aa r X i v : . [ m a t h . OA ] A ug A Groupoid Picture of Elek Algebras
Clemens Borys * Abstract
We describe a construction by Gábor Elek, associating C ∗ -algebras with uniformly recur-rent subgroups, in the language of groupoid C ∗ -algebras. This allows us to simplify severalproofs in the original paper and add a new characterisation of nuclearity. We furthermorerelate our groupoids to the dynamics of the group acting on its uniformly recurrent subgroup. The constructions of (reduced) C ∗ -algebras associated with groups or, more generally, crossedproducts associated with actions of groups on topological spaces are well-known in the field ofoperator algebras and continue to provide a handy tool for constructing and understanding ex-amples of C ∗ -algebras, as well as establishing ties to other branches of mathematics such asdynamics or geometric group theory. Defined by Glasner and Weiss [3], Uniformly recurrentsubgroups , or URS for short, have recently drawn a lot of attention in the world of C ∗ -algebras,when Kennedy [4] managed to characterise C ∗ -simplicity of a discrete group Γ as the absenceof non-trivial amenable uniformly recurrent subgroups. Another relation between uniformly re-current subgroups and C ∗ -algebras is given by a construction of G. Elek [2], who constructs a C ∗ -algebra that is closely tied to the dynamics of a finitely generated discrete group acting onone of its uniformly recurrent subgroups Z , but takes more of the combinatorial nature of Z intoaccount than the crossed product does. This construction is thereby very well-suited for finding C ∗ -algebras with desired properties by rephrasing such properties on the combinatorial level ofURSs described by their associated Schreier graphs, and Elek obtains, for example, a C ∗ -algebrawith a uniformly amenable and a nonuniformly amenable trace.In this paper we recast Elek’s construction from the viewpoint of groupoid C ∗ -algebras, sim-plifying and extending the ties between properties of the URS and its associated algebra. Usingthe new angle, we improve on Elek’s characterisation of when his algebras are nuclear.In addition to this introduction there are five sections. After recalling the necessary terminol-ogy in Section 2, we construct an étale groupoid for a given URS Z whose reduced C ∗ -algebrais canonically isomorphic to Elek’s C ∗ -algebra C ∗ r ( Z ) in Section 3. In Section 4 we relate thisgroupoid to the dynamics of the action of Γ on Z by conjugation. Finally, in Section 5, we use thenew framework to give simpler proofs for some of Elek’s results on simplicity and nuclearity ofthe associated C ∗ -algebras and add the converse implication to his characterisation of nuclearityof C ∗ r ( Z ).The author is grateful to his supervisors M. Musat and M. Rørdam for their continued helpand support, as well as to G. Elek for some exciting conversations. * Supported by a grant from the Danish Council for Independent Research, Natural Sciences. Preliminaries
We recall the definition of Elek’s C ∗ -algebras associated with uniformly recurrent subgroups. Let Γ be a finitely generated discrete group. Let Sub( Γ ) be the space of its subgroups, equipped withthe topology of pointwise convergence of the characteristic functions associated to the subsets andleft Γ -action by conjugation γ. H = γ H γ − for H ∈ Sub( Γ ) and γ ∈ Γ . For a discrete group Γ thistopology is also known as the Fell topology on Sub( Γ ). Recall that a uniformly recurrent subgroup or URS Z of Γ is a closed, Γ -invariant subspace of Sub( Γ ) on which the action is minimal , that is,on which every orbit is dense. The URS is called generic , if the stabiliser of any subgroup H ∈ Z is as small as possible, namely H itself.Fixing a finite, symmetric system of generators Q of Γ , to each H ∈ Sub( Γ ) we may assign arooted, labeled graph S Q Γ ( H ) called its Schreier graph , which has vertex set Γ / H , root H , and forevery γ H ∈ Γ / H and q ∈ Q an edge from γ H to q γ H labeled by q . We denote the shortest-pathmetric on a graph S by d , or d S if there is ambiguity, and likewise the balls of radius R around avertex x ∈ S by B R ( x ) or B R ( S , x ). On the space S Q Γ of Schreier graphs associated with URS’s of Γ , we introduce a metric by d S Q Γ ( S , S ) : = − r (1)for S = S Q Γ ( H ) and S = S Q Γ ( H ) two graphs in S Q Γ and r the largest integer such that B r ( S , H )and B r ( S , H ) are root-label isomorphic. This space carries a left Γ -action, where γ. S Q Γ ( H ) = S Q Γ ( γ H γ − ), that is, γ acts by changing the root. Elek identifies the graphs S Q Γ ( H ) for whichthe orbit closure of H forms a URS as those where any root-label isomorphism class of balls isrepeated with at most bounded distance from any point in the graph (see [2, Proposition 2.1]), andthe graphs for which it forms a generic URS as those where vertices with large isomorphic ballsare su ffi ciently far apart (see [2, Proposition 2.3]).To associate a C ∗ -algebra with a given URS Z , Elek considers its local kernel algebra C Z formed by the local kernels K : Γ / H × Γ / H → C of finite width on S Q Γ ( H ) for some subgroup H ∈ Z . A kernel K on S Q Γ ( H ) is of width R , if K ( x , y ) = x , y ∈ S Q Γ ( H ) with d ( x , y ) > R . It is furthermore local with width R , if K ( x , γ. x ) = K ( y , γ. y ) for any x and y withroot-label isomorphic R -balls and γ ∈ Γ of length at most R . Equipped with pointwise addition,convolution KL ( x , y ) = P z K ( x , z ) L ( z , y ), and involution K ∗ ( x , y ) = K ( y , x ) for local kernels K and L and x , y , and z in Γ / H , the local kernel algebra forms a ∗ -algebra. Up to isomorphism, thisalgebra does not depend on the choice of root H ∈ Z . A “regular” representation of C Z on ℓ ( Γ / H )is given by ( K f )( x ) = P K ( x , y ) f ( y ) for f ∈ ℓ ( Γ / H ). The reduced C ∗ -algebra C ∗ r ( Z ) of Z is thecompletion of C Z in the norm induced by this representation.Several C ∗ -algebraic properties of C ∗ r ( Z ) can be read o ff of the URS Z and its Schreier graph.Genericity of Z implies simplicity of C ∗ r ( Z ), as discussed in Section 5. Furthermore, a local versionof Yu’s property A for the Schreier graph S Q Γ ( H ), for any subgroup H in the URS Z , is equivalentto nuclearity of C ∗ r ( Z ). Recall that a Schreier graph S Q Γ ( H ) for H ∈ Z ⊆ Γ has Elek’s local propertyA , if there is a sequence of local functions ρ n : Γ / H → l ( Γ / H ) , x ρ nx , such that k ρ nx k =
1, while d ( x , y ) ≤ n implies k ρ nx − ρ ny k ≤ / n . As with kernels, locality of ρ n means that there is R n > ρ nx is supported in the R n -ball B R n ( x ) centred at x and whenever θ is a root-label isomorphism B R n ( x ) → B R n ( y ), we have ρ ny ◦ θ = ρ nx . For a given uniformly recurrent subgroup, we proceed to construct a groupoid whose regularrepresentation and reduced C ∗ -algebra model Elek’s construction on the local kernel algebra.2et Z be a uniformly recurrent subgroup of a discrete group Γ , fix H ∈ Z , and let S = S Q Γ ( H )be its Schreier graph. For any n ∈ N we define an equivalence relation on the vertices V ( S ) of S by p ∼ n q ⇔ B n ( S , p ) (cid:27) r , l B n ( S , q ) , that is, if the n -balls around p and q are isomorphic under an isomorphism preserving the rootsand labels. Such a root-label isomorphism is necessarily unique. Equivalently, if γ p and γ q areelements of Γ describing paths from the root of S to p and q , respectively, then p ∼ n q if and only if γ p . S and γ q . S are 2 − n -close in the metric d S Q Γ introduced on S Q Γ in Equation (1). Let E n = V ( S ) / ∼ n denote the finite set of equivalence classes of ∼ n equipped with the discrete topology and theobvious connecting maps e n + : E n + → E n . Then, as in [2, Lemma 6.1.4], it is easy to check thatlim ←−− E n is homeomorphic to Z as a subspace of Sub( Γ ), which in turn is a Cantor space or a finitediscrete set. It is noteworthy that under this identification the orbit of H in Z is exactly describedby those elements in lim ←−− E n , which can be represented by the equivalence classes [ p ] n of a fixed vertex p ∈ S . The other elements describe subgroups in the orbit closure, but not the orbit, of H . All elements of G x are therefore represented by We employ this description of the space Z to construct an ample Hausdor ff étale groupoid G with unit space G homeomorphic to Z , whosereduced groupoid C ∗ -algebra is C ∗ r ( Z ).Let G be given by lim ←−− E n and let x = ([ x ] , [ x ] , . . . ) ∈ G . In slight abuse of notation we willcontinue to write x n for a representing vertex of the class [ x n ] n in E n that forms the n -coordinateof x . The arrows of G x will be given by equivalence classes of pairs ( x , γ ) for γ ∈ Γ , where weidentify two such pairs ( x , γ ) and ( x , γ ′ ) for l ( γ ′ ) ≥ l ( γ ), if γ x l ( γ ′ ) = γ ′ x l ( γ ′ ) . Note that d ( x l ( γ ) , γ x l ( γ ) )might be strictly less that l ( γ ), in which case there is another γ ′ ∈ Γ of length d ( x l ( γ ) , γ x l ( γ ) ), suchthat ( x , γ ) and ( x , γ ′ ) denote the same arrow. We call such γ ′ of minimal length. The range map isconsequently defined as r ( x , γ ) = x . We fix the source of ( x , γ ) as γ x : = ([ γ x l ( γ ) ] , [ γ x l ( γ ) + ] , . . . ).Intuitively, an arrow with range x is thought of as a path from x to γ x seen as vertices in theSchreier graph, but γ x n is only well-defined for n ≥ l ( γ ), in which case γ x n determines a uniqueisomorphism class of ( n − d )-balls and thereby a class in E n − d where d = d ( x n , γ x n ) . All elementsof G x are therefore represented by a pair ( x , γ ) with γ of minimal length and we can describe ( x , γ )as consistent choices of vertices γ x n in the balls described by the classes [ x n ] n .More formally, we define G as a subset of the projective limit lim ←−− F n of the finite, discrete sets F n = G [ x n ] ∈ E n B n ( S , x n ) ⊔ {∞ n } (2) (cid:27) G [ x n ] ∈ E n { γ ∈ Γ | l ( γ ) ≤ n } / ≈ x n ⊔ {∞ n } , (3)where B n ( S , x n ) denotes the n -ball in S that is determined uniquely by [ x n ], even if there is a choicein the representing vertex x n .To avoid this choice of representing elements, a pair ([ x n ] , y ) with y ∈ B n ( S , x n ) as in Equation(2) can be more readily expressed as a pair ([ x n ] , γ ) as in Equation (3), where γ is any chosenpath from x n to y inside B n ( S , x n ), up to γ ≈ x n γ ′ if both paths lead to the same vertex in S , thatis, if γ. x n = γ ′ . x n . Implicit in this description is our later identification of G with a quotient ofthe transformation groupoid Z ⋊ Γ . In this picture, ∞ n fills in for choices of vertices that are notcontained in B n ( S , x n ) or respectively for γ whose lengths as words in the generators exceeds n .The connecting maps f n + : F n + → F n are then given by([ x n + ] , γ ) ( e n + ([ x n + ]) , γ ) if d ( x n + , γ x n + ) ≤ n ∞ n else ∞ n +
7→ ∞ n . G : = lim ←−− F n \ {∞} with ∞ = ( ∞ , ∞ , . . . ), equipped with the subspace topology of theprojective limit. Equivalently, with x and y in G and γ, γ ′ ∈ Γ , the topology on G is given by themetric d G (( x , γ ) , ( y , γ ′ )) = − N for N maximal such that ([ x N ] N , γ. x N ) and ([ y N ] N , γ. y N ) coincidein F N . To simplify notation, we write ( x , γ ) in place of the equivalence class it represents andidentify ([ x n ] , γ ) with ∞ n , whenever l ( γ ) > n .If ( x , γ ) and ( y , γ ′ ) are composable, that is, if x = γ ′ . y , then we define the composition( y , γ ′ )( x , γ ) to be ( y , γγ ′ ). Note, that s (( y , γγ ′ )) = s ( x , γ ′ ), since ([ x ] , [ x ] , . . . ) coincides with([ x N ] , [ x N + ] , . . . ) in G for any N ∈ N . We see that any x ∈ G is a unit in G when written as( x , e ) for e ∈ Γ the neutral element, and consequently the equivalence class represented by ( x , γ )has as inverse the class represented by ( γ. x , γ − ). Proposition 3.1.
Let Z be a uniformly recurrent subgroup of a discrete group Γ . The aboveturns G into an ample minimal Hausdor ff étale groupoid with unit space homeomorphic to Z . Proof.
It is easy to see that the operations above indeed turn G into a groupoid. Equipping G withthe locally compact Hausdor ff subspace topology of lim ←−− F n , we have to check that the definedoperations are continuous. Consider the basis open sets U e N ,γ = { ( x , γ ) ∈ G | [ x N ] N = e N } , that fix γ ∈ Γ and e N ∈ E N for some N . The range map is obviously continuous, as any basic openset of lim ←−− E n can be turned into a union of basic open sets of lim ←−− F n by letting γ vary. Inversion iscontinuous, since the action of Γ on Z is, which in turn makes the source map continuous. To seethat the composition is continuous, we fix e N ∈ E N and γ ∈ Γ and find the preimage of U e N ,γ underthe composition map. For ( x , η )( η − . x , η ′ ) we first need that x ∈ U e N ,η and secondly that η ′ = η − γ .Hence the desired preimage is described by the intersection of [ η ∈ Γ U e N ,η × U e ,η − γ with the subspace G (2) of composable pairs in G × G and therefore open in G (2) . We used e todenote the unique equivalence class in E .To see that the range map is a local homeomorphism, note that it restricts to a homeomorphismonto its image on every basic open set U e N ,γ , and G is therefore étale. Finally, G is ample as it isan étale groupoid with totally disconnected unit space.As the orbits of Z and G (0) coincide, G is minimal. (cid:3) It is noteworthy that the construction above does not depend on the choice of H ∈ Z : Proposition 3.2.
For di ff erent choices of H , H ′ ∈ Z the groupoids G , G ′ constructed above areisomorphic. Proof.
This amounts to showing that the root-label equivalence classes E n and E ′ n of n -balls in S = S Q Γ ( H ) and S ′ = S Q Γ ( H ′ ) are identical sets of rooted, labeled balls. That is, for every p ∈ V ( S )there is p ′ ∈ V ( S ′ ) such that B n ( S , p ) (cid:27) r , l B n ( S ′ , p ′ ). But as Z is uniformly recurrent, for p = γ H the subgroup γ H γ − is in the orbit closure of H ′ and therefore S Q Γ ( γ H γ − ) is in the orbit closureof S ′ . Hence there is γ ′ ∈ Γ such that B n ( S , p ) (cid:27) r , l B n ( S Q Γ ( γ H γ − ) , γ H γ − ) (cid:27) r , l B n ( S ′ , γ ′ H ′ ) and p ′ = γ ′ H ′ gives E n ⊆ E ′ n . Equality follows by symmetry. (cid:3) Next, we identify the local kernel algebra C Z of Z with a dense subset of C c ( G ). Given akernel K ∈ C Z as a function on V ( S ) × V ( S ), recall that there is a minimal N ∈ N called thewidth of K such that K vanishes on any pairs ( x , y ) where the distance of x and y is more than N B N ( S , x ) (cid:27) r , l B N ( S , y ) implies that K ( x , γ x ) = K ( y , γ y ) if l ( γ ) ≤ N , so that K onlydepends on the root-label isomorphism class of N -balls. To K we assign a function f K ∈ C c ( G )by f K (( x , γ )) = K ( x M , γ x M ) with M = max { N , l ( γ ) } . If γ is chosen of minimal length, we maypick M = N . Equivalently, f K evaluates ( x , γ ) at it’s component ([ x N ] N , γ ) in F N and assigns K ( x N , γ x N ), where K ( ∞ N ) =
0. This is well-defined, as the vertex x N is given up to root-labelisomorphism of N -balls. The function f K is continuous, because it is uniformly locally constant:It is constant on any 2 − N -ball in G . To see that f K is compactly supported, note that the embedding G ֒ → lim ←−− F n is the one-point compactification of G . Therefore, a set U ⊆ G is relatively compact,exactly if there is M ∈ N such that no element of U has F M -component ∞ M . Equivalently, U asa subset of lim ←−− F n does not intersect the 2 − M -ball centred at ∞ . As f K is supported outside of the2 − N -ball centred at ∞ for N the width of K , it is compactly supported.Conversely, any locally constant function f ∈ C c ( G ) defines a kernel K f ∈ C Z . As f is locallyconstant, for each g ∈ G we can pick a Ball centred at g , on which f is constant. Then finitelymany of such balls cover the support of f and we may pick N ∈ N such that these have radiusat least 2 − N . Since two 2 − N -balls in G or lim ←−− F n are either disjoint or equal, f is constant on any2 − N -ball and supported outside of the 2 − N -ball of ∞ . Given such f , we may now define a localkernel K f with width at most N as follows: For a fixed vertex p ∈ V ( S ) consider the unit [[ p ]] D ([ p ] , [ p ] , . . . ) ∈ G and define K f ( p , γ p ) : = f ([[ p ]] , γ ). If γ p = γ ′ p , then ([[ p ]] , γ ) = ([[ p ]] , γ ′ ),so K f is a well-defined kernel. As the F N -component of ([[ p ]] , γ ) is ∞ N if d ( p , γ p ) > N , wehave K f ( p , γ p ) = N -balls B N ( S , p ) and B N ( S , q ) are root-label-isomorphic for p , q ∈ V ( S ) and l ( γ ) ≤ N , then ([[ p ]] , γ ) and ([[ q ]] , γ ) are 2 − N -close, since theirfirst N components coincide. As f is constant on 2 − N -balls, we have K f ( p , γ p ) = K f ( q , γ q ), so K f is local of width N .It is easy to check that f K f = f and K f K = K , so the local kernel algebra C Z of Z is in bijectionwith the subset of locally constant functions in C c ( G ), which is dense in C c ( G ), as G is totallydisconnected. As the ∗ -algebra structure of C Z is preserved under this inclusion, we identify C Z with a ∗ -subalgebra of C c ( G ): For p ∈ V ( S ) and γ ∈ Γ we have s ([[ p ]] , γ ) = [[ γ. p ]] and the orbitof [[ p ]] in G is { [[ q ]] | q ∈ V ( S ) } . We calculate f L ∗ K ([[ p ]] , γ ) = L ∗ K ( p , γ p ) = X q ∈ V ( S ) L ( p , q ) K ( q , γ p ) = X ([[ p ]] ,γ ′ ) ∈G [[ p ]] f L ([[ p ]] , γ ′ ) f K ([[ γ ′ p ]] , γ ( γ ′ ) − ) = f L ∗ f K ([[ p ]] , γ )and ( f K ) ∗ ([[ p ]] , γ ) = f K ([[ γ p ]] , γ − ) = K ( γ p , p ) = K ∗ ( p , γ p ) = f K ∗ ([[ p ]] , γ ) . We conclude that f L ∗ K = f L ∗ f K and ( f K ) ∗ = f K ∗ on the subset { ([[ p ]] , γ ) | p ∈ V ( S ) , γ ∈ Γ } ⊆ G of arrows in G , whose range is given by the equivalence classes of a single, constant vertex p in S .This subset is dense, as the orbit closure of S is dense in S Q Γ ( Z ) and as the functions in questionare continuous, they coincide on all of G .Next, we assert that C Z is dense in C c ( G ) even in reduced norm and that the reduced norm on C Z ⊆ C c ( G ) coincides with the norm obtained by representing C Z on B ( l ( Γ / H )). Theorem 3.3:
Let Z be a uniformly recurrent subgroup of a finitely generated group Γ and G as above. The C ∗ -algebras C ∗ r ( Z ) and C ∗ r ( G ) are isomorphic with the isomorphism extending thecanonical construction above uniquely. G the reduced norm on C c ( G ) is given as k f k = sup x k π x ( f ) k withthe representations π x : C c ( G ) → B ( l ( G x )) by π x ( f ) δ g = X g ′ ∈G r ( g ) f ( g ′ ) δ g ′ g for x ∈ G and g ∈ G x . The (faithful) representation π of C Z on B ( l ( Γ / H )) is given by π ( K ) δ p = X q ∈ V ( S ) K ( q , p ) δ q . Proof.
Let us first consider x = [[ H ]], the unit represented by the root in S . We obtain a map V ( S ) → G [[ H ]] by γ H ([[ γ H ]] , γ − ), mapping a vertex γ H in S to the arrow in G that isdescribed by any path from H to γ H . This is obviously surjective and is well-defined, as thearrows ([[ γ H ]] , γ − ) and ([[ γ ′ H ]] , ( γ ′ ) − ) are identified if γ H = γ ′ H . It is furthermore injective: If([[ γ H ]] , γ − ) = ([[ γ ′ H ]] , ( γ ′ ) − ), then the N -balls centred at γ H and γ ′ H are root-label isomorphicwith the isomorphism mapping H to H if N > l ( γ ) + l ( γ ′ ). But a root-label-isomorphism of N -ballsin Schreier graphs is the identity if it has a fixed point, hence γ H = γ ′ H . We have thus establisheda bijection between G [[ H ]] and Γ / H = V ( S ). This yields a unitary T : l ( V ( S )) → l ( G [[ H ]] ) whichintertwines the representations π and π [[ H ]] on C Z : Let h ∈ l ( V ( S )). Then π ( K ) h ( γ H ) = X γ ′ H ∈ V ( S ) K ( γ H , γ ′ H ) h ( γ ′ H )and so T ( π ( K ) h )([[ γ H ]] , γ − ) = X γ ′ H ∈ V ( S ) K ( γ H , γ ′ H ) h ( γ ′ H ) = X γ ′ H ∈ V ( S ) f K ([[ γ H ]] , γ ′ γ − ) h ( γ ′ H ) = X γ ′ H ∈ V ( S ) f K ([[ γ H ]] , γ ′ γ − )( T h )([[ γ ′ H ]] , ( γ ′ ) − ) = X g ∈G [[ H ]] f k (([[ γ H ]] , γ − ) g − )( T h )( g ) = (cid:0) π [[ H ]] ( f K ) T h (cid:1) ([[ γ H ]] , γ − ) , and therefore the representations π and π [[ H ]] define identical reduced norms on C Z ⊆ C c ( G ).Morally, this already implies that all source-fibre representations π x for x ∈ G are unitarilyequivalent to π , since the groupoid G does not depend on the choice of H ∈ Z in its constructionand for every x ∈ G there is a unique subgroup in Z which is mapped to x under the homeomor-phism Z (cid:27) G . For completeness we nevertheless show that all reduced representations of G areequivalent. As in any groupoid, the representations π x are unitarily equivalent for any two units x that share an orbit. Therefore we only need to consider π x for x ∈ G that corresponds to asubgroup H ′ in Z \ Γ . H . For fixed K ∈ C Z of width R and ǫ >
0, we may pick h ∈ l ( G [[ H ]] ) ofnorm one such that k π [[ H ]] ( f K ) h k > k π [[ H ]] ( f K ) k − ǫ and h is supported on arrows ([[ γ H ]] , γ − )with l ( γ ) ≤ N for some N ∈ N . As the orbit of H ′ in Z is dense, there is a unit y in the orbit of x in G , such that y and [[ H ]] are 2 − ( N + R ) -close, that is, their F N + R -components coincide. Hence, forevery arrow ([[ γ H ]] , γ − ) ∈ G [[ H ]] with l ( γ ) ≤ N + R there is a unique arrow ( γ. y , γ − ) ∈ G y that isdescribed by the same γ − , since two paths of length less than N + R starting at H end in the samevertex, exactly if the analogous paths in the isomorphic N + R -ball of y do. This yields a bijectionbetween the subspaces of G [[ H ]] and G y described by elements of Γ with length at most N + R . Ex-tending by zero, we transport h ∈ l ( G [[ H ]] ) to a function h ′ ∈ l ( G y ) with 1 = k h k = k h ′ k alongthis bijection. Noting that π [[ H ]] ( f K ) h ∈ l ( G [[ H ]] ) is supported on ([[ γ H ]] , γ − ) with l ( γ ) ≤ N + R ,we may likewise transport this to a function in l ( G y ) of the same norm. It is now easy to seethat this function will just be π y ( f K ) h ′ , whence k π y ( f K ) k ≥ k π y ( f K ) h ′ k > k π [[ H ]] ( f K ) k − ǫ and by6nitary equivalence k π x ( f K ) k > k π [[ H ]] ( f K ) k − ǫ . By symmetry we obtain the converse direction,implying that all norms on C c ( G ) induced by reduced representations are identical and coincideon C Z ⊆ C c ( G ) with the reduced norm of C Z .We finally show that C Z is dense in C c ( G ) in this reduced norm, whence C ∗ r ( Z ) (cid:27) C ∗ r ( G ). Let f ∈ C c ( G ). As f is compactly supported, it vanishes on the 2 − R -ball around ∞ ∈ lim ←−− F n for some R , so that f ([[ η H ]] , γ − ) = γ is chosen of minimal length and yet l ( γ ) > R . We therefore findthat k π [[ H ]] ( f ) h k = X γ H ∈ Γ / H (cid:12)(cid:12)(cid:12)(cid:0) π [[ H ]] ( f ) h (cid:1) ([[ γ H ]] , γ − ) (cid:12)(cid:12)(cid:12) = X γ H ∈ Γ / H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X γ ′ H ∈ B R ( S ,γ H ) f ([[ γ H ]] , γ ′ γ − ) h ([[ γ ′ H ]] , ( γ ′ ) − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k f k ∞ X γ H ∈ Γ / H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X γ ′ H ∈ B R ( S ,γ H ) h ([[ γ ′ H ]] , ( γ ′ ) − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k f k ∞ (cid:13)(cid:13)(cid:13) X η ∈ Γ , l ( η ) ≤ R h (cid:13)(cid:13)(cid:13) ≤ ( | Q | + R k f k ∞ k h k , as there are less than ( | Q | + R elements of length at most R in Γ . Hence k f k r ≤ ( | Q | + R k f k ∞ for all f ∈ C c ( G ) supported outside of the 2 − R -ball around ∞ with k f k r = k π [[ H ]] ( f ) k the unique reducednorm. For any ǫ >
0, by partitioning G into open 2 − N -balls for large N , we may approximate f in C c ( G ) up to ǫ by a function f ǫ that is constant on every 2 − N -ball. As two balls are either disjointor one is contained in the other, we may choose f ǫ ( g ) supported outside of the 2 − R -ball of ∞ forlarge N , such that k f − f ǫ k r ≤ ( | Q | + R k f − f ǫ k ∞ = ǫ ( | Q | + R . (cid:3) To recap, for any uniformly recurrent subgroup Z we have constructed an ample minimalétale Hausdor ff groupoid with unit space homeomorphic to Z , such that the reduced C ∗ -algebras C ∗ r ( Z ) of Z and C ∗ r ( G ) of G coincide. This enables us to examine C ∗ r ( Z ) using tools for groupoid C ∗ -algebras. In this section we shed some light on the relationship between our newly defined groupoid G associated with a uniformly recurrent subgroup Z of Γ and the transformation groupoid Z ⋊ Γ associated with the action of Γ on Z by conjugation.As a space, the transformation groupoid Z ⋊ Γ is simply the cartesian product Z × Γ equippedwith the product topology. The unit space is given by the subspace Z × { e } and identified with Z ,the range and source of an arrow ( H , γ ) are respectively given by H and γ − . H , while the productof two composable arrows is ( H , γ )( γ − . H , η ) = ( H , γη ). This turns Z ⋊ Γ into a Hausdor ff étalegroupoid, which for a URS Z is furthermore ample and minimal.To distinguish our groupoid G from Z ⋊ Γ , we describe the range fibres G H further. Recallthat the homeomorphism between Z and lim ←−− E n describes every group H ∈ Z as a sequence ofisomorphism classes of balls in S Q Γ ( H ′ ) for any H ′ ∈ Γ . In particular, the isomorphism class in E n associated with H is given by the n -ball around the root in S Q Γ ( H ). Two arrows ( H , γ ) and ( H , η )in G will then coincide if and only if the paths in S Q Γ ( H ) that start at the root and are described by γ and η end in the same vertex so that γ H = η H or equivalently η − γ ∈ H . By this identificationwe obtain a surjective map q : Z ⋊ Γ → G that maps ( H , γ ) ([[ H ]] , γ ). To simplify this notationand remove the ambiguity in the choice of γ , we continue to denote ([[ H ]] , γ ) as ( H , γ H ). We seethat the range fibres G H are given by the quotients Γ / H while those of ( Z ⋊ Γ ) H are simply given7y Γ and G arises from the transformation groupoid describing the action of Γ on Z without takinglocality into account by dividing out the appropriate subgroup in every fibre.Indeed, this identification is not merely as sets, but as topological groupoids: Proposition 4.1.
The map q : Z ⋊ Γ → G given by ( H , γ ) ( H , γ H ) for a subgroup H ∈ Z and a group element γ ∈ Γ is a continuous, open, and surjective groupoid homomorphism. Inparticular, G is a quotient of the transformation groupoid Z ⋊ Γ . Proof.
As argued above, q is surjective and it is obviously a homomorphism.We fix a basis of the topology of G as U H , N ,γ = (cid:26) ( K , γ K ) | d S Q Γ ( S Q Γ ( H ) , S Q Γ ( K )) ≤ − N (cid:27) indexed by a subgroup H ∈ Z , an element γ ∈ Γ and N ∈ N with l ( γ ) ≤ N and consisting of allarrows ( K , γ K ) such that the N -balls around the root in S Q Γ ( H ) and S Q Γ ( K ) coincide. Equivalently,a group element η ∈ Γ with l ( η ) ≤ N is contained in K if and only if it is contained in H . Likewise,we fix a basis V H , N ,γ = (cid:26) ( K , γ ) | d S Q Γ ( S Q Γ ( H ) , S Q Γ ( K )) ≤ − N (cid:27) of the topology of Z ⋊ Γ .It is now easy to see that q is open. Let ( K , γ K ) ∈ q ( V H , N ,γ ). Then U K , M ,γ for M = max { N , l ( γ ) } is a neighbourhood of ( K , γ K ) contained in q ( V H , N ,γ ).Conversely, q − ( U H , N ,γ ) = (cid:26) ( K , γ k ) | d S Q Γ ( S Q Γ ( H ) , S Q Γ ( K )) ≤ − N , k ∈ K (cid:27) , and we claim that this is open in Z ⋊ Γ . Indeed, for every ( K , γ k ) ∈ q − ( U H , N ,γ ) the neighbourhood V K , M ,γ k is contained in q − ( U H , N ,γ ), where M = max { N , l ( k ) } , so that any subgroup in the 2 − M -ballaround K is guaranteed to contain the element k . (cid:3) We employ our description of the Elek algebras as groupoid algebras to give simplified proofs ofElek’s characterisations, explaining why they arise in the language of groupoids.
Proposition 5.1. If Z is generic, then G is principal. Proof.
Suppose G is not principal. Then there is a unit x = ([ x ] , [ x ] , . . . ) and an arrow in theisotropy G xx that is not a unit. Therefore there is a group element γ ∈ Γ such that x = γ. x but( x , e ) , ( x , γ ). Especially γ. x N , x N for large N, while B N − l ( γ ) ( S , x N ) (cid:27) r , l B N − l ( γ ) ( S , γ. x N ) and d ( x N , γ. x N ) ≤ l ( γ ). By [2, Proposition 2.3], Z is not generic. (cid:3) Corollary 5.2:
If Z is generic, then C ∗ r ( Z ) is simple. This reproduces [2, Theorem 7].
Proof.
By Proposition 5.1, G is a minimal principal étale groupoid, and every such groupoid hasa simple reduced C ∗ -algebra by [6, Proposition 4.3.7]. (cid:3) Regarding nuclearity, we are able to add the converse direction to Elek’s characterisation [2,Theorem 8]. We first describe when our groupoids are amenable.8 heorem 5.3:
Let Z be a uniformly recurrent subgroup of the finitely generated discrete group Γ and let G be the groupoid associated with Z. The Schreier graph S Q Γ ( H ) of any group H ∈ Zhas local property A, if and only if G is (topologically) amenable.Proof. We show that local property A of the Schreier graph S Q Γ ( H ) implies topological amenabilityof G . Let G be constructed from H ∈ Z and let ρ n : Γ / H → l ( Γ / H ) implement local property A of S Q Γ ( H ). Let S n describe the locality of ρ n as in Section 1. Let x S n denote any element of theequivalence class of E S n forming the S n -component of x and define f n ∈ C c ( G ) by f n ( x , γ ) = ρ nx Sn ( γ. x S n ). This is independent of the choice of x S n , as ρ n are locally defined and of width atmost S n , continuous as it is constant on 2 − S n -balls and compactly supported, as it vanishes on the2 − S n -ball of ∞ . For any unit u ∈ G we have X ( u ,γ ) ∈G u | f n ( u , γ ) | = X y ∈ B Sn ( u Sn ) | ρ nu Sn ( y ) | = k ρ nu Sn k = , using that ρ nx is supported in the S n -ball centred at u x . Furthermore, we calculate f n ∗ f ∗ n ( x , γ ) = X ( γ. x ,γ ′ ) ∈G γ. x f n (( x , γ )( γ. x , γ ′ )) f n ( γ. x , γ ′ ) , where we may restrict to l ( γ ′ ) ≤ S n , as f n ( γ. x , γ ′ ) vanishes otherwise, = X ( γ. x ,γ ′ ) ∈G γ. x , l ( γ ′ ) ≤ S n ρ nx Sn + l ( γ ) ( γ ′ γ x S n + l ( γ ) ) ρ n γ x Sn + l ( γ ) ( γ ′ γ x S n + l ( γ ) ) = X z ∈ B Sn ( γ x Sn + l ( γ ) ) ρ nx Sn + l ( γ ) ( z ) ρ n γ x Sn + l ( γ ) ( z ) = h ρ nx Sn + l ( γ ) , ρ n γ x Sn + l ( γ ) i , using again the assumptions on the support of ρ n γ x Sn + l ( γ ) . However, for fixed u , v ∈ Γ / H we have (cid:12)(cid:12)(cid:12) − h ρ nu , ρ nv i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) h ρ nu − ρ nv , ρ nv i (cid:12)(cid:12)(cid:12) ≤ k ρ nu − ρ nv k · k ρ nv k † ) ≤ / n · n →∞ −−−−→ , where the estimate ( † ) holds for large n such that d ( u , v ) ≤ n . But as l ( γ ) is bounded on compactsets, the same estimate may be used uniformly on any compact set for su ffi ciently large n , so that f n ∗ f ∗ n converges to one uniformly on compact subsets. Those functions witness the (topological)amenability of G as in the original definition [5, page 92].Conversely, suppose that G is topologically amenable. By an equivalent characterisation ofamenability [1, Prop 2.2.13], we may assume that there is a sequence f n ∈ C c ( G ), such that X ( u ,γ ) ∈G u | f n ( u , γ ) | n →∞ −−−−→ G as a function of u and X h ∈G r ( g ) | f n ( g − h ) − f n ( h ) | n →∞ −−−−→ G as a function of g .To obtain local maps ρ nx as in the definition of local property A, we first, for any ǫ >
0, approximate f n by (uniformly) locally constant functions f n ,ǫ ∈ C c ( G ) such that k f n − f n ,ǫ k ∞ < ǫ , choosing f n ,ǫ to be zero on the largest possible ball centred at ∞ .We next construct a sequence ǫ n ց
0, such that f n ,ǫ n satisfies the convergence properties above:As for fixed n there is T n ∈ N such that both f n and f n ,ǫ for every ǫ > − T n -ballof ∞ we may restrict summation to arrows described by group elements γ with length l ( γ ) at most9 n , such that the respective sums become finite with less than ( | Q | + T n terms. Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( u ,γ ) ∈G u | f n ( u , γ ) | − X ( u ,γ ) ∈G u | f n ,ǫ ( u , γ ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X ( u ,γ ) ∈G u , l ( γ ) ≤ T n ( | f n ( u , γ ) | + ǫ ) − | f n ( u , γ ) | ≤ ( | Q | + T n (2 k f n k ∞ + ǫ ) ǫ ǫ → −−−→ , with the convergence uniformly in u ∈ G . Likewise, we calculate X h ∈G r ( g ) | f n ,ǫ ( g − h ) − f n ,ǫ ( h ) | ≤ X h ∈G r ( g ) , l ( γ ) ≤ T n ( | f n ( g − h ) − f n ( h ) | + ǫ ) ≤ (4 ǫ + ǫ k f n k ∞ )( | Q | + T n + X h ∈G r ( g ) | f n ( g − h ) − f n ( h ) | ǫ → −−−→ X h ∈G r ( g ) | f n ( g − h ) − f n ( h ) | , with the convergence uniformly in g ∈ G . Picking ǫ n → / n ≥ ( | Q | + T n (2 k f n k ∞ + ǫ n ) ǫ n and 1 / n ≥ (4 ǫ n + ǫ n k f n k ∞ )( | Q | + T n does the trick. To such f n ,ǫ n we may assign ρ n : Γ / H → l ( Γ / H ) by ρ nx ( γ x ) = f n ,ǫ n ([[ x ]] , γ ).Picking S n such that f n ,ǫ n is constant on any 2 − S n -ball, we see that ρ nx is supported on B S n ( x ),since f n ,ǫ n vanishes on the 2 − S n -neighbourhood of ∞ . Similarly, ρ nx ( γ x ) = ρ ny ( γ y ) for l ( γ ) ≤ S n ifthe S n -balls around x and y are isomorphic, as ([[ x ]] , γ ) and ([[ y ]] , γ ) are 2 − S n -close in that caseand we conclude that ρ n is local of width S n .Furthermore, we compute that k ρ n γ H k = X γ ′ H ∈G / H | ρ n γ H ( γ ′ H ) | = X ([[ γ H ]] ,γ ′ ) ∈G u | f n ,ǫ n ([[ γ H ]] , γ ′ ) | n →∞ −−−−→ G . Likewise, k ρ n γ H − ρ n γ ′ H k = X η H ∈ Γ / H | ρ n γ H ( η H ) − ρ n γ ′ H ( η H ) | = X η H ∈ Γ / H | f n ,ǫ n ( γ [[ H ]] , ηγ − ) − f n ,ǫ n ( γ ′ [[ H ]] , η ( γ ′ ) − ) | = X η H ∈ Γ / H | f n ,ǫ n (cid:16) ( γ [[ H ]] , γ ′ γ − )( γ ′ [[ H ]] , η ( γ ′ ) − ) (cid:17) − f n ,ǫ n ( γ ′ [[ H ]] , η ( γ ′ ) − ) | = X η H ∈ Γ / H | f n ,ǫ n (cid:16) ( γ ′ [[ H ]] , γ ( γ ′ ) − ) − ( γ ′ [[ H ]] , η ( γ ′ ) − ) (cid:17) − f n ,ǫ n ( γ ′ [[ H ]] , η ( γ ′ ) − ) | = X h ∈G γ ′ [[ H ]] | f n ,ǫ n (cid:16) ( γ [[ H ]] , γ ′ γ − ) h (cid:17) − f n ,ǫ n ( h ) | n →∞ −−−−→ γ ′ [[ H ]] , γ ( γ ′ ) − ) in compact subsets of G . In particular, the convergenceis uniform, when the pair ( γ, γ ′ ) is taken from a subset with bounded di ff erence, that is, if thereis N > l ( γ ( γ ′ ) − ) < N . More importantly, this means that for any N ∈ N we may10hoose n ∈ N , such that k ρ n γ H − ρ n γ ′ H k ≤ / N , whenever d S Q Γ ( H ) ( γ H , γ ′ H ) ≤ N and n ≥ n . Theanalogous statement holds after replacing ρ nx with the normed function ˆ ρ nx = ρ nx / k ρ nx k , since k ˆ ρ n γ H − ˆ ρ n γ ′ H k ≤ k ρ n γ H k (cid:18) k ρ n γ H − ρ n γ ′ H k + (cid:12)(cid:12)(cid:12)(cid:12) k ρ n γ H k − k ρ n γ ′ H k (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , while k ρ n γ H k and (cid:12)(cid:12)(cid:12)(cid:12) k ρ n γ H k − k ρ n γ ′ H k (cid:12)(cid:12)(cid:12)(cid:12) converge uniformly to 1 and 0, respectively. Then, afterrelabelling, ˆ ρ n witnesses local property A of S Q Γ ( H ). (cid:3) For étale groupoids there is a clear relation between amenability and nuclearity of their reduced C ∗ -algebras, which directly translates to our case. Compare for example [7, Section 2] for a briefoverview of the di ff erent notions of groupoid amenability. Corollary 5.4:
Let Z be a uniformly recurrent subgroup and H ∈ Z. The graph S Q Γ ( H ) haslocal property A if and only if the C ∗ -algebra C ∗ r ( Z ) is nuclear. This reproduces and extends [2, Theorem 8].
Proof.
As the groupoid G associated with Z is étale, C ∗ r ( G ) is nuclear if and only if G is (topolog-ically) amenable by [1, Corollary 6.2.14]. By Theorem 5.3 this is exactly the case if the Schreiergraph S Q Γ ( H ) has local property A . (cid:3) Remark.
A minimal amenable second countable étale groupoid has simple reduced C ∗ -algebra,if and only if it is topologically principal. Hence, if the Schreier graphs of Z have local property A , then C ∗ r ( Z ) is simple if and only if Z is generic.More generally, C ∗ r ( Z ) is simple for arbitrary URS Z , if and only if C ∗ r ( G ) has the intersec-tion property relative to C ( G ), meaning that every (nontrivial) ideal of C ∗ r ( G ) intersects C ( G )nontrivially. References [1] C. Anantharaman-Delaroche and J. Renault,
Amenable groupoids , Monographies deL’Enseignement Mathématique, vol. 36, L’Enseignement Mathématique, Geneva, 2000.[2] G. Elek,
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An intrinsic characterization of C ∗ -simplicity , accepted to Annales scientifiquesde l’École normale supérieure (2018), arXiv:1509.01870.[5] J. Renault, A groupoid approach to C ∗ -algebras , Lecture Notes in Mathematics, vol. 793,Springer, Berlin, 1980.[6] A. Sims, Étale groupoids and their C ∗ -algebras , (2017), arXiv:1710.10897.[7] A. Sims and D. Williams, Amenability for Fell bundles over groupoids , Illinois J. Math.57