ρ y ( C ∗ r ( G )) ⊗ C ∗ r (Γ) defined by δ y ( a ) = W y ( a ⊗ W ∗ y , and then ( ρ y ⊗ ι ) ◦ δ = δ y ◦ ρ y . Taking the direct sum of the representations ρ y we conclude that the homomorphisms δ y define the required extension of δ . (cid:3) Lemma 4.7.
Let A be a C ∗ -algebra and A ⊂ A a dense ∗ -subalgebra. For h ∈ C c ( G xx ) ⊙ A , define k h k e = inf {k f k r : f ∈ C c ( G ) ⊙ A , ( η x ⊗ ι )( f ) = h } , where k f k r is the norm of f in the minimal tensor product C ∗ r ( G ) ⊗ A . Then k·k e extends uniquelyto a C ∗ -cross norm on C ∗ e ( G xx ) ⊙ A , and if ( q V ) V is a net as in Theorem 2.4, then k h k e = lim V k ( q V ⊗ f ( q V ⊗ k r (4.4) for every h ∈ C c ( G xx ) ⊙ A and any f ∈ C c ( G ) ⊙ A such that ( η x ⊗ ι )( f ) = h . We will need this lemma only for A = C G ⊂ A = C ∗ r ( G ) for a discrete group G , in which case alarge part of it is a consequence of Theorem 2.4 applied to the groupoid G × G . The general caseis, however, of some independent interest and the proof is not much longer. roof. Identity (4.4) and the C ∗ -seminorm property of k·k e are proved exactly as Lemmas 2.5and 2.6, if we view the elements of C c ( G ) ⊙ A as A -valued functions on G . Furthermore, identi-ties (2.3) and (4.4) imply that k h ⊗ a k e = k h k e k a k for all h ∈ C c ( G xx ) and a ∈ A .Next, the seminorm k·k e on C c ( G xx ) ⊙ A dominates the norm defined by the embedding C c ( G xx ) ⊙A ֒ → C ∗ e ( G xx ) ⊗ A . This is proved similarly to the proof of the inequality k·k ′ e ≥ k·k e in Lemma 2.6:since ϑ x,e ⊗ ι : C ∗ r ( G ) ⊗ A → C ∗ e ( G xx ) ⊗ A is a contraction, for all h ∈ C c ( G xx ) ⊙ A and f ∈ C c ( G ) ⊙ A such that ( η x ⊗ ι )( f ) = h , the norm of h in C ∗ e ( G xx ) ⊗ A is not larger than the norm of f in C ∗ r ( G ) ⊗ A . In particular, k·k e is a C ∗ -norm on C c ( G xx ) ⊙ A , and if we denote by B the k·k e -completion of C c ( G xx ) ⊙ A , then the identity map on C c ( G xx ) ⊙ A extends to a ∗ -homomorphism ϕ : B → C ∗ e ( G xx ) ⊗ A .On the other hand, the equality k h ⊗ a k e = k h k e k a k implies that the identity map on C c ( G xx ) ⊙ A extends to a ∗ -homomorphism ψ : C ∗ e ( G xx ) ⊙ A → B . By construction, ϕ ◦ ψ is the identity map on C ∗ e ( G xx ) ⊙ A , hence ψ is injective. Therefore C ∗ e ( G xx ) ⊙ A can be viewed as a subalgebra of B , so therestriction of the norm on B to this subalgebra gives the required extension of k·k e .Finally, C c ( G xx ) ⊙ A is dense in C ∗ e ( G xx ) ⊙ A with respect to any C ∗ -cross norm on C ∗ e ( G xx ) ⊙ A ,so any such norm is completely determined by its restriction to C c ( G xx ) ⊙ A . (cid:3) Therefore the completion of C c ( G xx ) ⊙ A with respect to k·k e is a C ∗ -tensor product of C ∗ e ( G xx ) and A . Lemma 4.8. If A is an exact C ∗ -algebra, then the completion of C G xx ⊙ A with respect to k·k e isthe minimal tensor product C ∗ e ( G xx ) ⊗ A .Proof. The lemma is obviously true for nuclear C ∗ -algebras, in particular, for finite dimensionalones. Assuming that A is only exact, represent it as a concrete C ∗ -algebra A ⊂ B ( H ) . Then theembedding map A → B ( H ) is nuclear. It follows that if we fix f ∈ C c ( G ) ⊙ A and ε > , thenwe can find contractive completely positive maps θ : A → Mat n ( C ) and ψ : Mat n ( C ) → B ( H ) suchthat k f − ( ι ⊗ ψ ◦ θ )( f ) k r < ε .Put h := ( η x ⊗ ι )( f ) ∈ C G xx ⊙ A . Let us denote by k·k the minimal norm on C ∗ e ( G xx ) ⊙ B for B = A and B = Mat n ( C ) . Since the lemma is true for Mat n ( C ) , by using (4.4) we get k h k ≥ k ( ι ⊗ θ )( h ) k = lim V k ( ι ⊗ θ ) (cid:0) ( q V ⊗ f ( q V ⊗ (cid:1) k r ≥ lim V k ( ι ⊗ ψ ◦ θ ) (cid:0) ( q V ⊗ f ( q V ⊗ (cid:1) k r ≥ lim V k ( q V ⊗ f ( q V ⊗ k r − ε = k h k e − ε. As ε was arbitrary, we thus have k h k ≥ k h k e . Since the opposite inequality holds by the previouslemma, we get the result. (cid:3) Proof of Theorem 4.5.
Since δ : C ∗ r ( G ) → C ∗ r ( G ) ⊗ C ∗ r (Γ) is isometric and δ ( q V ∗ f ∗ q V ) = ( q V ⊗ δ ( f )( q V ⊗ for every f ∈ C c ( G ) , identities (2.3) and (4.4) imply that δ induces an injective ∗ -homomorphism ∆ x from C ∗ e ( G xx ) into the k·k e -completion of C G xx ⊙ C ∗ r (Γ) . Put G := Φ( G xx ) ⊂ Γ . Then the image of ∆ x is contained in the closure of C G xx ⊙ C ∗ r ( G ) in the k·k e -completion of C G xx ⊙ C ∗ r (Γ) , hence (by (4.4))in the k·k e -completion of C G xx ⊙ C ∗ r ( G ) . Since G is exact by assumption, by Lemma 4.8 we concludethat ∆ x can be viewed as a homomorphism C ∗ e ( G xx ) → C ∗ e ( G xx ) ⊗ C ∗ r ( G ) .For a discrete group H and a subgroup H , denote by λ H/H the quasi-regular representationof H on ℓ ( H/H ) and by ε H the trivial representation of H . Let us also denote weak containmentand quasi-equivalence of representations by ≺ and ∼ , respectively. Now, put H := G xx ∩ ker Φ and et ρ be a unitary representation of G xx that integrates to a faithful representation of C ∗ e ( G xx ) . Thenthe injectivity of ∆ x means that ρ ≺ ρ ⊗ ( λ G ◦ Φ | G xx ) ∼ ρ ⊗ λ G xx /H ∼ ρ ⊗ (Ind ε H ) . Since H is amenable by assumption, we have ε H ≺ λ H , hence ρ ⊗ (Ind ε H ) ≺ ρ ⊗ (Ind λ H ) ∼ ρ ⊗ λ G xx ∼ λ G xx , where in the last step we used Fell’s absorption principle. Therefore ρ ≺ λ G xx . Since k·k e dominatesthe reduced norm, we conclude that k·k e = k·k r . (cid:3) Remark . If G is an exact group, then Lemma 4.8 for A = C ∗ r ( G ) can be rephrased by saying thatfor any locally compact étale groupoid G and any x ∈ G (0) we have C ∗ e (( G × G ) xx ) = C ∗ e ( G xx ) ⊗ C ∗ r ( G ) .If we could prove this for arbitrary G , the assumption of exactness in Theorem 4.5 would beunnecessary. ♦ Applying Theorem 4.5 to groupoids from Example 4.1 and recalling that the subgroups of exactgroups are exact, we get the following.
Corollary 4.10.
Assume an exact discrete group Γ acts on a locally compact étale groupoid G . Fora fixed x ∈ G (0) , assume that the isotropy group G xx is amenable. Then k·k e = k·k r on the groupalgebra of (Γ ⋉ G ) xx . Note that if G is topologically amenable (see [ADR00, Section 2.2.b]), then its isotropy groupsare amenable.Theorem 4.5 also applies to the injectively graded groupoids from Examples 4.2–4.4, when thegrading group Γ is exact. Although the assumption of exactness is very mild, it is still somewhat un-satisfactory, since Theorem 4.5 does not fully cover even the trivial case of transformation groupoids(Proposition 2.10). We will therefore prove the following result, which assumes much more aboutthe groupoid structure, but does not need exactness of the grading group. Theorem 4.11.
Suppose Γ is a discrete group and G is a Γ -graded locally compact étale groupoid,with grading Φ :
G → Γ . For a fixed x ∈ G (0) , assume Φ is injective on G xx and there is a family { U g } g ∈G xx of bisections such that: (1) g ∈ U g and Φ( U g ) = Φ( g ) for all g ∈ G xx ; (2) U x = G (0) and, for all g ∈ G xx , U g − = U − g ; (3) if g , . . . , g n ∈ G xx and g . . . g n = x , then U g U g . . . U g n ⊂ G (0) .Then k·k e = k·k r on C G xx . Again, we need some preparation to prove the theorem. Take y ∈ G (0) . Lemma 4.12.
Define a binary relation ∼ x on G y by g ∼ x h ⇐⇒ ∃ g , . . . , g n ∈ G xx with h = U g . . . U g n g. Then ∼ x is an equivalence relation.Proof. Since U x = G (0) , for every g ∈ G y we have that g = U x g , and hence g ∼ x g .Assume now that g ∼ x h , and let g , . . . , g n ∈ G xx with h = U g . . . U g n g . Then r ( h ) ∈ r ( U g ) = s ( U g − ) and hence the composition U g − h = U g − U g . . . U g n g gives an element in G y . By ourassumptions, U g − U g is contained in G (0) , so U g − U g U g . . . U g n g = U g . . . U g n g. Continuing like this we get that U g − n . . . U g − U g − h = g , and hence h ∼ x g . o complete the proof that ∼ x is an equivalence relation assume that g ∼ x h and h ∼ x k .There exist g , . . . , g n ∈ G xx and h , . . . , h m ∈ G xx such that h = U g . . . U g n g and k = U h . . . U h m h .Combining these two identities we get that k = U h . . . U h m h = U h . . . U h m U g . . . U g n g and hence g ∼ x k . (cid:3) Let K y := G y / ∼ x . For each κ ∈ K y , set H κ := span { δ g : g ∈ κ } ⊂ ℓ ( G y ) . Then ℓ ( G y ) = L κ ∈ K y H κ . Lemma 4.13. If f ∈ C c ( G ) is zero outside S g ∈G xx U g , then each subspace H κ ( κ ∈ K y ) is invariantunder the action of ρ y ( f ) .Proof. Take g ∈ κ . Since ρ y ( f ) δ g = X h ∈G r ( g ) f ( h ) δ hg , it suffices to prove that hg ∈ κ whenever f ( h ) = 0 . If f ( h ) = 0 , then by assumption h ∈ U k forsome k ∈ G xx . This means that hg = U k g , and hence hg ∼ x g . This proves the lemma. (cid:3) The following lemma will allow us to embed H κ into ℓ (Γ) . Lemma 4.14.
For every κ ∈ K y , the map κ ∋ g Φ( g ) ∈ Γ is injective.Proof. Assume g, h ∈ κ are such that Φ( g ) = Φ( h ) . By definition, there exist g , . . . , g n ∈ G xx suchthat g = U g U g . . . U g n h . Applying Φ to both sides we get that Φ( g ) = Φ( g ) . . . Φ( g n )Φ( h ) . Since Φ( g ) = Φ( h ) and Φ is injective on G xx , this implies that g . . . g n = x . By our assumptions wethen have that U g U g . . . U g n ⊂ G (0) , and hence g = h . (cid:3) Proof of Theorem 4.11.
Fix a nonzero function h ∈ C c ( G xx ) . Let { g , . . . , g n } ⊂ G xx be the supportof h . Choose an open set V ⊂ G (0) such that x ∈ V ⊂ T ni =1 r ( U g i ) , the closure of W i := r − ( V ) ∩ U g i is compact and contained in U g i for each i , and such that Φ( W i ) = Φ( g i ) for each i , which inparticular implies that the sets W , . . . , W n are disjoint. Next, choose functions f i ∈ C c ( U g i ) suchthat f i ( g ) = h ( g i ) for all g ∈ W i . Finally, choose a function q ∈ C c ( G (0) ) with supp q ⊂ V such that ≤ q ≤ and q ( x ) = 1 . We now define f := n X i =1 q ∗ f i ∈ C c ( G ) . Then η x ( f ) = h , and in order to prove the theorem it suffices to show that k ρ y ( f ) k ≤ k h k r for all y ∈ G (0) .By Lemma 4.13 it is then enough to show that k ρ y ( f ) | H κ k ≤ k h k r for all y ∈ G (0) and κ ∈ K y = G y / ∼ x . This can be reformulated as follows. By identifying G xx with the subgroup Φ( G xx ) of Γ , wecan extend the function h by zero to a function ˜ h on Γ , so that ˜ h (Φ( g )) = h ( g ) for g ∈ G xx and ˜ h = 0 on Γ \ Φ( G xx ) . Denote by λ Γ the regular representation of Γ . Since λ Γ ◦ Φ | G xx decomposes intoa direct sum of copies of the regular representation of G xx , we have k h k r = k λ Γ (˜ h ) k . Therefore weneed to show that k ρ y ( f ) | H κ k ≤ k λ Γ (˜ h ) k . (4.5)Let us show first that f ( gk − ) = q ( r ( g ))˜ h (Φ( gk − )) for all g, k ∈ κ. (4.6) f gk − ∈ S ni =1 W i , then this follows by the definition of f and ˜ h . If gk − ∈ ( S ni =1 U g i ) \ ( S ni =1 W i ) ,then both sides of (4.6) are zero, again by the definition of f and since q ( r ( g )) = 0 . Assumenow that gk − / ∈ S ni =1 U g i . Then f ( gk − ) = 0 , and we are done if r ( g ) / ∈ V . So assume that r ( g ) ∈ V . If Φ( gk − ) / ∈ { Φ( g i ) } ni =1 then the right-hand side of (4.6) is zero, so let us assumethat Φ( gk − ) = Φ( g l ) for some l ∈ { , . . . , n } . Since g, k ∈ κ , there are elements s , . . . , s m ∈ G xx such that g = U s . . . U s m k . Hence g = r . . . r m k for uniquely defined elements r i ∈ U s i . Since r ( gk − ) ∈ V ⊂ r ( U g l ) , we have ∅ 6 = U − g l gk − = U − g l r . . . r m ⊂ U − g l U s . . . U s m . But since Φ( s . . . s m ) = Φ( gk − ) = Φ( g l ) , the last set is contained in G (0) by assumption, hence gk − ∈ U g l , which is a contradiction. Thus (4.6) is proved.Now, by Lemma 4.14 we have an isometry u : H κ → ℓ (Γ) , uδ g := δ Φ( g ) . In terms of this isometry, identity (4.6) can be written as ( ρ y ( f ) δ k , δ g ) = q ( r ( g ))( λ Γ (˜ h ) uδ k , uδ g ) for all g, k ∈ κ. In other words, ρ y ( f ) | H κ = u ∗ m ˜ q λ Γ (˜ h ) u, where m ˜ q : ℓ (Γ) → ℓ (Γ) is the operator of multiplication by the function ˜ q : Γ → [0 , defined by ˜ q (Φ( g )) := q ( r ( g )) for g ∈ κ and ˜ q := 0 on Γ \ Φ( κ ) . This clearly implies (4.5). (cid:3) Corollary 4.15.
Let G be the étale groupoid associated with a partial action of a discrete group Γ ona locally compact Hausdorff space X . Then, for every x ∈ G (0) = X , we have k·k e = k·k r on C G xx .Proof. Using the notation from Example 4.2, for every g = ( x, γ, x ) ∈ G xx consider the bisection U g := { ( γy, γ, y ) : y ∈ X γ − } . Then the assumptions of Theorem 4.11 are satisfied, so we get the result. (cid:3)
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