A Cuntz-Pimsner Model for the C ∗ -algebra of a Graph of Groups
aa r X i v : . [ m a t h . OA ] M a y A CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OFGROUPS ALEXANDER MUNDEY AND ADAM RENNIE
Abstract.
We provide a Cuntz-Pimsner model for graph of groups C ∗ -algebras. This allowsus to compute the K -theory of a range of examples and show that graph of groups C ∗ -algebrascan be realised as Exel-Pardo algebras. We also make a preliminary investigation of whetherthe crossed product algebra of Baumslag-Solitar groups acting on the boundary of certain treessatisfies Poincar´e duality in KK -theory. By constructing a K -theory duality class we computethe K -homology of these crossed products. Introduction
In this article we provide a Cuntz-Pimsner model for the C ∗ -algebras of graphs of groupsintroduced in [BMPST17]. By associating an operator algebra to a graph of groups, we obtaininvariants of the dynamics via the K -theory and K -homology of the algebra.Graphs of groups were introduced by Bass [Bas93] and Serre [Ser80] to study the structure ofgroups via their action on trees. They have since become an important tool in geometric grouptheory and low-dimensional topology, see eg. [DD89].The relationship between graphs of groups and associated operator algebras has developedsporadically. A construction which parallels ours, for graphs of groups of finite type, was consid-ered by Okayasu, [Oka05]. Graphs of groups of finite type correspond to actions of virtually freegroups on trees [SW79, Theorem 7.3]. Like the algebra considered in [BMPST17], our construc-tion works for a much larger class of examples. Pimsner [Pim86] has used graphs of groups todecompose KK -groups for crossed products algebras, while Julg and Valette [JV84] used similarideas to investigate K -amenability of groups via their action on trees. More recently, graphs of C ∗ -algebras [FF14] have been introduced to investigate the K -theory of quantum groups.In [BMPST17], a generator and relations picture of an operator algebra associated to a graphof groups was presented, and we recall this in Section 2. The associated algebra was shown tobe Morita equivalent to a crossed product, and so has a groupoid model. The crossed productarises from the action of the fundamental group of the graph of groups on the boundary ofits universal covering tree. Groupoid models are helpful because the associated algebra can bedescribed in terms of a dense subalgebra of functions, and the KMS-states can be describedusing the machinery of [Ren80].Groupoid models are not so helpful for computing K -theory, and in this paper we supplementthe groupoid picture with a Cuntz-Pimsner description in Section 4. The correspondence fromwhich the Cuntz-Pimsner algebra is constructed is simple enough to make K -theory computa-tions tractable, as we show in Section 6. This correspondence, described in Section 3, is builtfrom direct summands related to both Rieffel imprimitivity bimodules [Rie74] and Kaliszewski-Larsen-Quigg’s subgroup correspondences [KLQ18]. The construction of the correspondence isalso reminiscent of the correspondences associated to the (dual of) a directed graph.In addition to K -theory computations, identifying a Cuntz-Pimsner model for graph of groupsalgebras allows one to import considerable existing technology for their study. For example, thegauge invariant ideals can be classified [Kat07]. Cuntz-Pimsner algebras have a distinguishedsubalgebra, the fixed point of the gauge action called the core. The core is often more amenable C ∗ -ALGEBRA OF A GRAPH OF GROUPS 2 to study, being a direct limit, and “supports” many of the invariants of dynamical interest,[Kat04, Pim97, CNNR11]. A Cuntz-Pimsner model also provides a canonical choice of “Toeplitzgraph of groups algebra”, namely the Toeplitz algebra of the associated correspondence. It isalso worthy of note that having a Cuntz-Pimsner model presents a crossed product by a fairlycomplicated group as a “generalised crossed product” by the integers.In Section 5, by constructing a Morita equivalent correspondence, we obtain a Morita equiv-alent Cuntz-Pimsner algebra, which can be directly shown to be an Exel-Pardo algebra [EP17].In particular, from a graph of groups we construct a directed graph acted upon by a group witha relevant cocycle, so that the associated Exel-Pardo algebra agrees with our Morita equivalentCuntz-Pimsner algebra.In addition to K -theory, one can also consider K -homology as an invariant of a C ∗ -algebra.To access K -homological information one often uses index theory and the pairing with K -theory.Many of the graphs of groups C ∗ -algebras we consider have torsion in their K -theory and so thismethod does not apply [HR00, Chapter 7]. We show in Section 7, using the Poincar´e dualitytechniques of [RRS19], that we can access the K -homology of an important class of examples,Baumslag-Solitar groups acting on the boundary of certain regular trees. Acknowledgements.
Both authors would like to thank Aidan Sims for discussions aboutamenability, and the first author would like to thank Nathan Brownlowe and Jack Spielbergfor their many insights into graph of groups C ∗ -algebras.2. Preliminaries
Graphs of groups.
We begin by recalling the notion of an (undirected) graph.
Definition 2.1. A graph Γ = (Γ , Γ , r, s ) consists of a countable sets of vertices Γ and edges Γ together with a range map r : Γ → Γ and source map s : Γ → Γ . We also assume thatthere is an involutive “edge reversal” map e e on Γ satisfying e = e and r ( e ) = s ( e ). An orientation of a graph Γ is a collection of edges Γ ⊆ Γ such that for each e ∈ Γ precisely oneof e or e is contained in Γ .We always assume that a graph comes with an orientation, but all constructions we considerare independent of this choice of orientation. With a choice of orientation, a graph can beconsidered as a directed graph in which each edge has a ‘ghost’ edge pointing in the oppositeorientation.Graphs are usually represented by a diagram. For a graph Γ and e ∈ Γ we either draw theedge e with a dashed line, as on the left of following figure, or omit it entirely as on the right: ee e . We say that a graph Γ is locally finite if | r − ( v ) | < ∞ for all v ∈ Γ , and nonsingular r − ( r ( e )) = { e } for all e ∈ Γ . In particular, a graph is nonsingular if and only if every vertexhas valence strictly greater than 1. For each n ∈ N we write Γ n = { e e · · · e n | e i ∈ Γ , s ( e i ) = r ( e i +1 ) } for the collection of paths of length n . Our convention for the direction of paths is thesame as that of [Rae05]. A circuit is a path e · · · e k ∈ Γ k for some k ∈ N such that r ( e ) = s ( e k )and e i = e i +1 for all 1 ≤ i ≤ k − e k − = e . A tree is a non-empty connected graphwithout circuits.A group H is said to act on the left of a tree X = ( X , X , r, s ) if it acts on the left ofboth X and X in such a way that r ( h · e ) = h · r ( e ) and s ( h · e ) = h · s ( e ) for all h ∈ H and e ∈ X . We say that H acts without inversion if h · e = e for all h ∈ H and e ∈ X .By performing barycentric subdivision on the edges of X , the group H can always be made CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 3 to act without inversion [Ser80, Section 3.1]. If H acts without inversion on X , then there isa well-defined quotient graph H \ X = ( H \ X , H \ X , r H \ X , s H \ X ), with r H \ X ( He ) = Hr ( e ), s H \ X ( He ) = Hs ( e ), and He = He for all e ∈ X .Bass-Serre Theory [Ser80, Bas93] asks what additional data on the quotient graph H \ X isrequired to reconstruct the original action of H on X . The central idea is that this additionaldata can be captured in a “graph of groups” which we now define. Definition 2.2. A graph of groups G = (Γ , G ) consists of a connected graph Γ together with:(i) a discrete group G x for each vertex x ∈ Γ ;(ii) a discrete group G e for each edge e ∈ Γ , with G e = G e ; and(iii) a monomorphism α e : G e → G r ( e ) for each e ∈ Γ .We give a brief summary of the duality between graphs of groups and group actions ontrees. The interested reader is directed towards [Ser80, Bas93, DD89], or the introduction of[BMPST17] for further details.Suppose that H acts on a tree X = ( X , X , r, s ) without inversions. Set Γ := H \ X . Fixa fundamental domain Y = ( Y , Y , r, s ) of the action of H on X . That is, Y is a connectedsubgraph of X such that Y contains precisely one edge from each edge orbit. Note that Y could contain more than one vertex from each vertex orbit.For each v ∈ Y we associate to the vertex Hv ∈ Γ the group G Hv := stab H ( v ), and for each e ∈ Y we associate to He ∈ Γ the group G He := stab H ( e ). Note that if v, w ∈ Y are in thesame orbit under the action of H , then stab H ( v ) ∼ = stab H ( w ) so up to isomorphism our choiceof vertex groups is consistent. For each e ∈ Y the inclusion of stab H ( e ) in stab H ( r ( e )) inducesa monomorphism α He : G He → G r ( He ) . The graph of groups G = (Γ = H \ X, G ) constructed inthis way is called the quotient graph of groups for the action of H on X .Conversely, if one starts with a graph of groups G = (Γ , G ), then there is a group π ( G ), calledthe fundamental group of G , and a universal covering tree X G , both constructed from paths in G . The fundamental group π ( G ) acts naturally, without inversions, on the tree X G . We will notmake explicit use of the construction of the fundamental group nor the universal covering tree,and as such direct the interested reader to [Ser80, Bas93, BMPST17] for their construction. The crux of Bass-Serre theory is that studying discrete group actions on trees is equivalentto studying graphs of groups. In particular, the quotient graph of groups construction and theconstruction of the fundamental group acting on the universal covering tree are mutually inverse [Ser80, Theorem 13] . We impose conditions on the graphs of groups that we work with.
Definition 2.3.
A graph of groups G = (Γ , G ) is said to be locally finite if the underlyinggraph Γ is locally finite and for all e ∈ Γ , the subgroup α e ( G e ) has finite index in G r ( e ) . Agraph of groups G = (Γ , G ) is said to be nonsingular if for all e ∈ Γ with r − ( r ( e )) = { e } themonomorphism α e : G e → G r ( e ) is not an isomorphism.Local finiteness and nonsingularity of G is equivalent to local finiteness and nonsingularity ofthe associated universal covering tree X G . All graphs of groups considered in this article will belocally finite and nonsingular.
CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 4 Example . The modular group H := P SL ( Z ) ∼ = Z ∗ Z = h a, b | a = b = 1 i acts withoutinversion on the (2 , wvebb b aa . The generator a acts by rotation around a 2-valent vertex w , while b acts by rotation aroundan adjacent 3-valent vertex v . The orbits of vertices are coloured either red or blue in theabove figure. As such, the quotient graph Γ consists of 1 edge and 2 vertices. We also havestab H ( v ) = h b | b = 1 i and stab H ( w ) = h a | a = 1 i . The associated graph of groups thequotient graph of groups G = (Γ , G ) is drawn in the following way: Z Z { } . Note that we have suppressed the inclusion maps from { } into both Z and Z . Here G is bothlocally finite and nonsingular. Example . The Baumslag-Solitar group BS (1 ,
2) = h a, t | tat − = a i , acts on the infinite 3-regular tree X in a manner which we now spend some time describing.Fix a vertex v ∈ X and an edge e ∈ X with r ( e ) = v . Consider the infinite rooted binarytree X ′ with edge set equal to all the edges of X \ { e, e } which are in the same connectedcomponent as v . We label the vertices of X ′ with words from the alphabet { , } using thebinary structure of the tree. In particular, the vertices adjacent to v in X ′ are labelled by 0and 1, and inductively if a vertex is labelled by w · · · w k ∈ { , } k , then it is adjacent to thevertices labelled by w · · · w k w · · · w k
1, and w · · · w k − . The labelling can be observed in thefollowing diagram: v
01 00011011 000001010011100101110111 fet t t taa aa aa .
CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 5 Fix a doubly infinite path µ in X containing e and passing through the vertices labelled by0 , , , . . . . The generator t of G acts hyperbolically on X by translating X along the infinitepath µ in such a way that t · v = 0 and t · w · · · w k = 0 w · · · w k in the subtree X ′ .The generator a of BS (1 ,
2) fixes each edge in X \ ( X ′ ) and acts via a so-called “odometeraction” on the remaining vertices. In particular, the action is defined recursively by a · w w · · · w k = ( w · · · w k if w = 00( a · w · · · w k ) if w = 1 . It is relatively straightforward to check that the actions of t and a on X respect the relationsof BS (1 , BS (1 ,
2) on X . The action is transitiveon both edges and vertices. Consequently, the quotient graph BS (1 , \ X is a loop. One checksthat with v and f as in the diagramstab BS (1 , ( v ) = h a i ∼ = Z , andstab BS (1 , ( f ) = stab BS (1 , (0) = h tat − i ∼ = Z . Using additive notation for the integers, in the quotient graph of groups for the action of BS (1 , X we have α BS (1 , e = id and α BS (1 , e : k k which we represent diagrammatically as, Z id k k Z . Similar considerations show that for all m, n ∈ Z the group BS ( m, n ) = h a, t | ta m t − = a n i acts on the ( | m | + | n | )-regular tree, with the quotient graph of groups given by Z k mkk nk Z . C ∗ -algebras associated to graphs of groups. In [BMPST17] the authors introduceda C ∗ -algebra associated to a graph of groups using the notion of G -families. A G -family isanalogous to a Cuntz-Krieger-family, which are used to define directed graph C ∗ -algebras (see[Rae05]). Definition 2.6.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups. Foreach e ∈ Γ choose a transversal (a system of distinct representatives) Σ e for G r ( e ) /α e ( G e ) andsuppose that 1 G r ( e ) ∈ Σ e . A G -family in a C ∗ -algebra B is a collection of partial isometries { S e | e ∈ Γ } in B together with representations U x : G x → B , g U x,g of G x by partialunitaries for each x ∈ Γ satisfying the relations:(G1) U x, U y, = 0 for each x, y ∈ Γ with x = y ;(G2) U r ( e ) ,α e ( g ) S e = S e U s ( e ) ,α e ( g ) for each e ∈ Γ and g ∈ G e ;(G3) U s ( e ) , = S ∗ e S e + S e S ∗ e for each e ∈ Γ ; and(G4) S ∗ e S e = X r ( f )= s ( e ) , µ ∈ Σ f hf =1 e U s ( e ) ,µ S f S ∗ f U ∗ s ( e ) ,µ for each e ∈ Γ . CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 6 We note that the relation (G4) is independent of the choice of transversals (see [BMPST17,Remark 3.2]). The relations (G3) and (G4) are similar to the Cuntz-Krieger relations of directedgraph C ∗ -algebras [Rae05, p. 6]. In contrast to the Cuntz-Krieger relations, (G3) implies that S e S e = 0 despite the fact that r ( e ) = s ( e ). On the other hand, (G2) can be interpreted asintertwining the representations of G e induced by α e and α e . Notation 2.7. If e ∈ Γ and g ∈ G r ( e ) , then we will write S ge := U r ( e ) ,g S e . For the remainder of this article we will always assume that for a graph of groups G = (Γ , G ) we have chosen transversals Σ e for G r ( e ) /α e ( G e ) such that G r ( e ) ∈ Σ e . We use G -families toassociate a C ∗ -algebra to a graph of groups. Definition 2.8.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups. The graph of groups C ∗ -algebra C ∗ ( G ) is the unique (up to isomorphism) C ∗ -algebra generated bya G -family { u x , s e | x ∈ Γ , e ∈ Γ } which is universal in the following sense: if { U x , S e | x ∈ Γ , e ∈ Γ } is a G -family in a C ∗ -algebra B , then there is a unique ∗ -homomorphism fromΦ : C ∗ ( G ) → B such that Φ( u x ) = U x and Φ( s e ) = S e .This definition of a graph of groups C ∗ -algebra is analogous to the definition of the full group C ∗ -algebra. We will never consider a “reduced” version of the definition. A concrete G -familyis constructed in [BMPST17, Remark 3.4]. In general, the C ∗ -algebras of graphs of groupsare not C ∗ -algebras of directed graphs (see [BMPST17, Example 3.13]). On the other hand,if G is a locally finite, nonsingular graph of groups with trivial edge groups, then accordingto [BMPST17, Theorem 3.6] there is a directed graph E G such that C ∗ ( E G ) ∼ = C ∗ ( G ). As weshall see in Section 6, if the edge groups are non-trivial then the K -groups of C ∗ ( G ) often havetorsion and are therefore not the C ∗ -algebras of directed graphs.We recall the definition of the boundary ∂X of a tree X from [BMPST17, Definition 2.18]. Definition 2.9.
Let X be a locally finite non-singular tree and fix x ∈ X . The boundary of X (at x ) is the collection of infinite paths x∂X = { e e e · · · | e i ∈ X , r ( e ) = x, r ( e i +1 ) = s ( e i ) , and e i = e i +1 for all i ∈ N } equipped with the topology generated by the cylinder sets Z ( e e · · · e k ) = { f f f · · · ∈ x∂X | f i = e i for all 1 ≤ i ≤ k } ranging over all finite paths e e · · · e k ∈ X k such that r ( e ) = x and e i = e i +1 .The boundary x∂X is a totally disconnected compact Hausdorff space. Moreover, if X isconnected, then up to homeomorphism the boundary is independent of the choice of x ∈ X .In this case we simply write ∂X for the boundary. As noted in [BMPST17, Remark 2.19], theabove notion of boundary agrees with the Gromov boundary of X .The action of a group on a tree extends naturally to an action by homeomorphisms on theboundary. In particular, the fundamental group π ( G ) of a graph of groups G acts naturally onthe boundary ∂X G of its universal covering tree.Finally, we mention the main theorem of [BMPST17] which can be interpreted as a C ∗ -algebraic analogue of the Bass-Serre Theorem. Theorem 2.10 ([BMPST17, Theorem 4.1]) . Let G = (Γ , G ) be a locally finite nonsingular graphof countable groups. Then C ∗ ( G ) is isomorphic to K ( ℓ (Γ )) ⊗ ( C ( ∂X G ) ⋊ τ π ( G )) , where τ isthe action on C ( ∂X G ) induced by the action of π ( G ) on the universal covering tree X G . Group action cocycles.
To streamline computations in C ∗ ( G ), we introduce group actioncocycles. They allow us to “pass group elements through edges” in the sense of Lemma 2.12below. CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 7 Definition 2.11.
Let G and H be groups and suppose that G acts on a set X . An H -valued1-cocycle for the action of G on X is a map c : G × X → H such that c ( gh, x ) = c ( g, h · x ) c ( h, x ) (1)for all g, h ∈ G and x ∈ X .Let G = (Γ , G ) be a graph of groups and recall that Σ e ⊆ G r ( e ) is a transversal for thecoset space G r ( e ) /α e ( G e ). For each e ∈ Γ the group G r ( e ) acts canonically by left translationon G r ( e ) /α e ( G e ), and this induces an action of G r ( e ) on Σ e . There is a G e -valued cocycle c e : G r ( e ) × Σ e → G e for the action of G r ( e ) on Σ e given by c e ( g, µ ) := α − e (( g · µ ) − gµ ) (2)for all g ∈ G r ( e ) and µ ∈ Σ e . In the product ( g · µ ) − gµ we are considering both µ and g · µ aselements of G r ( e ) . Then for each g ∈ G r ( e ) and µ ∈ Σ e we have gµ = ( g · µ ) α e ( c e ( g, µ )) . (3)In particular, g = ( g · α e ( c e ( g, Lemma 2.12.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups andsuppose that { U x | x ∈ Γ } ∪ { S e | e ∈ Γ } is a G -family. Then for all e ∈ Γ , g ∈ G r ( e ) , and µ ∈ Σ e we have, U r ( e ) ,g S µe = S ( g · µ ) e U s ( e ) ,c e ( g,µ ) . (4) Proof.
This follows immediately from (3) and (G2). (cid:3)
Now suppose that f ∈ Γ is such that s ( f ) = r ( e ) and f = e . Then the cocycle c e restricts toa cocycle c fe : G f × Σ e → G e for the induced action of G f on Σ f via α f . Explicitly, for g ∈ G f and µ ∈ Σ e we define c fe ( g, µ ) := c e ( α f ( g ) , µ ) . (5)In the case where f = e , the set Σ e \ { } is invariant under the left action of G e = G e on Σ e . Inparticular, c e restricts to a cocycle c ee : G e × (Σ e \ { } ) → G e given by c ee ( g, µ ) = c e ( α e ( g ) , µ ) . C ∗ -correspondences and Cuntz-Pimsner algebras. We refer to [Lan95] or [RW98]for background on C ∗ -correspondences and their C ∗ -algebras. Given a right Hilbert A -module E , we write End A ( E ) for the C ∗ -algebra of adjointable operators on E . For ξ, η ∈ E we writeΘ ξ,η for the rank-one operator defined by Θ ξ,η ( ζ ) = ξ · ( η | ζ ) A for all ζ ∈ E . The closed 2-sidedideal of generalised compact operators is denotedEnd A ( E ) := span { Θ ξ,η | ξ, η ∈ E } ⊳ End A ( E ) . A right Hilbert A -module E is said to be full if ( E | E ) A = A . Definition 2.13. An A – B -correspondence is a pair ( φ, E ) consisting of a right Hilbert B -module E together with a ∗ -homomorphism φ : A → End B ( E ), which defines a left action of A on E .We say that ( φ, E ) is faithful if φ is injective and ( φ, E ) is nondegenerate if φ ( A ) E = E .An A – B -correspondence ( φ, E ) is said to be an imprimitivity bimodule if E is full and comeswith a left A -linear inner product A ( · | · ) making it a full left Hilbert A -module which satisfiesthe imprimitivity relation ξ · ( η | ζ ) B = φ ( A ( ξ | η )) ζ for all ξ, η, ζ ∈ E . CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 8 If E is a right Hilbert A -module then we denote by E ∗ = { ξ ∗ | ξ ∈ E } the conjugate vectorspace of E . Then E ∗ becomes a left Hilbert A -module with left action a · ξ ∗ = ( ξ · a ∗ ) ∗ andinner product A ( ξ ∗ | η ∗ ) = ( ξ | η ) A . If ( φ, E ) is an A – B -imprimitivity bimodule then E ∗ becomes a B – A -imprimitivity bimodule such that E ⊗ B E ∗ ∼ = A and E ∗ ⊗ A E ∼ = B [RW98,Proposition 3.28]. Definition 2.14. A representation of an A – A -correspondence ( φ, E ) in a C ∗ -algebra B , denoted( π, ψ ) : ( φ, E ) → B , consists of a ∗ -homomorphism π : A → B together with a linear map ψ : E → B such that:(i) ψ ( ξ ) π ( a ) = ψ ( ξ · a ) for all ξ ∈ E and a ∈ A ,(ii) π ( a ) ψ ( ξ ) = ψ ( φ ( a ) ξ ) for all a ∈ A and ξ ∈ E , and(iii) ψ ( ξ ) ∗ ψ ( η ) = π (( ξ | η ) A ) for all ξ, η ∈ E .We denote by C ∗ ( π, ψ ) the C ∗ -algebra generated by π ( A ) ∪ ψ ( E ) in B and call this the C ∗ -algebragenerated by ( π, ψ ) . Pimsner [Pim97] introduced two universal C ∗ -algebras for representations of correspondences.The first is the Toeplitz algebra, introduced next. The second is the Cuntz-Pimsner algebra,corresponding to a restricted class of representations. Subsequent work by Muhly-Solel [MS00]and Katsura [Kat04] expanded the class of correspondences to which these algebras can beassociated. Definition 2.15.
The
Toeplitz algebra T E of an A – A -correspondence ( φ, E ) is the unique (upto isomorphism) C ∗ -algebra generated by a representation ( j A , j E ) of ( φ, E ) satisfying the fol-lowing universal property: if ( π, ψ ) : ( φ, E ) → B is a representation then there is a unique ∗ -homomorphism π × ψ : T E → C ∗ ( π, ψ ) such that ( π × ψ ) ◦ j A = π and ( π × ψ ) ◦ j E = ψ .Now suppose that ( φ, E ) is a full, faithful, and nondegenerate A – A -correspondence, andsuppose that φ ( A ) ⊆ End A ( E ). Given a representation ( π, ψ ) : ( φ, E ) → B there is an induced ∗ -homomorphism ψ (1) : End B ( E ) → B satisfying ψ (1) (Θ ξ,η ) = ψ ( ξ ) ψ ( η ) ∗ for all ξ, η ∈ E . Wesay that a representation ( π, ψ ) of ( φ, E ) is Cuntz-Pimsner covariant if ( ψ (1) ◦ φ )( a ) = π ( a ) forall a ∈ A . Definition 2.16.
The
Cuntz-Pimsner algebra O E of an A – A -correspondence ( φ, E ) is the unique(up to isomorphism) C ∗ -algebra generated by a Cuntz-Pimsner covariant representation ( i A , i E )of E satisfying the following universal property: if ( π, ψ ) : ( φ, E ) → B is a Cuntz-Pimsnercovariant representation then there is a unique ∗ -homomorphism π × ψ : O E → C ∗ ( π, ψ ) suchthat ( π × ψ ) ◦ i A = π and ( π × ψ ) ◦ i E = ψ .We show in Theorem 4.1 that the C ∗ -algebra of a graph of groups C ∗ ( G ) can be realised as aCuntz-Pimsner algebra. In the next section we describe the correspondence that the aforemen-tioned Cuntz-Pimsner algebra is built from.3. The graph of groups correspondence
One standard method of building a Cuntz-Pimsner model for directed graph algebras, [RRS19,Proposition 3.8], starts with functions on vertices as the coefficient algebra, and builds a C ∗ -correspondence from functions on edges. The coefficient algebra for our Cuntz-Pimsner modelof C ∗ ( G ) is instead the direct sum of matrices over the group C ∗ -algebras of the edge groups. Inthe case that our edge groups are all trivial, our correspondence is akin to the directed graphcorrespondence for the dual graph, [Rae05, Corollary 2.6]. CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 9 The graph of groups correspondence.
We begin by considering a path of length two, f e ∈ Γ . We associate to the path f e a C ∗ ( G f )– C ∗ ( G e )-correspondence D fe . The corre-spondence D fe is analogous to the discrete case of the subgroup correspondences considered in[KLQ18]; the difference being that we do not assume that G f = G e . Definition 3.1 (The algebras) . For each e ∈ Γ let B e := C ∗ ( G e ), the full group C ∗ -algebra of G e . Let A e := M Σ e ( B e ) denote the C ∗ -algebra of | Σ e | × | Σ e | matrices over B e .We define a right B e -module F e as follows. Consider the ∗ -algebra of compactly supportedfunctions C c ( G e ) on G e , equipped with the convolution product. Define a right action of a ∈ C c ( G e ) on ξ ∈ C c ( G r ( e ) ) by ( ξ · a )( h ) = X g ∈ G e ξ ( hα e ( g )) a ( g − ) , and define a C c ( G e )-valued inner product on ξ, η ∈ C c ( G r ( e ) ) by( ξ | η ) C c ( G e ) ( h ) := X g ∈ G r ( e ) ξ ( g ) η ( gα e ( h )) = X µ ∈ Σ e X k ∈ G e ξ ( µα e ( k )) η ( µα e ( kh )) . (6)A norm on C c ( G e ) is given by k ξ k = k ( ξ | ξ ) C c ( G e ) k / full , where k · k full denotes the full C ∗ -normon C c ( G e ). Then C c ( G r ( e ) ) can be completed into a right B e -module (see [Lan95, p. 4] or[RW98, Lemma 2.16] for details). Definition 3.2 (An imprimitivity bimodule) . Let F e denote the right B e -module given bycompleting C c ( G r ( e ) ) in the norm defined by the inner product (6).Recall that for each e ∈ Γ the vertex group G r ( e ) acts canonically on the set of transversalsΣ e for G r ( e ) /α e ( G e ). As such, we can consider the associated (full) crossed-product C ∗ -algebra C (Σ e ) ⋊ G r ( e ) . We remark that the module F e is the C (Σ e ) ⋊ G r ( e ) – C ∗ ( G e )-imprimitivy bimoduleof Green’s Imprimitivity Theorem [RW98, Theorem C.23]. In particular, F e comes equipped witha left action ψ : C (Σ e ) ⋊ G r ( e ) → End B e ( F e ) which we spend a moment to describe.Since G is locally finite, Σ e is a finite set for each e ∈ Γ . So we may view C (Σ e ) ⋊ G r ( e ) as acompletion of C c ( G r ( e ) × Σ e ), where C c ( G r ( e ) × Σ e ) is a ∗ -algebra under the operations( a ∗ b )( g, µ ) = X h ∈ G r ( e ) a ( h, µ ) b ( h − g, h − · µ ) and a ∗ ( g, µ ) = a ( g − , g − · µ ) , for a, b ∈ C c ( G r ( e ) × Σ e ). The left action ψ : C (Σ e ) ⋊ G r ( e ) → End B e ( F e ) then satisfies ψ ( a ) ξ ( h ) = X k ∈ G r ( e ) a ( k − , h · ξ ( kh ) (7)for all a ∈ C c ( G r ( e ) × Σ e ) and ξ ∈ C c ( G r ( e ) ), where h · ∈ Σ e is the transversal representingthe coset hα e ( G e ). The module F e is also full and comes equipped with a left C (Σ e ) ⋊ G r ( e ) -valued inner product so that F ∗ e is a B e – C (Σ e ) ⋊ G r ( e ) -correspondence. Although it may not beimmediately apparent, F e is a free right B e -module. Lemma 3.3.
Let e ∈ Γ and consider the right B e -module L µ ∈ Σ e B e . Denote the left actionof A e ∼ = End A ( L µ ∈ Σ e B e ) on L µ ∈ Σ e B e by ℓ . Then ( ℓ, L µ ∈ Σ e B e ) and ( ψ, F e ) are isomorphic C ∗ -correspondences.Proof. Identify C c (Σ e × G e ) with L µ ∈ Σ E C c ( G e ). A straightforward computation shows thatthe linear map Φ : C c ( G r ( e ) ) → C c (Σ e × G e ) defined byΦ( ξ )( µ, k ) = ξ ( µα e ( k )) , µ ∈ Σ e , k ∈ G e , (8) CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 10 preserves both inner products and right actions, and therefore extends to an isometric linearmap Φ : F e → L µ ∈ Σ e B e of right B e -modules. Since Φ sends the point mass at µα e ( k ) to thepoint mass at k in the µ -th copy of B e , Φ is surjective.Since F e is finitely generated, Green’s Imprimitivity Theorem [RW98, Theorem C.23] impliesthat End B e ( F e ) ∼ = C (Σ e ) ⋊ G r ( e ) . Consequently, C (Σ e ) ⋊ G r ( e ) ∼ = A e , the left actions arepreserved, and the C ∗ -correspondences ( ψ, F e ) and ( ℓ, L µ ∈ Σ e B e ) are isomorphic. (cid:3) In the sequel, we frequently identify the correspondences ( ψ, F e ) and ( ℓ, L µ ∈ Σ e B e ) in themanner described by Lemma 3.3. Using (8) to identify C c (Σ e × G e ) as a dense subspace of F e ,the left action ψ : C (Σ e ) ⋊ G r ( e ) → End B e ( F e ) of (7) satisfies ψ ( a ) ξ ( µ, h ) = X k ∈ G r ( e ) a ( k − , h · ξ ( k · µ, c e ( k, µ ) h ) µ ∈ Σ e , h ∈ G e , (9)for all a ∈ C c ( G e × Σ e ) and ξ ∈ C c (Σ e × G e ).Our next goal is to construct a B f – B e -correspondence D fe from F e by restricting the leftaction of C (Σ e ) ⋊ G e to an action of B f . In the special case where f = e , the construction variesslightly, and so we introduce the following notation. Notation 3.4.
For each f e ∈ Γ let ∆ fe ⊆ Σ e be given by∆ fe := ( ∅ if f = e { G r ( e ) } if f = e. Definition 3.5.
For each f e ∈ Γ define the right Hilbert B e -module D fe := M µ ∈ Σ e \ ∆ fe B e considered as a direct sum of right Hilbert B e -modules. Note that if f = e and α e is surjective(so Σ e = { G r ( e ) } ), then D ee = { } . Proposition 3.6.
Let f e ∈ Γ , and suppose that either f = e , or α e is not surjective. Let c fe : G f × Σ e → G e be the cocycle for the action of G f on Σ e given by (5) . Then ( ϕ fe , D fe ) isa B f – B e -correspondence with left action ϕ fe : B f → End B e ( D fe ) satisfying ϕ fe ( a ) ξ ( µ, h ) = X k ∈ G f a ( k − ) ξ ( α f ( k ) · µ, c fe ( k, µ ) h ) (10) for all a ∈ C c ( G f ) and ξ ∈ C c ((Σ e \ ∆ fe ) × G e ) . Moreover, ϕ fe is unital and faithful.Proof. Since Σ e is finite the universal property of C ∗ ( G r ( e ) ) induces an injective ∗ -homomorphism π : C ∗ ( G r ( e ) ) → C (Σ e ) ⋊ G r ( e ) such that π ( a )( g, µ ) = a ( g ) for all a ∈ C c ( G r ( e ) ). As G r ( e ) isdiscrete [Rie74, Proposition 1.2] implies that the inclusion C c ( G f ) ⊆ C c ( G r ( e ) )—induced by α f : G f → G r ( e ) —extends to an injective ∗ -homomorphism ι : B f → C ∗ ( G r ( e ) ). Hence, thecomposition ψ ◦ π ◦ ι defines a left action of C ∗ ( G f ) on F e by compact operators. Moreover, thisaction is unital and faithful. It follows from (9) and the definitions of π and ι that ψ ◦ π ◦ ι = ϕ fe when f = e . If f = e , then the discussion in Section 2.3 implies that the action of G e on Σ e restricts to an action on Σ e \ { } . In particular, the action ψ ◦ π ◦ ι on F e restricts to an actionon L µ ∈ Σ e \{ } B e satisfying (10). (cid:3) When f = e the copy of C ∗ ( G e ) corresponding to the identity coset is removed from F e in order take care of the orthogonality of S ∗ e S e and S e S ∗ e introduced by condition (G3). Thiswill become more apparent later. Conceptually, we have passed to the dual graph in order toaccommodate this orthogonality. CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 11 We assemble the C ∗ -correspondences ( ϕ fe , D fe ) into a single C ∗ -correspondence. Definition 3.7 (The graph of groups module) . Let B := M e ∈ Γ B e and D := M fe ∈ Γ D fe . We remind the reader that for each e ∈ Γ we also have e ∈ Γ . Here we think of D fe as a rightHilbert B -module, with B l acting trivially on D fe unless l = e . So D is a direct sum of rightHilbert B -modules. We call D the graph of groups module associated to G . Proposition 3.8.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups.Extend ϕ fe to a left action ϕ fe : B → End B ( D fe ) , where ϕ fe ( B l ) = { } unless l = f . Then D is full and ( ϕ := ⊕ fe ∈ Γ ϕ fe , D ) is a B – B -correspondence with a faithful, non-degenerate leftaction satisfying ϕ ( B ) ⊆ End B ( D ) .Proof. Nonsingularity of G guarantees that for each edge e ∈ Γ there exists some f ∈ Γ with s ( f ) = r ( e ) and D fe = { } . Since each summand D fe of D is full as a right Hilbert B e -moduleby Lemma 3.15, it follows that D is full as a right Hilbert B -module. Nonsingularity of G alsoimplies that for each f ∈ Γ there always exists e ∈ Γ such that either f = e or such that α e is not surjective. It now follows from Proposition 3.6 that ϕ fe is injective for each f e ∈ Γ , andso ϕ is also injective. Since each ϕ fe : B f → End B e ( D fe ) is unital ϕ is non-degenerate.To see that ϕ ( B ) ⊆ End B ( D ), recall from Proposition 3.6 that ϕ fe ( B f ) ⊆ End B e ( D fe ). Since G is locally finite, for each f ∈ Γ and a ∈ B f we have ϕ ( a ) = P r ( e )= s ( f ) ϕ fe ( a ), which belongsto L r ( e )= s ( f ) End B ( D fe ) ⊆ End B ( D ). In general, each a ∈ B can be approximated in norm bya finite sum of a f ∈ B f , so it follows that ϕ ( a ) ∈ End B ( D ). (cid:3) Definition 3.9 (The graph of groups correspondence) . We call ( ϕ, D ) of Proposition 3.8 the graph of groups correspondence associated to G .3.2. The amplified graph of groups correspondence.
The graph of groups correspondence D proves to be very useful for the K -theory calculations in Section 6 and will be analysed furtherin Section 5, but to facilitate the construction of a Cuntz-Pimsner model of C ∗ ( G ), we pass to aMorita equivalent correspondence, which we call the amplified graph of groups correspondence .By [MS00], the associated Cuntz-Pimsner algebras will also be Morita equivalent. To begin weamplify D fe to an A f – A e -correspondence. Recall that F ∗ e is a B e – A e -correspondence. Definition 3.10 (Amplified correspondences) . For each f e ∈ Γ define a right A e -module E fe as the balanced tensor product, E fe := F f ⊗ B f D fe ⊗ B e F ∗ e . Let ϕ fe : A f → End A e ( E fe ) denote the left action induced by the left action of A f ∼ = End B f ( F f )on F f . Then ( ϕ fe , E fe ) is an A f – A e -correspondence.The graph of groups correspondence is now built in a similar manner to ( ϕ, D ). Definition 3.11 (The amplified graph of groups correspondence) . Define A := M e ∈ Γ A e = M e ∈ Γ M Σ e ( C ∗ ( G e )) and E := M fe ∈ Γ E fe . Then E is a right Hilbert A -module with right action analogous to that of B on D . A leftaction ϕ : A → End A ( E ) is given by ϕ = ⊕ fe ∈ Γ ϕ fe . We call the A – A -correspondence ( ϕ, E )the amplified graph of groups correspondence associated to G . CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 12 Proposition 3.12.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups.Then E is full and the left action ϕ : A → End A ( E ) is faithful, non-degenerate, and by compactoperators.Proof. This follows from Proposition 3.8 and the fact that each F e is a finitely generated A e – B e -imprimitivity bimodule. (cid:3) Passing to the amplified graph of groups correspondence induces a Morita equivalence on theassociated Cuntz-Pimsner algebras.
Proposition 3.13.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups.Then O E is Morita equivalent to O D .Proof. Let F = L e ∈ Γ F e . Using the (completed) direct sum of C ∗ -modules, we have E ∼ = (cid:16) M f ∈ Γ F f (cid:17) ⊗ B (cid:16) M fe ∈ Γ D fe (cid:17) ⊗ B (cid:16) M e ∈ Γ F ∗ e (cid:17) = F ⊗ B D ⊗ B F ∗ . The Morita equivalence of D and E then passes to a Morita equivalence of O D and O E , by[MS00]. (cid:3) We now give a second description of the module ( ϕ fe , E fe ) in terms of the conjugate module F ∗ e which, for computational purposes, has the advantage of not being built from balanced tensorproducts. For each e ∈ Γ let A e := C c (Σ e × G e × Σ e ). We think of A e as a dense ∗ -subalgebraof A e with multiplication defined for each a, b ∈ A e by( ab )( µ, h, ν ) = X σ ∈ Σ f k ∈ G f a ( µ, k, σ ) b ( σ, k − h, ν ) . Remark . If Σ e × G e × Σ e is considered as a groupoid with product ( µ, g, ν ) · ( ν, h, σ ) =( µ, gh, σ ), then A e is the full groupoid C ∗ -algebra of Σ e × G e × Σ e , and the product definedabove is just the convolution product on A e .Now suppose that f e ∈ Γ and consider the right Hilbert A e -module( F ∗ e ) Σ f × (Σ e \ ∆ fe ) := M µ ∈ Σ f M ν ∈ Σ e \ ∆ fe F ∗ e with the standard right A e -valued inner product and coordinate-wise right action of A e . To keeptrack of the conjugate module structure we identify L µ ∈ Σ f C c ( G f ) ⊆ F f with C c (Σ f × G f ), and L µ ∈ Σ e C c ( G e ) ⊆ F ∗ e with C c ( G e × Σ e ). Let E fe := C c (Σ f × (Σ e \ ∆ fe ) × G e × Σ e ) which weconsider as a dense subspace of ( F ∗ e ) Σ f × (Σ e \ ∆ fe ) . Lemma 3.15.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups. Then foreach f e ∈ Γ the modules E fe and ( F ∗ e ) Σ f × (Σ e \ ∆ fe ) are isomorphic as right Hilbert A e -modules.In addition, if either f = e or α e is not surjective, then there is a faithful non-degenerate leftaction ψ fe : A f → End A e (( F ∗ e ) Σ f × (Σ e \ ∆ fe ) ) satisfying ( ψ fe ( a ) ξ )( µ, ν, h, σ ) = X k ∈ G f ρ ∈ Σ f a ( µ, k − , ρ ) ξ ( ρ, k · ν, c fe ( k, ν ) h, σ ) (11) for all a ∈ A e and ξ ∈ E fe , which makes ( ϕ fe , E fe ) and ( ψ fe , ( F ∗ e ) Σ f × (Σ e \ ∆ fe ) ) isomorphic as A f – A e -correspondences. CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 13 Proof.
Define a linear map Φ : C c (Σ f × G f ) ⊗ C c ( G f ) C c ((Σ e \ ∆ fe ) × G e ) ⊗ C c ( G e ) C c ( G e × Σ e ) → E fe by Φ( a ⊗ ξ ⊗ b )( µ, σ, h, ν ) = X g ∈ G f X k ∈ G e a ( µ, g − ) ξ ( g · σ, c fe ( g, σ ) k ) b ( k − h, ν ) . Using the inner product on E fe we compute for a ∈ C c (Σ f × G f ), ξ ∈ C c ((Σ e \ ∆ fe ) × G e ), and b ∈ C c ( G e × Σ e ),( a ⊗ ξ ⊗ b | a ⊗ ξ ⊗ b ) A e ( µ, h, ν )= ( b | ( ξ | ϕ fe (( a | a ) C c ( G f ) ) ξ ) C c ( G e ) · b ) A e ( µ, h, ν )= X k ∈ G e g ∈ G e b ( k, µ )( ξ | ϕ fe (( a | a ) C c ( G f ) ) ξ ) C c ( G e ) ( g ) b ( g − kh, ν )= X k ∈ G e g ∈ G e X ρ ∈ Σ e \ ∆ fe l ∈ G e ξ ( µ, l ) b ( k, µ ) ϕ fe (( a | a ) C c ( G f ) ) ξ ( ρ, lg ) b ( g − kh, ν )= X k ∈ G e g ∈ G e X ρ ∈ Σ e \ ∆ fe l ∈ G e X m ∈ G f ξ ( ρ, l ) b ( k, µ )( a | a ) C c ( G f ) ( m − ) ξ ( m · ρ, c fe ( m, ρ ) lg ) b ( g − kh, ν )= X k ∈ G e g ∈ G e X ρ ∈ Σ e \ ∆ fe l ∈ G e X m ∈ G f X σ ∈ Σ f r ∈ G f a ( σ, r ) ξ ( ρ, l ) b ( k, µ ) a ( σ, rm − ) ξ ( m · ρ, c fe ( m, ρ ) lg ) b ( g − kh, ν ) . Now make the following sequence of substitutions: rm − s , r · ρ λ , l c fe ( r − , λ ) d , dg t , and dk u . Upon performing these substitutions, the cocycle condition (1) impliesthat ( a ⊗ ξ ⊗ b | a ⊗ ξ ⊗ b ) A e ( µ, h, ν )= X σ ∈ Σ f λ ∈ Σ e \ ∆ fe X u ∈ G e (cid:16) X r ∈ G f d ∈ G e a ( σ, r ) ξ ( r − · λ, c fe ( r − , λ ) d ) b ( d − u, µ ) (cid:17) × (cid:16) X s ∈ G f t ∈ G e a ( σ, s ) ξ ( s − · λ, c fe ( s − , λ ) t ) b ( t − uh, ν ) (cid:17) = X σ ∈ Σ f λ ∈ Σ e \ ∆ fe X u ∈ G e Φ( a ⊗ ξ ⊗ b )( σ, λ, u, µ )Φ( a ⊗ ξ ⊗ b )( σ, λ, uh, ν )= (Φ( a ⊗ ξ ⊗ b ) | Φ( a ⊗ ξ ⊗ b )) C c ( G e ) ( µ, h, ν ) . As such, Φ extends to an isometric linear map Φ : E fe → ( F ∗ e ) Σ f × (Σ e \ ∆ fe ) . Let ǫ fµ,g denote thepoint mass at ( µ, g ) ∈ Σ f × G f ; let χ feν,h denote the point mass at h ∈ G e in the ν -th copy of C ∗ ( G e ) in D fe ; and let ǫ ek,σ denote the point mass at ( k, σ ) ∈ G e × Σ e . Then Φ( ǫ eµ, ⊗ χ feσ, ⊗ ǫ eh,ν )is the point mass at ( µ, ν, h, σ ) in E fe . Since E fe is spanned by point masses, Φ is surjective.It is routine to check that Φ respects the right action of A e . By pulling the left action ϕ fe of A f on E fe over to a left action on ( F ∗ e ) Σ f × (Σ e \ ∆ fe ) we see that (11) is satisfied, and as such( ϕ fe , E fe ) and ( ψ fe , ( F ∗ e ) Σ f × (Σ e \ ∆ fe ) ) are isomorphic. (cid:3) In the sequel we will always identify ( ϕ fe , E fe ) with ( ψ fe , ( F ∗ e ) Σ f × (Σ e \ ∆ fe ) ). CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 14 Notation 3.16.
We introduce the following notation for point masses. For each e ∈ Γ wewrite ǫ eµ,g,ν for the point mass at ( µ, g, ν ) ∈ Σ e × G e × Σ e , considered as element of the dense ∗ -subalgebra A e = C c (Σ e × G e × Σ e ) of A e . For each f e ∈ Γ let X fe = Σ f × (Σ e \ ∆ fe ) × G e × Σ e (12)so that E fe = C c ( X fe ). We let χ feµ,ν,g,σ denote the point mass at ( µ, ν, g, σ ) ∈ X fe . Remark . For the reader familiar with [BMPST17, Definition 2.4], it is possible to viewelements of X fe as a triple consisting of a “ G -path” of length 2, a group element in G e , and a“ G -path” of length 1. To be more precise, let Y fe = { ( g f g e, k, g e ) ∈ G × G e × G } . Thenthe map π : Y fe → X fe given by π ( g f g e, k, g e ) = ( g , g , k, g ) is a bijection, even in the casewhere f = e .We record some identities for point masses which can be deduced from the structures of A and ( F ∗ e ) Σ f × (Σ e \ ∆ fe ) , together with Lemma 3.15. With δ the Kronecker delta we have ǫ fµ,g,ν ǫ ea,h,b = δ νf,ae ǫ fµ,gh,b (13)( χ feµ,ν,g,σ | χ pqa,b,h,c ) A = δ µfνe,apbq ǫ eσ,g − h,c (14) χ feµ,ν,g,σ · ǫ ta,h,b = δ σe,at χ feµ,ν,gh,b (15) ϕ ( ǫ ta,h,b ) χ feµ,ν,g,σ , and = δ bt,µf χ fea,h · ν,c fe ( h,ν ) ,σ . (16)Together, (14) and (15) imply thatΘ χ feµ,ν,g,σ ,χ pqa,b,h,c = δ σe,cq Θ χ feµ,ν,g,σ ,χ pea,b,h,σ . (17)We finish this section with a description of a frame for E , and consequently a descriptionof ϕ ( a ) in terms of rank-1 operators. Frames were introduced by Frank and Larson [FL02]for Hilbert C ∗ -modules as a generalisation of orthonormal bases. We take the definition from[RRS19] that frame for E consists of a sequence ( u i ) ∞ i =1 in E such that P ∞ i =1 Θ u i ,u i convergesstrictly to the identity operator on E . Lemma 3.18.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups. Then ( χ feµ,ν, , ) fe ∈ Γ , µ ∈ Σ f , ν ∈ Σ e \ ∆ fe constitutes a frame for E . In particular, for each f ∈ Γ and a ∈ A f we have ϕ ( a ) = X µ ∈ Σ f X r ( e )= s ( f ) ν ∈ Σ e \ ∆ fe Θ ϕ ( a ) χ feµ,ν, , ,χ feµ,ν, , . (18) Proof.
Let ǫ eg,σ ∈ F ∗ e denote the point mass at ( g, σ ) ∈ G e × Σ e . Observe that for each e ∈ Γ the right A e -module F ∗ e admits a frame consisting of just ǫ e , . Since Lemma 3.15 implies that E is a direct sum of copies of F ∗ e , the point masses χ feµ,ν, , constitute a frame for E .The second statement follows from the first: if ( u i ) ∞ i =1 is a frame for E and a ∈ A , then since ϕ ( a ) is compact, ϕ ( a ) = ϕ ( a ) P ∞ i =1 Θ u i ,u i = P ∞ i =1 Θ ϕ ( a ) u i ,u i with convergence in norm. (cid:3) A Cuntz-Pimsner model for C ∗ ( G )In this section we prove our first main result: that C ∗ ( G ) is isomorphic to the Cuntz-Pimsneralgebra of the amplified graph of groups correspondence ( ϕ, E ) associated to G . Cuntz-Pimsneralgebras can be considered a “generalised crossed products by Z ” and so in light of Theorem 2.10,we transform a crossed product by a potentially complicated group acting on the boundary ofa tree to a generalised crossed product by Z . CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 15 We recall from [Kat04] that Cuntz-Pimsner algebras admit a gauge action . That is an actionof the circle group γ : T → Aut( O E ) such that γ z ( i E ( ξ )) = zi E ( ξ ) and γ z ( i A ( a )) = i A ( a ) forall ξ ∈ E and a ∈ A , where ( i A , i E ) : ( ϕ, E ) → O E is the universal Cuntz-Pimsner covariantrepresentation. Similarly, it is stated in [BMPST17, Proposition 3.7] that C ∗ ( G ) admits a gaugeaction γ ′ : T → Aut( C ∗ ( G )) satisfying γ ′ z ( s e ) = zs e and γ ′ z ( u x,g ) = u x,g for all e ∈ Γ , x ∈ Γ and g ∈ G x . Theorem 4.1.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups andlet ( ϕ, E ) be the associated amplified graph of groups correspondence. Then the Cuntz-Pimsneralgebra O E is gauge-equivariantly isomorphic to C ∗ ( G ) . For the remainder of this section we fix a locally finite nonsingular graph of countable groups G = (Γ , G ). Recall that C ∗ ( G ) is generated by a universal G -family { u x , s e | x ∈ Γ , e ∈ Γ } .For each e ∈ Γ and g ∈ G r ( e ) write s ge for the partial isometry u r ( e ) ,g s e . In order to proveTheorem 4.1 we first construct a Cuntz-Pimsner covariant representation ( π, ψ ) of the C ∗ -correspondence E inside C ∗ ( G ). The universal property of Cuntz-Pimsner algebras then inducesa ∗ -homomorphism Ψ : O E → C ∗ ( G ).To simplify some of the computations in C ∗ ( G ) we have the following lemma. Lemma 4.2.
Let e, f ∈ Γ and suppose that g ∈ Σ e and h ∈ Σ f . Then s ∗ ge s hf = δ ge,hf s ∗ e s e , (19) and s ∗ e s e s hf = δ r ( f ) ,s ( e ) (1 − δ hf, e ) s hf . (20) Proof.
The identities (G3) and (G4) imply that the projections s ge s ∗ ge and s hf s ∗ hf are orthogonalwhenever ge = hf . It then follows that, s ∗ ge s hf = s ∗ ge s ge s ∗ ge s hf s ∗ hf s hf = δ ge,hf s ∗ ge s ge = δ ge,hf s ∗ e u ∗ r ( e ) ,g u r ( e ) ,g s e = δ ge,hf s ∗ e s e . For the second identity, we again compute using (G3) and (G4), s ∗ e s e s hf = (cid:16) X r ( t )= s ( e ) a ∈ Σ t \ ∆ te s at s ∗ at (cid:17) s hf s ∗ hf s hf = δ r ( f ) ,s ( e ) (1 − δ hf, e ) s hf . (cid:3) Lemma 4.3.
Let A = L A e be the algebra from Definition 3.11. There is a ∗ -homomorphism π : A → C ∗ ( G ) satisfying π ( ǫ eµ,g,ν ) = s µe u s ( e ) ,α e ( g ) s ∗ νe (21) for each e ∈ Γ and ( µ, g, ν ) ∈ Σ e × G e × Σ e .Proof. For each e ∈ Γ consider the C ∗ -subalgebra C e := span { s µe u s ( e ) ,α e ( g ) s ∗ νe | µ, ν ∈ Σ e , g ∈ G e } of C ∗ ( G ). Note that (G2) implies s µe u s ( e ) ,α e ( g ) s ∗ νe = s µe s ∗ νe u r ( e ) ,α e ( g ) . It follows from (19) and(G2) that ( s µe u s ( e ) ,α e ( g ) s ∗ νe )( s ae u s ( e ) ,α e ( h ) s ∗ be ) = δ ν,a s µe u s ( e ) ,α e ( gh ) s ∗ be , and ( s µe u s ( e ) ,α e ( g ) s ∗ νe ) ∗ = s νe u s ( e ) ,α e ( g − ) s ∗ µe . In particular, the elements { s µe s ∗ νe | µ, ν ∈ Σ e } form a system of matrix units in C e . Hence, there is a ∗ -homomorphism θ : M Σ e ( C ) → span { s µe s ∗ νe | µ, ν ∈ Σ e } satisfying θ ( ǫ eµ, ,ν ) = s µe s ∗ νe . The map and g P µ ∈ Σ e s µe s µe u r ( e ) ,α e ( g ) defines a unitary representation of G e in C e which commutes with the range of θ . Consequently,there is a ∗ -homomorphism π e : A e → C e satisfying π e ( ǫ eµ,g,ν ) = s µe u s ( e ) ,α e ( g ) s ∗ νe .The identity (19) implies that π e ( A e ) π f ( A f ) = π f ( A f ) π e ( A e ) = 0 for e = f . As such, π := ⊕ e ∈ Γ π e defines a ∗ -homomorphism from A to C ∗ ( G ) such that π ( ǫ eµ,g,ν ) = s µe u s ( e ) ,α e ( g ) s ∗ νe for all e ∈ Γ , µ, ν ∈ Σ e , and g ∈ G e . (cid:3) CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 16 Lemma 4.4.
There is a norm-decreasing linear map ψ : E → C ∗ ( G ) satisfying ψ ( χ feµ,ν,g,σ ) = s µf s νe u s ( e ) ,α e ( g ) s ∗ σe (22) for all f e ∈ Γ and ( µ, ν, g, σ ) ∈ X fe . Moreover, ψ ∗ ( ξ ) ψ ( η ) = π (( ξ | η ) A ) for all ξ, η ∈ E .Proof. For each f e ∈ Γ define ψ fe : C c ( X fe ) → C ∗ ( G ) by ψ fe ( χ feµ,ν,g,σ ) = s µf s νe u s ( e ) ,α e ( g ) s ∗ σe .Let ψ := ⊕ fe ∈ Γ ψ fe denote the induced map from the algebraic direct sum L fe ∈ Γ C c ( X fe ) to C ∗ ( G ). Fix f e, tr ∈ Γ , ( µ, ν, g, σ ) ∈ X fe , and ( a, b, h, c ) ∈ X tr . It follows from Lemma 4.2,(G2), and (14) that ψ ( χ feµ,ν,g,σ ) ∗ ψ ( χ tra,b,h,c ) = s σe u s ( e ) ,α e ( g − ) s ∗ νe s ∗ µf s at s br u s ( r ) ,α r ( h ) s ∗ cr = δ µf,at s σe u s ( e ) ,α e ( g − ) s ∗ νe s ∗ f s f s br u s ( r ) ,α r ( h ) s ∗ cr = δ µf,at (1 − δ br, f ) s σe u s ( e ) ,α e ( g − ) s ∗ νe s br u s ( r ) ,α r ( h ) s ∗ cr = δ µf,at (1 − δ br, f ) δ νe,br s σe u s ( e ) ,α e ( g − ) s ∗ e s e u s ( r ) ,α r ( h ) s ∗ cr = δ µf,at δ νe,br (1 − δ νe, f ) s σe u s ( e ) ,α e ( g − h ) s ∗ ce = δ µf,at δ νe,br (1 − δ νe, f ) π ( ǫ eσ,g − h,c )= π (( χ feµ,ν,g,σ | χ tra,b,h,c ) A ) . Consequently, ψ ( ξ ) ∗ ψ ( η ) = π (( ξ | η ) A ) for all ξ, η ∈ L fe ∈ Γ C c ( X fe ). This in turn gives k ψ ( ξ ) k = k π (( ξ | ξ ) A ) k ≤ k ( ξ | ξ ) A k = k ξ k , so that ψ extends to a bounded linear map ψ : E → C ∗ ( G ) with the property that ψ ( ξ ) ∗ ψ ( η ) = π (( ξ | η ) A ) for all ξ, η ∈ E . (cid:3) Proposition 4.5.
The pair ( π, ψ ) defines a Cuntz-Pimsner covariant representation of E in C ∗ ( G ) . In particular, if ( i A , i E ) : ( ϕ, E ) → O E is the universal Cuntz-Pimsner covariant rep-resentation, then there is an induced ∗ -homomorphism Ψ : O E → C ∗ ( G ) such that π ( a ) =(Ψ ◦ i A )( a ) and ψ ( ξ ) = (Ψ ◦ i E )( ξ ) for all a ∈ A and ξ ∈ E .Proof. Given Lemma 4.3 and Lemma 4.4 all that remains is to show that ( π, ψ ) respects boththe right and left actions of A on E , and satisfies Cuntz-Pimsner covariance. Using Lemma 4.2,(G2), and (15) we compute ψ ( χ feµ,ν,g,σ ) π ( ǫ ta,h,b ) = ( s µf s νe u s ( e ) ,α e ( g ) s ∗ σe )( s at u s ( t ) ,α t ( h ) s ∗ bt )= δ σe,at s µf s νe u s ( e ) ,α e ( gh ) s ∗ be = δ σe,at ψ ( χ feµ,ν,gh,b )= ψ ( χ feµ,ν,g,σ · ǫ ta,h,b ) . It follows that ψ ( ξ ) π ( a ) = ψ ( ξ · a ) for all ξ ∈ E and a ∈ A . For the left action we use Lemma 2.12and (16) to see that π ( ǫ ta,h,b ) ψ ( χ feµ,ν,g,σ ) = ( s at u s ( t ) ,α t ( h ) s ∗ bt )( s µf s νe u s ( e ) ,α e ( g ) s ∗ σe )= δ bt,µf s af u s ( f ) ,α f ( h ) s ∗ f s f s νe u s ( e ) ,α e ( g ) s ∗ σe = δ bt,µf s af u s ( f ) ,α f ( h ) s νe u s ( e ) ,α e ( g ) s ∗ σe = δ bt,µf s af s ( h · ν ) e u s ( e ) ,α e ( c fe ( h,ν ) g ) s ∗ σe = δ bt,µf ψ ( χ feµ,h · ν,c fe ( h,ν ) g,σ )= ψ (cid:0) ϕ ( ǫ ta,h,b ) χ feµ,ν,g,σ (cid:1) . Hence, π ( a ) ψ ( ξ ) = ψ ( ϕ ( a ) ξ ) for all a ∈ A and ξ ∈ E . Consequently, ( π, ψ ) is a representationof the C ∗ -correspondence ( ϕ, E ) in C ∗ ( G ). CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 17 We now claim that ( π, ψ ) is Cuntz-Pimsner covariant. Using (G4) and Lemma 3.18 wecompute ( ψ (1) ◦ ϕ )(1 A f ) = X µ ∈ Σ f X r ( e )= s ( f ) ν ∈ Σ e \ ∆ fe ψ ( χ feµ,ν, , ) ψ ( χ feµ,ν, , ) ∗ = X µ ∈ Σ f X r ( e )= s ( f ) ν ∈ Σ e \ ∆ fe s µf s νe s ∗ e s e s ∗ νe s ∗ µf = X µ ∈ Σ f s µf (cid:16) X r ( e )= s ( f ) ν ∈ Σ e \ ∆ fe s νe s ∗ νe (cid:17) s ∗ µf = X µ ∈ Σ f s µf s ∗ f s f s ∗ µf = X µ ∈ Σ f s µf s ∗ µf = π (cid:16) X µ ∈ Σ f ǫ eµ, ,µ (cid:17) = π (1 A f ) . With (18) the preceding computation can be modified to show that ( ψ (1) ◦ ϕ )( a ) = π ( a ) for all a ∈ A f and therefore all a ∈ A f . Since each a ∈ A can be approximated by a finite sum of a e ∈ A e , it follows that ( ψ (1) ◦ ϕ )( a ) = π ( a ) for all a ∈ A . The universal property of O E nowinduces a ∗ -homomorphism Ψ : O E → C ∗ ( G ) such that π ( a ) = (Ψ ◦ i A )( a ) and ψ ( ξ ) = (Ψ ◦ i E )( ξ )for all a ∈ A and ξ ∈ E . (cid:3) To see that Ψ is an isomorphism we construct an inverse Φ. To this end, we identify a G -family { V x , T e | x ∈ Γ , e ∈ Γ } within O E and then use the universal property of C ∗ ( G ) to obtain a ∗ -homomorphism Φ : C ∗ ( G ) → O E that is inverse to Ψ. Lemma 4.6.
Let ( i A , i E ) : E → O E denote the universal Cuntz-Pimsner covariant representa-tion. For each f ∈ Γ let T f := X r ( e )= s ( f ) µ ∈ Σ e \ ∆ fe i E (cid:0) χ fe ,µ, ,µ (cid:1) . (23) Then T f is a partial isometry in O E with source and range projections given by T ∗ f T f = X r ( e )= s ( f ) µ ∈ Σ e \ ∆ fe i A ( ǫ eµ, ,µ ) and T f T ∗ f = i A ( ǫ f , , ) . (24) Proof.
The first identity of (24) follows from (14) as for all e, t ∈ r − ( s ( f )), µ ∈ Σ e and ν ∈ Σ t , i E ( χ fe ,µ, ,µ ) ∗ i E ( χ ft ,ν, ,ν ) = i A (( χ fe ,µ, ,µ | χ ft ,ν, ,ν ) A ) = δ µe,νt i A ( ǫ eµ, ,µ ) . For the second identity we use (17) to see that for all e, t ∈ r − ( s ( f )), µ ∈ Σ e and ν ∈ Σ t , i E ( χ fe ,µ, ,µ ) i E ( χ ft ,ν, ,ν ) ∗ = i (1) E (cid:16) Θ χ fe ,µ, ,µ ,χ ft ,ν, ,ν (cid:17) = δ µe,νt i (1) E (cid:16) Θ χ fe ,µ, ,µ ,χ fe ,µ, ,µ (cid:17) . CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 18 It now follows from Cuntz-Pimsner covariance of ( i A , i E ), (16), and (18) that T f T ∗ f = X r ( e )= s ( f ) µ ∈ Σ e \ ∆ fe (cid:16) Θ χ fe ,µ, ,µ ,χ fe ,µ, ,µ (cid:17) = X ν ∈ Σ f X r ( e )= s ( f ) µ ∈ Σ e \ ∆ fe (cid:16) Θ ϕ fe ( ǫ f , , ) χ fe ,ν, ,µ ,χ fe ,ν, ,µ (cid:17) = ( i (1) E ◦ ϕ )( ǫ f , , )= i A ( ǫ f , , ) . (cid:3) Our aim now is to establish partial unitary representations of G x in O E for each x ∈ Γ . Toaccomplish this, let x ∈ Γ and g ∈ G x and consider the element of O E defined by, V x,g := X r ( f )= xµ ∈ Σ f i A (cid:0) ǫ fg · µ,c f ( g,µ ) ,µ (cid:1) . (25)Note that when g = 1 we have V x, = P r ( f )= x i A (1 A f ). Lemma 4.7.
For each x ∈ Γ the map G x ∋ g V x,g defines a partial unitary representationof G x in O E .Proof. Fix x ∈ Γ and g, h ∈ G x . The cocycle condition (1) implies that for all e, f ∈ r − ( x ), µ ∈ Σ f and ν ∈ Σ e , ǫ fg · µ,c f ( g,µ ) ,µ ǫ eh · ν,c e ( h,ν ) ,ν = δ f,e δ µ,h · ν ǫ fg · ( h · ν ) ,c f ( g,h · ν ) c f ( h,ν ) ,ν = δ f,e δ µ,h · ν ǫ f ( gh ) · ν,c f ( gh,ν ) ,ν . As such, V x,g V x,h = V x,gh . Making the substitution µ = g − · ν at the second equality, and usingthe cocycle condition (1) to see that c f ( g, g − · ν ) c f ( g − , ν ) = 1, we have V ∗ x,g = X r ( f )= xµ ∈ Σ f i A ( ǫ fµ,c f ( g,µ ) − ,g · µ )= X r ( f )= xν ∈ Σ f i A ( ǫ fg − · ν,c f ( g,g − · ν ) − ,ν )= X r ( f )= xν ∈ Σ f i A ( ǫ fg − · ν,c f ( g − ,ν ) ,ν )= V x,g − . (26)Consequently, V x,g is a partial unitary and g V x,g is a partial unitary representation of G x . (cid:3) Proposition 4.8.
The collection { V x , T e | x ∈ Γ , e ∈ Γ } defines a G -family in O E . Inparticular, there is a unique ∗ -homomorphism Φ : C ∗ ( G ) → O E such that Φ( s e ) = T e for all e ∈ Γ , and Φ( u x,g ) = V x,g for all x ∈ Γ and g ∈ G x .Proof. We will show that { V x , T e | x ∈ Γ , e ∈ Γ } satisfies (G1)–(G4) and then use the uni-versal property of C ∗ ( G ) to give Φ. The identity (G1) is follows immediately from direct sumdecomposition of A : if x = y ∈ Γ and e, f ∈ Γ are such that r ( e ) = x and r ( f ) = y then ǫ eµ, ,µ ǫ fν, ,ν = 0 for any µ ∈ Σ e and ν ∈ Σ f . The identity (24) yields (G3). Both (24) and (26) CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 19 imply that for f ∈ Γ and µ ∈ Σ f , V r ( f ) ,µ T f T ∗ f V ∗ r ( f ) ,µ = X r ( e )= r ( f ) ν ∈ Σ e X r ( t )= r ( f ) σ ∈ Σ t i A ( ǫ eµ · ν,c e ( µ,ν ) ,ν ) i A ( ǫ f , , ) i A ( ǫ tσ,c e ( µ,σ ) − ,µ · σ )= i A ( ǫ fµ, , ) i A ( ǫ f , , ) i A ( ǫ f , ,µ )= i A ( ǫ fµ, ,µ ) . The identity (G4) now follows from (24).We now claim that (G2) is satisfied. For each f ∈ Γ and g ∈ G f , V r ( f ) ,α f ( g ) T f = X r ( e )= r ( f ) ν ∈ Σ e X r ( t )= s ( f ) µ ∈ Σ t \ ∆ ft i A ( ǫ eα f ( g ) · ν,c e ( α f ( g ) ,ν ) ,ν ) i E ( χ ft ,µ, ,µ )= X r ( t )= s ( f ) µ ∈ Σ t \ ∆ ft i E ( ϕ ( ǫ fα f ( g ) · ,c f ( α f ( g ) , , ) χ ft ,µ, ,µ )= X r ( t )= s ( f ) µ ∈ Σ t \ ∆ ft i E ( ϕ ( ǫ f ,g, ) χ ft ,µ, ,µ )= X r ( t )= s ( f ) µ ∈ Σ t \ ∆ ft i E ( χ ft ,α f ( g ) · µ,c ft ( g,µ ) ,µ ) , where we have used the fact that α f ( g ) · c f ( α f ( g ) , ν ) = g . On the other hand, T f V s ( f ) ,α f ( g ) = X r ( t )= s ( f ) µ ∈ Σ t \ ∆ ft X r ( e )= r ( f ) ν ∈ Σ e i E ( χ ft ,µ, ,µ ) i A ( ǫ eα f ( g ) · ν,c e ( α f ( g ) ,ν ) ,ν )= X r ( t )= s ( f ) α f ( g ) · ν ∈ Σ t \ ∆ ft i E ( χ ft ,α f ( g ) · ν, ,α f ( g ) · ν · ǫ tα f ( g ) · ν,c ft ( g,ν ) ,ν )= X r ( t )= s ( f ) α f ( g ) · ν ∈ Σ t \ ∆ ft i E ( χ ft ,α f ( g ) · ν,c ft ( g,ν ) ,ν ) . After observing that α f ( g ) · ν ∈ Σ t \ ∆ ft if and only if ν ∈ Σ t \ ∆ ft we see that (G2) is satisfied.The existence of Φ now follows immediately from the universal property of C ∗ ( G ). (cid:3) We can now prove the main result of this section.
Proof of Theorem 4.1.
We claim that Ψ and Φ are mutually inverse isomorphisms. Recall thatΨ ◦ i A = π and Ψ ◦ i E = ψ . It follows that for each f ∈ Γ we have,(Ψ ◦ Φ)( s f ) = Ψ( T f ) = X r ( e )= s ( f ) µ ∈ Σ e \ ∆ fe s f s µe s ∗ µe = s f s ∗ f s f = s f , CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 20 and for each x ∈ Γ and g ∈ G x ,(Ψ ◦ Φ)( u x,g ) = Ψ( V x,g ) = X r ( e )= xµ ∈ Σ f s ( g · µ ) e u s ( e ) ,c e ( g,µ ) s µe = X r ( e )= xµ ∈ Σ f u r ( e ) ,g s µe s ∗ µe = u x,g (cid:16) X r ( e )= xµ ∈ Σ f s µe s ∗ µe (cid:17) = u x,g . Now suppose that ( µ, g, ν ) ∈ Σ e × G e × Σ e . Since { T e | e ∈ Γ } ∪ { V x | x ∈ Γ } forms a G -family,(Φ ◦ Ψ)( i A ( ǫ eµ,g,ν )) = Φ( s µe u s ( e ) ,α e ( g ) s ∗ νe ) = T µe V s ( e ) ,α e ( g ) T ∗ νe = V r ( e ) ,µα e ( g ) T e T ∗ e V ∗ r ( e ) ,ν = V r ( e ) ,µα e ( g ) i A ( ǫ e , , ) V ∗ r ( e ) ,ν , where we have used (24) for the last equality. Now using (26) we see that, V r ( e ) ,µα e ( g ) i A ( ǫ e , , ) V ∗ r ( e ) ,ν = X r ( f )= r ( e ) σ ∈ Σ f X r ( t )= r ( e ) ρ ∈ Σ t i A ( ǫ fµα e ( g ) · σ,c f ( µα e ( g ) ,σ ) ,σ ) i A ( ǫ e , , ) i A ( ǫ tρ,c t ( ν,ρ ) − ,ν · ρ )= i A ( ǫ eµα e ( g ) · ,c e ( µα e ( g ) , c e ( ν, − ,ν · )= i A ( ǫ eµ,g,ν ) . Now suppose that ( µ, ν, g, σ ) ∈ X fe . Then,(Φ ◦ Ψ)( i E ( χ feµ,ν,g,σ )) = Φ( s µf s νe u s ( e ) ,α e ( g ) s ∗ σe ) = T µf T νe V s ( e ) ,α e ( g ) T ∗ σe . Using the computation of (Φ ◦ Ψ)( i A ( ǫ eµ,g,ν )) above we find that, T µf T νe V s ( e ) ,α e ( g ) T ∗ σe = T µf V r ( e ) ,να e ( g ) T e T ∗ e V ∗ r ( e ) ,σ = T µf i A ( ǫ eν,g,σ )= V r ( f ) ,µ X r ( t )= s ( f ) ρ ∈ Σ t \ ∆ ft i E ( χ ft ,ρ, ,ρ ) i A ( ǫ eν,g,σ )= V r ( f ) ,µ i E ( χ fe ,ν,g,σ )= X r ( t )= r ( e ) ρ ∈ Σ t i A ( ǫ tµ · ρ,c f ( µ,ρ ) ,ρ ) i E ( χ fe ,ν,g,σ )= i A ( χ fµ · ,c f ( µ, , ) i E ( χ fe ,ν,g,σ )= i E ( χ feµ,ν,g,σ ) . It now follows that Φ and Ψ are mutually inverse isomorphisms. Gauge-equivariance followsfrom the definition of either Φ or Ψ. (cid:3) O D as an Exel-Pardo algebra We now make a return the graph of groups correspondence ( ϕ, D ) and its Cuntz-Pimsneralgebra O D . In many ways, O D is more elementary than O E , and since the algebras are Moritaequivalent by Proposition 3.13, for many purposes we can instead consider O D . Cuntz-Pimsneralgebras often admit a description in terms of generators and relations, and O D is no exception. Proposition 5.1.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups.Then O D is the universal C ∗ -algebra generated by a collection of partial isometries { s fµe | f e ∈ CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 21 Γ , µ ∈ Σ e \ ∆ fe } and a family of partial unitary representations u e : g u e,g of G e for each e ∈ Γ satisfying the relations:(D1) u e, u f, = 0 for each e, f ∈ Γ with e = f ;(D2) u f,g s fµe = s f ( α f ( g ) · µ ) e u e,c fe ( g,µ ) for each f e ∈ Γ and g ∈ G f ;(D3) s ∗ fµe s fµe = u e, for all f e ∈ Γ and µ ∈ Σ e \ ∆ fe ;(D4) X r ( e )= s ( f ) µ ∈ Σ e \ ∆ fe s fµe s ∗ fµe = u f, for all f ∈ Γ .Proof. Let ( i B , i D ) denote the universal Cuntz-Pimsner covariant representation of ( ϕ, D ) in O D . For each e ∈ Γ , let ǫ eg ∈ C c ( G e ) ⊆ C ∗ ( G e ) denote the point mass at g ∈ G e . For each f e ∈ Γ and µ ∈ Σ e \ ∆ fe let χ feµ denote the identity for the µ -th copy of C ∗ ( G e ) in the directsum D fe . Then set s fµe := i E ( χ feµ ) and u e,g := i A ( ǫ eg ) . Using the fact that ( i B , i D ) is a Cuntz-Pimsner covariant representation of ( ϕ, D ) it is straight-forward to check that each s fµe is a partial isometry, each u e : g u e,g is a partial uni-tary representation of G e in D , and the relations (D1)–(D4) are satisfied. Since the module D = span { χ feµ · a | f e ∈ Γ , µ ∈ Σ e \ ∆ fe , a ∈ C ∗ ( G e ) } , it follows that O D is generated by { s fµg | f e ∈ Γ , µ ∈ Σ e \ ∆ fe } ∪ { u e,g | e ∈ Γ , g ∈ G e } . Now let C be a C ∗ -algebra generated by partial isometries { t fµe | f e ∈ Γ , µ ∈ Σ e \ ∆ fe } anda family of partial unitary representations v e : g v e,g of G e satisfying (D1)–(D4). Since B isa direct sum of full group C ∗ -algebras, universality yields a ∗ -homomorphism π : B → C suchthat π ( ǫ eg ) = v e,g .Let f e, pq ∈ Γ , µ ∈ Σ e \ ∆ fe , ν ∈ Σ q \ ∆ pq , g ∈ G e , and h ∈ G q . It follows from (D1) and (D4)that t fµe t ∗ fµe and t pνq t ∗ pνq are mutually orthogonal projections unless f µe = pνq . Accordingly,(D3) and (6) imply that( t fµe u e,g ) ∗ t pνq u q,h = v ∗ e,g t ∗ fµe t pνq v q,h = v ∗ e,g t ∗ fµe t fµe t ∗ fµe t pνq t ∗ pνq t pνq v q,h = δ fµe,pνq v ∗ e,g t ∗ fµe t fµe v e,h = δ fµe,pνq v e,g − h = δ fµe,pνq π ( ǫ eg − h )= π (( χ feµ · ǫ eg | χ pqν · ǫ qh ) B ) . (27)In particular, k ξ feµ · ǫ eg k ≤ k t fµe v e,g k . It follows that there is a norm-decreasing linear map ψ : D → C satisfying ψ ( χ feµ · ǫ eg ) = t fµe v e,g for all f e ∈ Γ , µ ∈ Σ e \ ∆ fe and g ∈ G e . As ψ ( χ feµ · ǫ eg ) = ψ ( χ feµ ) π ( ǫ eg ) it follows from a standard argument that ψ ( ξ · a ) = ψ ( ξ ) π ( a ) for all ξ ∈ D and a ∈ B . Similarly, (27) implies that ψ ( ξ ) ∗ ψ ( η ) = π (( ξ | η ) B ) for all ξ, η ∈ D .To see that the left action is respected, observe that (10) together with (D2) imply that π ( ǫ qg ) ψ ( χ feµ ) = δ q,f v f,g t fµe = δ q,f t f ( α f ( g ) · µ ) e v e,c fe ( g,µ ) = δ q,f ψ ( χ feα f ( g ) · µ · ǫ ec fe ( g,µ ) ) = ψ ( ϕ ( ǫ qg ) χ feµ ) . Again, a standard argument shows that ψ ( ϕ ( a ) ξ ) = π ( a ) ψ ( ξ ) for all ξ ∈ D and a ∈ B . Hence,( π, ψ ) is a representation of ( ϕ, D ) in C .For Cuntz-Pimsner covariance, we use the fact that ( χ feµ ) fe ∈ Γ ,µ ∈ Σ e \ ∆ fe constitutes a framefor D . So ϕ ( a ) = P fe ∈ Γ P µ ∈ Σ e \ ∆ fe Θ ϕ ( a ) χ feµ ,χ feµ . It then follows from (D4) that for all f ∈ e CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 22 and g ∈ G e , ψ (1) ◦ ϕ ( ǫ eg ) = ψ (1) (cid:16) X r ( e )= s ( f ) µ ∈ Σ e \ ∆ fe Θ ϕ ( ǫ eg ) χ feµ ,χ feµ (cid:17) = X r ( e )= s ( f ) µ ∈ Σ e \ ∆ fe π ( ǫ eg ) ψ ( χ feµ ) ψ ( χ feµ ) ∗ = X r ( e )= s ( f ) µ ∈ Σ e \ ∆ fe v e,g t fµe t ∗ fµe = v e,g = π ( ǫ eg ) . It once again follows from a standard argument that ψ (1) ◦ ϕ ( a ) = π ( a ) for all a ∈ B , so ( ϕ, D )is Cuntz-Pimsner covariant.The universal property of O D as a Cuntz-Pimsner algebra yields a unique ∗ -homomorphismΦ : O D → C such that Φ ◦ i D = ψ and Φ ◦ i B = π . Moreover, Φ is the unique ∗ -homomorphismsatisfying Φ( s fµe ) = t fµe and Φ( u e,g ) = v e,g for all f e ∈ Γ , µ ∈ Σ e \ ∆ fe , and g ∈ G e . Hence, O D is universal for the relations (D1)–(D4). (cid:3) Remark . Similar considerations to the proof of Proposition 5.1 can also provide a set ofgenerators and relations for O E ∼ = C ∗ ( G ) which differ from the usual G -family generators. Weomit this description as it is more complicated than the corresponding one for O D , and we donot make use of it.Exel and Pardo [EP17] introduced a class of C ∗ -algebras, now referred to as Exel-Pardoalgebras , which simultaneously generalise Nekrashevych’s algebras [Nek04, Nek09] for self-similaractions, as well as class of C ∗ -algebras introduced by Katsura [Kat08] which describe all UCTKirchberg algebras. Definition 5.3 ([EP17]) . Let G be a countable discrete group acting on a finite directed graphΛ = (Λ , Λ , r, s ), and suppose that c : G × Λ → G is a cocycle for the action satisfying c ( g, e ) · x = g · x for all g ∈ G , e ∈ Λ , and x ∈ Λ . The Exel-Pardo algebra O Λ ,G is the universalunital C ∗ -algebra generated by { p x | x ∈ Λ } ∪ { v e : e ∈ Λ } ∪ { w g : g ∈ G } subject to relations:(EP1) { p x | x ∈ Λ } ∪ { v e : e ∈ Λ } is a Cuntz-Krieger Λ-family in the sense of [Rae05];(EP2) g w g is a unitary representation of G ;(EP3) w g v e = v g · e w c ( g,e ) for all g ∈ G and e ∈ Λ ; and(EP4) w g p x = p g · x w g for all g ∈ G and x ∈ Λ .We claim that for a graph of groups with finite underlying graph, the algebra O D is an Exel-Pardo algebra and give an explicit construction. To this end, let G = (Γ , G ) be a locally finitenonsingular graph of countable discrete groups with a fixed choice of transversals Σ e for each e ∈ Γ . Moreover, suppose that Γ is actually finite.Consider the directed graph Λ G = (Λ G , Λ G , r Λ , s Λ ) with vertices Λ G = Γ , edges Λ G = { f µe | f e ∈ Γ , µ ∈ Σ e \ ∆ fe } , range map r Λ ( f µe ) = f , and source map s Λ ( f µe ) = e . Local finitenessof G implies that Λ is locally finite, and nonsingularity of G implies that Λ has no sources. Notethat if Σ e = { } , then there are no edges from the vertex e to e .Define an action of g = ( g k ) k ∈ Γ ∈ Q e ∈ Γ G e on Λ G by g · e = e and g · f µe = ( f ( α f ( g f ) · µ ) e if k = f,f µe otherwise . (28) CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 23 for all e ∈ Λ G and f µe ∈ Λ G . The structure of the directed graph Λ G and the action of Q G e on Λ G are independent of the choice of transversals. For each f ∈ Γ let ι f : G f → Q G e denotethe natural inclusion and define a map c : Q G e × Λ G → Λ G by c ( g, f µe ) = ι e ◦ c fe ( g f , µ ) . (29)Since each c fe is a cocycle it follows that c is a cocycle for the action of Q G e on Λ G . Moreover,since the action of Q G e fixes Λ G , the cocycle trivially satisfies c ( g, f µe ) · e = g · e for all e ∈ Λ G . Theorem 5.4.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable discrete groupswith Γ a finite graph. Then O D is isomorphic to the Exel-Pardo algebra O Λ G , Q e ∈ Γ1 G e withcocycle c given by (29) .Proof. We show that the defining relations for O D from Proposition 5.1 agree with the definingrelations of the Exel-Pardo algebra O Λ G , Q e ∈ Γ1 G e with cocycle c . Suppose that { u e,g : e ∈ Γ , g ∈ G e } ∪ { s fµe : f µe ∈ Λ G } satisfy the hypotheses of Proposition 5.1, and that { p x | x ∈ Λ G } ∪{ v e : e ∈ Λ G } ∪ { w g : g ∈ G } satisfy the hypotheses of Definition 5.3.Clearly (D1), (D3), and (D4) are equivalent to { u e, : e ∈ Γ } ∪ { s fµe : f µe ∈ Λ G } beinga Cuntz-Krieger Λ G -family. Moreover, we have the following correspondence between the twofamilies of elements: p e ←→ u e, v fµe ←→ s fµe w g ←→ X e ∈ Γ u e,g e p e w ι e ( h ) p e ←→ u e,h . Since Γ is finite, P e ∈ Γ u e, = 1. So g P e ∈ Γ u e,g e is a unitary representation of Q G e ,and h p e w ι e ( h ) p e is a partial unitary representation of G e . The condition (D2) correspondsdirectly to the condition (EP3), and (EP4) is trivally satisfied since the action of Q G e on Λ G fixes vertices. (cid:3) Example . If G = (Γ , G ) is a graph of groups with trival edge groups, then Q G e is trivial and O Λ G , { } is just the graph C ∗ -algebra C ∗ (Λ G ). Note that Λ G is typically not the directed graph E G of [BMPST17, Theorem 3.6]. Example . Let G = (Γ , G ) be the graph of groups associated to BS (1 ,
2) from Example 2.5.Then Λ G is the directed graph defined by the diagram ee e e e ee ee e . The action of ( a, b ) ∈ G e × G e ∼ = Z on Λ G is such that ( a,
0) acts trivially, while (0 , b ) fixes both e e and e e , and (0 , b ) · eke = e (( b + k ) mod 2) e. The cocycle c is given by c (( a, b ) , x ) = (cid:0) , b + k − (( b + k ) mod 2)2 (cid:1) if x = eke, (0 , a ) if x = e e, (2 a,
0) if x = e e. CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 24 For this example, the triple ( Z , Λ G , c ) shares similarities with both [EP17, Example 3.4] and[EP17, Example 3.3], but is not an instance of either example.We finish this section with a summary of results. Recall that the Bass-Serre Theorem [Ser80,Theorem 13] implies that if G is a group acting on a tree X , then G is isomorphic to thefundamental group of the associated quotient graph of groups G , and X is G -equivariantlyisomorphic to the universal covering tree of G . The following is now a synthesis of Theorem 5.4,Proposition 3.13, Theorem 4.1, and Theorem 2.10. Corollary 5.7.
Suppose that G is a countable group acting without inversions on a locally finitenonsingular tree X , and let τ denote the induced action on C ( ∂X ) . Let G = (Γ , G ) denotequotient graph of groups associated to the action of G on X . Then with ( ϕ, D ) the graph ofgroups correspondence for G and ( ϕ, E ) as the amplified graph of groups correspondence, O D ∼ me O E ∼ = C ∗ ( G ) ∼ me C ( ∂X G ) ⋊ τ π ( G ) ∼ = C ( ∂X ) ⋊ τ G, where ∼ me denotes Morita equivalence. In addition, if Γ = G \ X is finite, then O Λ G , Q e ∈ Y G e is isomorphic to O D . K -theory The K -theory of Cuntz-Pimsner algebras is well-understood. In his seminal paper, Pimsner[Pim97] showed that for a large class of C ∗ -correspondences, the K -theory of the associatedCuntz-Pimsner algebra could be computed using the Kasparov class of the correspondence.Although we make use of KK -theory in this section, the Kasparov products we consider arerelatively tame, and we will not need Z -graded C ∗ -algebras. Definition 6.1. An odd Kasparov A – B -module ( A, φ E B , V ) consists of a countably generatedungraded right Hilbert B -module E , with φ : A → End B ( E ) a ∗ -homomorphism, together with V ∈ End B ( E ), such that a ( V − V ∗ ) , a ( V − , [ V, a ] are all compact endomorphisms for all a ∈ A .An even Kasparov A – B -module has, in addition, a grading by a self-adjoint endomorphism Γsatisfying Γ = 1, φ ( a )Γ = Γ φ ( a ) for all a ∈ A , and V Γ + Γ V = 0.We will not describe the equivalence relation on (even) Kasparov A – B -modules which yields KK ( A, B ), referring the reader to [Kas80]. We note that correspondences with compact leftaction define even KK -classes, by taking the operator V to be 0. The Kasparov product of suchcorrespondences is just the balanced tensor product.For each f e ∈ Γ let [ E fe ] := [( A f , E fe , ∈ KK ( A f , A e ) and [ D fe ] := [( B f , D fe , ∈ KK ( B f , B e ) denote the elements induced by the correspondences E fe and D fe . Recall that A e and B e are Morita equivalent via the bimodule F e and E fe = F f ⊗ B f D fe ⊗ B e F ∗ e . Moreover,since E and D are direct sums, Proposition 3.13 implies that A and B are also Morita equivalent.Conjugating with F = L e ∈ Γ F e gives an isomorphism KK ( B, B ) → KK ( A, A ) which carries[ D ] to [ E ].For i = 0 , · ⊗ B [ D ] : K i ( B ) → K i ( B ) can be thought of as an“adjacency matrix” of group homomorphisms indexed by Γ × Γ : the entry correspondingto ( f, e ) ∈ Γ × Γ is the group homomorphism · ⊗ A f [ D fe ] : K i ( C ∗ ( G f )) → K i ( C ∗ ( G e )) if s ( f ) = r ( e ), and the zero map otherwise.Recall that Proposition 3.13 together with Theorem 4.1 imply that O D is Morita equivalent to C ∗ ( G ). We have the following six-term sequence in K -theory which is analogous to the sequencefor directed graph C ∗ -algebras (cf. [Rae05, Theorem 7.16]). Theorem 6.2.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups. For i = 0 , let Λ i = · ⊗ A (id − [ D ]) : K i ( B ) → K i ( B ) . Let m : K ∗ ( O D ) → K ∗ ( C ∗ ( G )) denote the CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 25 isomorphism induced by Morita equivalence of Corollary 5.7 and let i B : B ֒ → O D denote theuniversal inclusion. Then the following six-term sequence of Abelian groups is exact: L e ∈ Γ K ( C ∗ ( G e )) L e ∈ Γ K ( C ∗ ( G e )) K ( C ∗ ( G )) K ( C ∗ ( G )) L e ∈ Γ K ( C ∗ ( G e )) L e ∈ Γ K ( C ∗ ( G e ))Λ m ◦ ( i B ) ∗ ∂ Λ m ◦ ( i B ) ∗ ∂ .Proof. This follows immediately from [Pim97, Theorem 4.9]. (cid:3)
Remark . The corresponding six-term sequences in KK -theory (cf. [Pim97, Theorem 4.9])also hold with the caveat of nuclearity being inserted where appropriate.Corollary 5.7 implies that Theorem 6.2 can be reinterpreted in terms of group actions on theboundary of a tree. Corollary 6.4.
Suppose that G is a countable group acting without inversions on a locallyfinite nonsingular tree X and let τ denote the induced action on C ( ∂X ) . Suppose that Y =( Y , Y , r Y , s Y ) is a fundamental domain for the action of G on X . For each e ∈ Y let G e = stab G ( e ) . Let m : K ∗ ( O D ) → K ∗ ( C ( ∂X ) ⋊ τ G ) denote the isomorphism induced by Moritaequivalence and let ι B : B → O D denote the universal inclusion. Then the following six-termsequence of abelian groups is exact: L e ∈ Y K ( C ∗ ( G e )) L e ∈ Y K ( C ∗ ( G e )) K ( C ( ∂X ) ⋊ τ G ) K ( C ( ∂X ) ⋊ τ G ) L e ∈ Y K ( C ∗ ( G e )) L e ∈ Y K ( C ∗ ( G e ))Λ m ◦ ( i B ) ∗ ∂ Λ m ◦ ( i B ) ∗ ∂ .Proof. Let G denote the quotient graph of groups associated to the action of G on X as outlinedin Section 2. In particular, Γ ≃ G \ X and for each e ∈ Γ the group G e is the stabiliser of theunique lifted edge e ∈ Y . Since O D is Morita equivalent to C ( ∂X ) ⋊ τ G , Theorem 6.2 yieldsthe result. (cid:3) Remark . Corollary 6.4 bears a resemblance to a special case of [Pim86, Theorem 16]. How-ever, here only the edge stabilizers are explicitly used to compute the K -theory of C ( ∂X ) ⋊ G .The data concerning the vertex stabilizers is encoded within the module E . It is not immediatelyclear to the authors how to translate between these two pictures.We now move to compute the K -theory of some examples. Although the maps Λ i involvetaking a Kasparov product, the actual computations are relatively tame in simple examples. Inwhat follows, we let 1 e denote the identity in C ∗ ( G e ) for all e ∈ Γ . Lemma 6.6.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups. For each f e ∈ Γ , [1 f ] ⊗ B [ D fe ] = ( | Σ e | − δ f,e )[1 e ] , where [1 f ] ∈ K ( C ∗ ( G f )) ∼ = KK ( C , C ∗ ( G f )) .Proof. Since the left action of C ∗ ( G f ) on D fe is unital, it follows that[1 f ] ⊗ B [ D fe ] = h(cid:16) C , M µ ∈ Σ e \ ∆ fe C ∗ ( G e ) , (cid:17)i = ( | Σ e | − δ f,e )[1 e ] . (cid:3) CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 26 Lemma 6.7.
Let G = (Γ , G ) be a locally finite nonsingular graph of countable groups. Then foreach e ∈ Γ we have [ D ee ] = ( | Σ e | −
1) id KK ( C ∗ ( G e ) ,C ∗ ( G e )) in KK ( C ∗ ( G e ) , C ∗ ( G e )) , where we identify C ∗ ( G e ) = C ∗ ( G e ) .Proof. The result follows immediately from (10), since the left and right actions of C ∗ ( G e ) on D ee = L µ ∈ Σ e \{ } C ∗ ( G e ) agree after identifying G e with G e . Consequently,[( C ∗ ( G e ) , D e , | Σ e |− X i =1 id KK ( C ∗ ( G e ) ,C ∗ ( G e )) . Note that in the case where α e : G e → G r ( e ) is surjective we have [ D ee ] = 0. (cid:3) Edges of groups.
We can now say something about the K -theory of the C ∗ -algebra ofan edge of groups, that is a graph of groups consisting of a single edge. Consider the edge ofgroups, G = G v G w G e α e α e . Assuming that G is non-singular it follows that | Σ e | ≥ | Σ e | ≥
2. Then K i ( B ) ∼ = K i ( C ∗ ( G e )) ⊕ K i ( C ∗ ( G e )). The only paths of length two in G are ee and ee so [ D ] = [ D ee ] ⊕ [ D ee ].Lemma 6.7 then implies that for a class ( x, y ) ∈ K i ( B ) we have( x, y ) ⊗ B (id ∗ − [ D ]) = (cid:0) x − ( | Σ e | − y, y − ( | Σ e | − x (cid:1) . (30)We consider a few specific cases of edges of groups. Example . Fix m, n ≥ G = ( G, Γ) be an edge of groups with G v = Z /n Z , G w = Z /m Z , G e = { } , and monomorphisms α e and α e being the inclusion of { } . Then | Σ e | = n and | Σ e | = m . In this case we have C ∗ ( G e ) = C ∗ ( G e ) ∼ = C so K ( B ) = Z [1 e ] ⊕ Z [1 e ] and K ( B ) = 0 . Applying the six-term sequence of Theorem 6.2, it follows that K ( C ∗ ( G )) ∼ = coker(Λ ) and K ( C ∗ ( G )) ∼ = ker(Λ ) where Λ : Z [1 e ] ⊕ Z [1 e ] → Z [1 e ] ⊕ Z [1 e ] is the Z -linear map given by(30). Treating [1 e ] as the column ( ) T and [1 e ] as the column ( ) T the map Λ has matrixrepresentation (cid:18) n − m − (cid:19) with Smith normal form (cid:18) − ( n − m − (cid:19) . It now follows that K ( C ∗ ( G )) ∼ = Z (1 − ( n − m − Z and K ( C ∗ ( G )) ∼ = ( Z if m = n = 2;0 otherwise.Since the edge groups are trivial in this example, [BMPST17, Theorem 3.6] shows that C ∗ ( G )is isomorphic to the C ∗ -algebra of a directed graph E G . One can readily check that the K -theorycomputed above agrees with that of the directed graph C ∗ -algebra C ∗ ( E G ). For the specific caseof the action of P SL ( Z ) ∼ = Z ∗ Z on the boundary ∂X of the tree X considered in Example 2.4we have K i ( C ( ∂X ) ⋊ P SL ( Z )) = 0 for i = 0 , Example . Fix m, n ∈ Z such that | m | , | n | ≥ G = ( G, Γ) be an edge of groups with G v = G w = G e = Z . Let α e denote multiplication by n and let α e denote multiplication by CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 27 m . Then | Σ e | = | n | , | Σ e | = | m | , and C ∗ ( G e ) = C ∗ ( G e ) ∼ = C ( T ). Letting u e denote the unitary z z in K ( C ∗ ( G e )), it follows that K ( B ) = Z [1 e ] ⊕ Z [1 e ] and K ( B ) = Z [ u e ] ⊕ Z [ u e ] . For i = 0 , i ) are free abelian since they are subgroups of the free abelian group K i ( B ). It now follows from Theorem 6.2 that K i ( C ∗ ( G )) ∼ = ker(Λ − i ) ⊕ coker(Λ i ) . A similarargument to Example 6.8 now shows that for i = 0 , K i ( C ∗ ( G )) ∼ = ( Z if | n | = | m | = 2; Z (1 − ( | n |− | m |− Z otherwise.6.2. Graphs of trivial groups.
Suppose that G = (Γ , G ) is a locally finite nonsingular graphof groups for which each edge group and vertex group is trivial. If Γ is also a finite graph, thenthe algebra C ∗ ( G ) is the Cuntz-Krieger algebra associated to Γ by Cornelissen, Lorscheid, andMarcolli in [CLM08] (see [BMPST17, Remark 3.10]). In this case C ∗ ( G e ) ∼ = C for all e ∈ Γ ,and for all f e ∈ Γ , D fe ∼ = ( C if f = e ;0 if f = e. In particular, K ( B ) = L e ∈ Γ Z [1 e ]. Since K ( B ) = 0, it follows from the six-term sequence ofTheorem 6.2 that K ( C ∗ ( G )) ∼ = coker(Λ ) and K ( C ∗ ( G )) ∼ = ker(Λ ) . On generating elements [1 f ] ∈ K ( B ) we have[1 f ] ⊗ B [ D ] = [1 f ] ⊗ B X r ( e )= s ( f ) [ D fe ] = (cid:16) X r ( e )= s ( f ) [1 e ] (cid:17) − [1 f ] , so that Λ ([1 f ]) = [1 f ] + [1 f ] − X r ( e )= s ( f ) [1 e ] . In particular, the product Kasparov product · ⊗ B [ D ] : K ( B ) → K ( B ) agrees with the operator T of [CLM08, Section 2.2]. Accordingly, if Γ is a finite graph then [CLM08, Theorem 1] can beused to compute K ( C ∗ ( G )) and K ( C ∗ ( G )) in terms of the number of cycles in the graph Γ.6.3. Generalised Baumslag-Solitar graphs of groups. A generalised Baumslag-Solitar(GBS) graph of groups is a graph of groups where all of the edge and vertex groups areisomorphic to Z . The fundamental group of such a graph of groups is called a generalisedBaumslag-Solitar (GBS) group . A survey of results concerning GBS groups has been compiledby Robinson [Rob15].For the remainder of this subsection, let G = (Γ , G ) be a GBS graph of groups. For each e ∈ Γ we have G e ∼ = Z , and so C ∗ ( G e ) ∼ = C ( T ), and K i ( C ∗ ( G e )) ∼ = Z for i = 0 , Proposition 6.10.
Let G = (Γ , G ) be a locally finite nonsingular GBS graph of groups. Let f e ∈ Γ with f = e and suppose that α e is given by multiplication by n , and α f is given by mul-tiplication by m for some n, m ∈ Z \ { } . Then the Kasparov product · ⊗ A [ D fe ] : K i ( C ∗ ( G f )) → K i ( C ∗ ( G e )) acts as multiplication by | n | on K and multiplication by sgn( n ) m on K , where sgn( x ) = x/ | x | is the sign function.Proof. The K -statement follows from Lemma 6.6 and the fact K ( C ( T )) is generated by theclass of the unit. For K fix a transversal Σ e = { , , . . . , | n | − } . The action of x ∈ G r ( e ) on µ ∈ Σ e is given by x · µ = ( x + µ ) mod | n | . Using (2) the corresponding cocycle is given by c e ( x, µ ) = x + µ − x · µn : the result of integer division of x + µ by n . The induced action of x ∈ G f CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 28 via f on µ ∈ Σ e is given by x · µ = ( mx + µ ) mod | n | . The cocycle for the action of G f is givenby c fe ( x, µ ) = mx + µ − x · µn .It follows from Equation (10) that the left action of the point mass ǫ fn ∈ C c ( Z ) ⊂ C ∗ ( G f ) on ξ ∈ C c ( Z ) | n | ⊂ D fe is given by ϕ fe ( ǫ fn ) ξ ( µ, k ) = ξ (cid:0) ( − n ) · µ, c fe ( − n, µ ) + k (cid:1) = ξ ( µ, k − m ) = ( ξ · ǫ em )( µ, k ) . Let u f and u e denote the unitary z z in C ∗ ( G f ) and C ∗ ( G e ), respectively. The precedingcomputation implies that ϕ fe ( u nf ) ξ = ξ · u me for all ξ ∈ D fe .Recall that the group K ( C ( T )) ∼ = Z is generated by the class of the unitary z z . Tofacilitate the computation of Kasparov products we utilise a special case of [BKR19, Section 3].By utilising the isomorphism K ( C ( T )) ∼ = KK ( C , C ( T )) we obtain a new generator, the classof the (unbounded) Kasparov module ( C , Ξ , C ( u )), where C ( u ) denotes the Cayley transform ofthe unitary u : z z . In particular, C ( u ) is the unbounded regular self-adjoint operator withdomain dom( C ( u )) = ( u − C ( T ) satisfying C ( u ) v = i u +1 u − v for all v ∈ dom( C ( u )); and Ξ is theright C ( T )-module given by the closure of dom( C ( u )) in the right C ( T )-module C ( T ).For each n ∈ G f denote the closure of ( u nf − C ( T ) in C ( T ) by Ξ f,n . In C ∗ ( G f ) ⊗ C ∗ ( G f ) D fe we have ( u nf − ⊗ ξ = 1 ⊗ ϕ fe ( u nf − ξ = (1 ⊗ ξ ) · ( u me − ξ ∈ D fe . Consequently, Ξ f,n ⊗ C ∗ ( G f ) D fe is isomorphic to Ξ | n | e,m . Moreover, it follows thatin K ( C ∗ ( G e )), n [ u f ] ⊗ A [ D fe ] = [ u nf ] ⊗ A [ D fe ]= [( C , Ξ f,n , C ( u nf )] ⊗ C ∗ ( G f ) [( C ∗ ( G f ) , D fe , C , Ξ f,n ⊗ C ∗ ( G f ) D fe , C ( u nf ) ⊗ C , Ξ | n | e,m , C (diag( u me ))]= | n | [ u me ]= | n | m [ u e ] . Dividing through by n gives [ u f ] ⊗ A [ D fe ] = sgn( n ) m [ u e ]. (cid:3) Example . Fix m, n ∈ Z \ { } and consider the loop of groups G from Example 2.5 given by, G e = Z α e : k nkα e : k mk G v = Z . Then G is the quotient graph of groups for an action of the Baumslag-Solitar group BS ( n, m ) = h a, t | ta n t − = a m i acting on a regular ( | m | + | n | )-valent tree. The case where n = 2 and m = 1was considered in detail in Example 2.5.Observe that B = C ∗ ( G e ) ⊕ C ∗ ( G e ) ∼ = C ( T ) , so K ( B ) ∼ = Z [1 e ] ⊕ Z [1 e ] and K ( B ) ∼ = Z [ u e ] ⊕ Z [ u e ] . Treat [1 e ] as the column ( ) T and [1 e ] as the column ( ) T . Then according to bothProposition 6.10 and Lemma 6.7 the map Λ : K ( A ) → K ( A ) of Theorem 6.2 can be identifiedwith left multiplication by the matrix (cid:18) − | n | | n | − | m | − − | m | (cid:19) which has Smith normal form (cid:18) gcd(1 − | n | , − | m | ) 00 0 (cid:19) CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 29 by [Sta16, Theorem 2.4]. Here we recall that gcd(0 , a ) = a for a ∈ Z and that the gcd isdefined up to sign. As convention, we take the positive gcd, but this does not affect any furthercomputations. Consequently,ker(Λ ) ∼ = ( Z if | m | = | n | = 1; Z otherwise; and coker(Λ ) ∼ = Z gcd(1 − | n | , − | m | ) Z ⊕ Z . Now, treat [ u e ] as the column ( ) T and [ u e ] as the column ( ) T . Then Proposition 6.10together with Lemma 6.7 imply that the map Λ : K ( B ) → K ( B ) can be identified with leftmultiplication by the matrix (cid:18) − sgn( n ) m | n | − | m | − − sgn( m ) n (cid:19) . We consider separately the cases where mn > mn < mn >
0, in which case sgn( n ) m = | m | and sgn( m ) n = | n | . A similarcomputation to that of Λ shows that for mn ≥ ) ∼ = ( Z if mn = 1; Z otherwise; and coker(Λ ) ∼ = Z gcd(1 − | n | , − | m | ) Z ⊕ Z . Now suppose that mn < m ) n = −| n | and sgn( n ) m = −| m | . It follows from [Sta16,Theorem 2.4] that the matrix associated to Λ has Smith normal form, gcd(1 + | m | , − | m | , | n | , − | n | ) 00 | m | + | n | )gcd(1+ | m | , −| m | , | n | , −| n | ) ! . Observe that gcd(1 + | m | , − | m | , | n | , − | n | ) = ( mn is odd1 if mn is even,so ker(Λ ) = 0 and coker(Λ ) ∼ = ( Z Z ⊕ Z ( | m | + | n | ) Z if mn is odd; Z | m | + | n | ) Z if mn is even.Since K i ( B ) is free Abelian for i = 0 , K i ( C ∗ ( G )) ∼ = ker(Λ − i ) ⊕ coker(Λ i ). To summarise, K ( C ∗ ( G )) ∼ = Z if mn = 1; Z ⊕ Z gcd(1 −| n | , −| m | ) Z if mn > Z ⊕ Z gcd(1 −| n | , −| m | ) Z if mn <
0; and K ( C ∗ ( G )) ∼ = Z if mn = 1; Z ⊕ Z gcd(1 −| n | , −| m | ) Z if mn > Z ⊕ Z Z ) if mn = − Z ⊕ Z Z ⊕ Z ( | m | + | n | ) Z if mn < − Z ⊕ Z | m | + | n | ) if mn < − m = n = 1 then the universal covering tree X G is aninfinite 2-regular tree with 2 boundary points. Hence, C ∗ ( G ) ∼ = C ( ∂X G ) ⋊ π ( G ) ∼ = C ⋊ id Z ∼ = C ⊗ C ∗ ( Z ) ∼ = C ( T ) . CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 30 The K -groups for C ( T ) agree with the m = n = 1 case above. Similarly, if m = 1 and n = − C ∗ ( G ) ∼ = C ( ∂X G ) ⋊ π ( G ) ∼ = C ⋊ id BS (1 , − ∼ = C ⊗ C ∗ ( BS (1 , − ∼ = ( C ( T ) ⋊ γ Z ) , where γ ∈ Aut( C ( T )) is induced by conjugation on T . An application of the Pimsner-Voiculescusequence for C ( T ) ⋊ γ Z shows that the K -groups above when m = 1 and n = − K -groups of ( C ( T ) ⋊ γ Z ) .Finally, we remark that [BMPST17, Corollary 7.15] states that C ∗ ( G ) is a Kirchberg algebraif and only if | m | , | n | ≥ | m | 6 = | n | . Hence the preceding K -theory computations can beused to classify C ∗ ( G ) ∼ = O D in this case.For a general locally finite nonsingular GBS graph of groups we have the following result. Theorem 6.12.
Let G = (Γ , G ) be a locally finite nonsingular GBS graph of groups. Supposethat for each e ∈ Γ the map α e is given by multiplication by m e ∈ Z . Let e denote the identityin C ∗ ( G e ) ∼ = C ( T ) and let u e denote the unitary z z in C ∗ ( G e ) . Then K ( C ∗ ( G )) ∼ = coker(Λ ) ⊕ ker(Λ ) and K ( C ∗ ( G )) ∼ = coker(Λ ) ⊕ ker(Λ ) , where Λ : L e ∈ Γ Z [1 e ] → L e ∈ Γ Z [1 e ] and Λ : L e ∈ Γ Z [ u e ] → L e ∈ Γ Z [ u e ] are given by Λ ([1 f ]) = [1 f ] − ( | m f | − f ] − X r ( e )= s ( f ) f = e | m e | [1 e ] and Λ ([ u f ]) = [ u f ] − ( | m f | − u f ] − X r ( e )= s ( f ) f = e sgn( m e ) m f [ u e ] . Proof.
Since K ( B ) = L e ∈ Γ Z [1 e ] and K ( B ) = L e ∈ Γ Z [ u e ] are free Abelian it follows that K i ( C ∗ ( G )) ∼ = coker(Λ i ) ⊕ ker(Λ − i ). The descriptions of Λ and Λ follow immediately fromProposition 6.10 together with Lemma 6.7. (cid:3) Poincar´e Duality
In this section we focus our attention on the case where G is the loop of groups considered inExample 6.11. In this case Theorem 2.10 asserts that C ∗ ( G ) is isomorphic to C ( ∂X G ) ⋊ BS ( n, m ),where X G is an ( | m | + | n | )-regular tree acted on by the Baumslag-Solitar group BS ( n, m ) = h a, t | ta n t − = a m i , [BMPST17, p. 126].We ask the question whether C ( ∂X G ) ⋊ BS ( n, m ) satisfies Poincar´e duality in Kasparov theory.Poincar´e duality for hyperbolic groups acting on their Gromov boundary has been proved byEmerson [Eme03], and similar results have been obtained for the Ruelle algebras of a quitegeneral class of hyperbolic dynamical systems [KPW17]. The Baumslag-Solitar groups are quitedifferent from the class of hyperbolic groups, and so too is our method of proof.Here we utilise the methods of [RRS19]. There a systematic method for deciding if funda-mental classes in K -theory and K -homology exist for a given Cuntz-Pimsner algebra whosecoefficient algebra satisfies Poncar´e duality.In our case the coefficient algebra will be (Morita equivalent to) a direct sum of copies of C ∗ ( Z ) ∼ = C ( T ), which does indeed satisfy Poincar´e duality. Then [RRS19] provides necessaryand sufficient conditions on the dynamics to be able to lift the K -theory fundamental class for C ( T ) to a K -theory fundamental class for C ∗ ( G ). The existence of a K -homology fundamentalclass we leave open.Nevertheless, even with ‘half’ of the Poincar´e duality, the methods of [RRS19] give isomor-phisms between K -theory and K -homology. Thus we obtain a computation of the K -homology CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 31 for the C ∗ -algebras of the loops of groups from Example 6.11, in spite of the large amount oftorsion in the K -theory preventing the use of duality (i.e. index pairing) methods.Given an A – B -correspondence ( φ, X ), taking the conjugate module X ∗ gives a B – A -module.We write ♭ ( x ) ∈ X ∗ for the copy of x ∈ X . Denoting the opposite algebra of A by A op andsimilarly for B op , we may regard X ∗ as an A op – B op -module ( φ op , X ∗ ) via φ op ( a op ) ♭ ( x ) b op := ♭ ( φ ( a ∗ ) xb ∗ ) , a ∈ A, b ∈ B, x ∈ X. (31)The map ♭ : X → X ∗ is anti-linear. As shown in [RRS19, Section 2], the left B -valued innerproduct on X ∗ can also be reinterpreted as a right B op -valued inner product. Thus the conjugate( φ op , X ∗ ) of an A – B -correspondence is an A op – B op -correspondence.Analogous to the conjugate module, one can define the conjugate algebra. Given a C ∗ -algebra A , the conjugate algebra A has the same product and adjoint, but the conjugate scalarmultiplication, [RW98, p. 157]. The map a a ∗ gives a (complex linear) isomorphism A → A op .If A is commutative we of course have A ∼ = A op , and this proves that if A is a commutative C ∗ -algebra then A ∼ = A . Lemma 7.1.
With the convention C ( T ) = { } and defining D = C ( T ) | n | ⊕ C ( T ) | m | ⊕ C ( T ) | n |− ⊕ C ( T ) | m |− , (32) let ( ϕ, D ) be the graph of groups correspondence over B = C ( T ) ⊕ C ( T ) for the loop of groupsfrom Example 6.11, and let ( ϕ op , D ∗ op ) be the conjugate correspondence. Then D ∼ = D ∗ op , wherewe use the fact that C ( T ) op = C ( T ) ∼ = C ( T ) .Proof. As above, the conjugate correspondence ( ϕ op , D ∗ op ) is a C ( T )– C ( T )-correspondence. Weobserve that the left action on D is highly non-trivial, while the right action is just componen-twise multiplication, and this remains true (up to a conjugation) for D ∗ op , as Equation (31) or[RRS19, p. 1121] shows.Rather than using the map ♭ : D → D ∗ op we utilise the fact that D is trivial (as a rightmodule) to define S : D → D ∗ op to be ♭ ◦ ∗ where ∗ is the componentwise C ∗ -adjoint. Then S is a linear map, and intertwines the actions of C ( T ) ⊕ C ( T ) since S ( ϕ ( a ) xb ) = ♭ ( b ∗ ( x ) ∗ i ϕ ( a ∗ ) ji ) = ♭ ( ϕ ( a ∗ ) ji ( x ) ∗ i b ∗ ) = ϕ op ( a op ) ♭ ( x ∗ ) b op using the commutativity of C ( T ). That S is a bijection is clear. (cid:3) The same argument shows that [ D ⊗ A op ] = [ A ⊗ D ∗ op ] ∈ KK ( C , A ⊗ A op ). Theorem 7.2.
Consider a loop of groups G as in Example 6.11, so that C ∗ ( G ) ∼ = C ( ∂X G ) ⋊ BS ( m, n ) where BS ( m, n ) is a Baumslag-Solitar group. Then there exists a K -theory class δ ∈ KK ( C , C ∗ ( G ) ⊗ C ∗ ( G )) such that the Kasparov product with δδ ⊗ C ∗ ( G ) · : K ∗ ( C ∗ ( G )) → K ∗ ( C ∗ ( G )) is an isomorphism.Proof. First of all C ∗ ( G ) ∼ = C ∗ ( G ) op via the inverse map of the transformation groupoid un-derlying the crossed product C ( ∂X G ) ⋊ BS ( n, m ), [BS17, Theorem 2.1]. The algebra B forthese examples is C ( T ) ⊕ C ( T ), which satisfies Poincar´e duality, [Kas88], [RRS19, Lemma 3.5],having fundamental class the direct sum of the fundamental classes for the summands. Let[ z ] ∈ KK ( C , C ( T )) denote the class of the unitary z z , and let [ ι C ,C ( T ) ] ∈ KK ( C , C ( T ))denote the class of the unital inclusion of C into C ( T ). The K -theory fundamental class for thecircle is β = [ z ] ⊗ [ ι C ,C ( T ) ] − [ ι C ,C ( T ) ] ⊗ [ z ] ∈ KK ( C , C ( T ) ⊗ C ( T )), and we let β = β ⊕ β be thecorresponding class for B = C ( T ) ⊕ C ( T ). CUNTZ-PIMSNER MODEL FOR THE C ∗ -ALGEBRA OF A GRAPH OF GROUPS 32 We summarise the argument of [RRS19, Theorem 4.8]. Using [ D ] = [ D ∗ op ], [ D ⊗ B ] =[ B ⊗ D ∗ op ] and the definition β ⊗ B [ D ] := β ⊗ B ⊗ B [ D ⊗ C ( T )] , β ⊗ B [ D ∗ op ] := β ⊗ B ⊗ B [ C ( T ) ⊗ D ∗ op ] , we see that β ⊗ B [ D ] = β ⊗ B [ D ∗ op ]. Hence, with ι : B → C ∗ ( G ) the inclusion and ∂ ∗ , ∂ ∗ denoting boundary maps, the diagram · · · / / K ( B ) − D / / β ⊗ (cid:15) (cid:15) K ( B ) ∂ ∗ / / β ⊗ (cid:15) (cid:15) K ( C ∗ ( G )) ι ∗ / / δ ⊗ (cid:15) (cid:15) K ( B ) − D / / β ⊗ (cid:15) (cid:15) K ( B ) ∂ ∗ / / β ⊗ (cid:15) (cid:15) · · ·· · · / / K ( B ) − D / / K ( B ) ι ∗ / / K ( C ∗ ( G )) ∂ ∗ / / K ( B ) − D / / K ( B ) ι ∗ / / · · · commutes if and only if( δ ⊗ · ) ◦ ∂ ∗ = ι ∗ ◦ ( β ⊗ · ) and ∂ ∗ ◦ ( δ ⊗ · ) = ( β ⊗ · ) ◦ ι ∗ . (33)As the Kasparov product β ⊗ B · : K j ( B ) → K j +1 ( B ) is an isomorphism for j = 0 ,
1, the fivelemma tells us that if if these two conditions on δ are satisfied, the Kasparov product with δ will provide an isomorphism.Finally, since β ⊗ B [ D ] = β ⊗ B [ D ∗ op ], a class δ satisfying the two conditions (33) can beexplicitly constructed. The recipe is given in [RRS19, Lemma 4.7]. (cid:3) Corollary 7.3.
The K -homology of B ( m, n ) is given by K ( C ∗ ( G )) ∼ = Z if mn = 1; Z ⊕ Z gcd(1 −| n | , −| m | ) Z if mn > Z ⊕ Z gcd(1 −| n | , −| m | ) Z if mn < and K ( C ∗ ( G )) ∼ = Z if mn = 1; Z ⊕ Z gcd(1 −| n | , −| m | ) Z if mn > Z ⊕ Z Z ) if mn = − Z ⊕ Z Z ⊕ Z ( | m | + | n | ) Z if mn < − is odd ; and Z ⊕ Z | m | + | n | ) if mn < − is even.Proof. The class δ provides an isomorphism K j ( C ∗ ( G )) ∼ = K j ( C ∗ ( G op )) ∼ = K j ( C ∗ ( G op )). (cid:3) No method employing duality (index) pairings between K -theory and K -homology can obtainthis result when mn = 1 due to the torsion subgroups: see for instance [HR00, Chapter 7]. References [Bas93] H. Bass, “Covering theory for graphs of groups,”
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School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW2522, Australia
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A. Mundey: [email protected]