Notes on C*-algebras of graphs A. Kumjian Abstract. C*-algebras of graphs include up to strong Morita equivalence all af algebras, Cuntz-Krieger algebras and the Toeplitz algebra. Our survey discusses recent developments and related constructions.

§1 Introduction 1.1: Since the introduction of the Bratteli diagram in [Br], graphs have been used both as a tool to study large classes of C*-algebras and as a source of an inexhaustible supply of examples. Bratteli introduced these diagrams to study inductive systems of finite dimensional C*-algebras and their limits, the so called af algebras. Though not itself an invariant of an af algebra, a diagram may easily be used to find the dimension range, the complete invariant for af algebras introduced by Elliott in [El] (for a unital af algebra the dimension range consists of ordered K0 together with the position of the unit). Moreover, it is much easier to construct an af algebra from a Bratteli diagram than from its dimension range. Somewhat later Cuntz and Krieger (see [CK]) introduced a new class of C*algebras generated by partial isometries generalizing examples studied by Cuntz a few years earlier (see [C1]). Briefly, if A is an n × n matrix with Aij = 0, 1 (with no zero row or column), then the associated Cuntz-Krieger algebra OA is defined to be the universal C*-algebra generated by n partial isometries S1 , . . . , Sn with orthogonal range projections subject to the relations: n X Si ∗ Si = Aij Sj Sj ∗ for i = 1, . . . , n. j=1

Under mild hypotheses these C*-algebras were shown to be simple (see [CK, 2.14]) and purely infinite (see [C2, 1.6]). These were regarded as an intriguing class of algebras and immediately became the focus of considerable attention (see [Ev2] for a detailed overview). As described in [CK, 2.16], the construction extends to matrices A with non-negative entries. Such a matrix may be viewed as the incidence matrix of a graph, so it is natural to view the Cuntz-Krieger algebra as arising from the graph; this point of view informs the discussion in §2 of these notes (see example 2.3d and [FW, EW, W]). 1.2: More recently, remarkable advances in our understanding of purely infinite simple C*-algebras have occurred, much of it inspired by attempts to classify 1991 Mathematics Subject Classification. Primary 46L05; Secondary 46L55. Research supported by nsf grant DMS-9706982. 1

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simple Cuntz-Krieger algebras. Rørdam showed that these C*-algebras may be classified by their K-theory (see [Rr1]) many years after their introduction. He further showed that one could obtain a purely infinite simple C*-algebra of arbitrary (countable) K-theory as a crossed product of an at algebra by a trace scaling automorphism (see [Rr2, 3.6]). Then Kirchberg and Phillips proved their striking classification theorem (see [Ki, Ph]): separable nuclear simple purely infinite C*algebras which are in the bootstrap class to which the uct applies (see [RSc]) are classified by their K-theory. 1.3: Cuntz-Krieger algebras have been generalized in a bewildering number of ways. We will focus on generalizations that relate most closely to C*-algebras of graphs (which will be discussed in more detail in §2). There are important parallels between the theory of these algebras and certain dynamical systems (e.g. shifts of finite type) as was made evident in the seminal paper [CK]. By analogy with the construction in [CK], Matsumoto has defined a class of C*-algebra associated to subshifts and has shown that under natural assumptions the associated C*-algebras are simple and purely infinite (see [Ma1, 7.5]). In [Ma2] he computed the K-theory of these C*-algebras and showed that under mild conditions the Kirchberg-Phillips classification theorem applies (see [Ma2, 4.11]). See also [Sa] for related work on sofic shifts. In [Pt1] (see also [Pt2, KPS]) Putnam constructed a C*-algebra from a Smale space (i.e. a compact metric space endowed with an expansive homeomorphism with canonical coordinates) which he called a Ruelle algebra. Since the Ruelle algebra associated to the shift of finite type with matrix A discussed in [CK] is stably isomorphic to OA , it is appropriate to regard Ruelle algebras as higher dimensional versions of Cuntz-Krieger algebras. If the Smale space is topologically mixing, then the associated Ruelle algebra is simple and purely infinite (see [PtS], [KPS, 3.5]). In [D1] Deaconu constructed a C*-algebra from an endomorphism (self-covering map) of a compact Hausdorff space using a groupoid generalizing Renault’s Cuntz groupoid (see [Rn1, §III.2]); his analysis includes a number of intriguing higher dimensional examples (involving tori and Heisenberg manifolds) as well as zero dimensional examples (which give the Cuntz-Krieger algebras). Under mild hypotheses these C*-algebras are shown to be simple. Endomorphisms of compact spaces may be interpreted as topological graphs. In [D2] Deaconu continued his analysis by introducing the topological analog of Bratelli diagrams for at algebras and in [D3] he determined the structure and K-theory of C*-algebras associated to certain topological graphs. Arzumanian and Renault study the notion of topological graph under the name polymorphism and show that expansive endomorphisms give rise to Smale spaces (see [AR]). 1.4: Given a Hilbert bimodule X over a C*-algebra A (that is, X is a right Hilbert A-module and the left action is given by bounded adjointable operators), Pimsner constructed a C*-algebra OX generated by copies of X and A subject to certain natural relations arising from the bimodule structure. Pimsner algebras include the Cuntz-Krieger algebras and crossed products by Z as special cases. Deaconu has shown (see [D4]) that C*-algebras of certain topological graphs may be realized as algebras of this type (that is, as OX for appropriate choice of Hilbert bimodule X and C*-algebra A). It may be appropriate to think of a Hilbert bimodule over

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a C*-algebra as the non-commutative analog of a graph. In recent work, the simplicity of OX has been established under varying assumptions (see [MS1, Ths. 1, 5], [KPW, Cor. 12], [DPZ, 4.7], [Pn, 4.2]). Nonselfadjoint analogs of Pimsner algebras have been studied in [MS2]. 1.5: Certain related constructions have at least as much claim to our consideration, but as our focus is on C*-algebras of graphs it will suffice to direct the interested reader’s attention to the longish bibliography. In [DR3] C*-algebras are associated to objects in certain C*-categories also studied in [DR1, DR2]; in [DPZ, 3.2] the authors compare the C*-algebra of a Hilbert bimodule resulting from this construction to the Pimsner algebra and show that they are isomorphic if the bimodule is full and projective. The Doplicher-Roberts algebras in [DR1, DR2] arising as fixed point algebras of Cuntz algebras under certain compact group actions determined by a finite dimensional unitary representation of the group may also be viewed as C*-algebras of pointed graphs (see [MRS, 3.3], [KPRR, 7.1]). Group actions on buildings have given rise to intriguing constructions of C*algebras generated by partial isometries that seem to be related to higher dimensional shifts of finite type (see [RSt1, RSt2]). Graphs have appeared in the theory of subfactors as invariants of embeddings (see [GHJ], [O]). Cuntz-Krieger algebras or their analogs have also arisen in this setting especially in connection with endomorphisms (see for example [I1, Ka2, Rh]). The recent book by Evans and Kawahigashi (see [EK]) should provide a good account of these and related matters. A brief discussion of early uses of graphs in algebra may be found in [Mu1, §6]; directed graphs (called quivers by the algebraists) were involved in the classification of finite dimensional algebras up to Morita equivalence. 1.6: The reader interested in a recent survey on C*-algebras associated to graphs would do well to consult [Pk]. We hope that the point of view taken in these notes is sufficiently different to justify their existence. My original intention was to assemble a large bibliography on material related to C*-algebras of directed graphs and relate in abbreviated form the results of most interest to me. I hope these notes have evolved beyond this stage if only a little. As with most surveys of this kind the point of view taken is personal. I regret that the bibliography is somewhat incomplete and that due to want of space (and expertise) not all references have been mentioned in the text. In the following we review some basic facts about C*-algebras of graphs and consider a number of examples (see 2.3) that illustrate phenomena discussed later. We view the path groupoid of a graph (see 2.4) as the key tool in our analysis of the associated C*-algebra; for a good introduction to the use of groupoids in C*theory consult [Rn1] or [Mu2]. The K-theory of graph C*-algebras is discussed in 2.5. The books by Pedersen ([Pd]) and Blackadar ([Bl]) are standard references for C*-algebras and their K-theory. In 2.6 we explore how the stucture of a graph, especially the presence of loops, is reflected in the structure of the C*-algebra. In 2.7 we relate the C*-algebra of a product graph to the C*-algebras of its component graphs (this is the only place in the paper where something approximating a proof appears). Finally in 2.8 and 2.9, we review the main results of [KP] dealing with free group actions on graphs. This was the subject matter of my talk in Shanghai.

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1.7: Acknowledgements: Thanks are due David Pask for his helpful comments on an earlier draft of these notes, to Valentin Deaconu and Ian Putnam for remarks regarding the relationships the various generalizations of Cuntz-Krieger algebras bear to one another, and to Doug Drinen, Paul Goldstein, Paul Muhly, John Quigg and Jonathan Samuel for providing me with references. I wish to thank Erik Christensen, Gert Pedersen and their colleagues at the University of Copenhagen for their hospitality and support during my brief sojourn there on the way to Shanghai. Lastly, I would like to thank the organizers for hosting a memorable conference. §2 C*-algebras of directed graphs 2.1: A (directed) graph E is a quadruple (E 0 , E 1 , r, s) where E 0 and E 1 are countable sets and r, s : E 1 → E 0 are maps. Elements in E 0 and E 1 are called vertices and edges respectively while r and s are called the range and source maps. A graph is said to be row-finite if s−1 (v) is finite for all v ∈ E 0 and locally finite if, in addition r−1 (v) is finite for all v ∈ E 0 . Unless otherwise specified all graphs considered in the following are assumed to be locally finite (although many of the results discussed below remain valid for row-finite graphs and perhaps more generally). An element v ∈ E 0 is said to be a sink if v ∈ / s(E 1 ) and a source if 1 v∈ / r(E ). It will further be assumed in 2.4 and following that there are no sinks (unless otherwise specified). For n > 1 a path of length n is an n-tuple x = (x1 , . . . , xn ) with xi ∈ E 1 and r(xi ) = s(xi+1 ) for i < n; let E n denote the set of such paths (we regard E 0 and E 1 as paths of length 0 and 1). The collection of all paths (of finite length) ∪n≥0 E n is denoted E ∗ ; for a path x ∈ E n write |x| = n. The range and source maps extend to E ∗ in the obvious way (s(x1 , . . . , xn ) = s(x1 ) and r(x1 , . . . , xn ) = r(xn )). Given paths x, y ∈ E ∗ with r(x) = s(y), one obtains a new path xy by concatenation so that |xy| = |x| + |y| (also, s(xy) = s(x) and r(xy) = r(y)). A path x = (x1 , . . . , xn ) with |x| > 0 is said to be a loop if s(x) = r(x) and a simple loop if, in addition, the elements s(xi ) are distinct. The space of infinite paths is given by: E ∞ = {(x1 , x2 , . . . ) : xi ∈ E 1 and r(xi ) = s(xi+1 ) for all i}. The source map on E ∞ is given by s(x1 , x2 , . . . ) = s(x1 ); if x ∈ E ∗ and y ∈ E ∞ so that r(x) = s(y), one may concatenate to obtain an element xy ∈ E ∞ (with s(xy) = s(x)). For x ∈ E ∗ define the cylinder set Z(x) = {xy : y ∈ E ∞ with r(x) = s(y)}; the collection of cylinder sets form a basis of compact open sets for a topology on E ∞ . Note that E ∞ may be empty. 2.2: Given a graph E, a *-representation of E (on some Hilbert space) is given by a family of mutually orthogonal projections indexed by vertices, {Pv : v ∈ E 0 }, and a family of partial isometries indexed by edges, {Se : e ∈ E 1 }, subject to the following conditions: Se ∗ Se = Pr(e) for all e ∈ E 1 and Pv =

X

Se S e ∗

for all v ∈ s(E 1 ).

s(e)=v

The universal C*-algebra generated by a *-representation of E is denoted by C ∗ (E) (see [KPR, 1.2]). By universality and the above conditions, there is a canonical

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action of the circle group, the gauge action, α : T → Aut C ∗ (E) given by αt (Pv ) = Pv 0

and αt (Se ) = tSe

1

for all v ∈ E , e ∈ E , t ∈ T. The crossed product by the gauge action is af (see §2.8) and therefore C ∗ (E) is stably isomorphic to the crossed product of an af algebra by Z (by the Takesaki-Takai duality theorem [Pd, 7.9.3]). Hence, C ∗ (E) is nuclear and in the bootstrap class to which the uct applies (see [RSc]). Thus, by the Kirchberg-Phillips classification theorem (see [Ki, Ph]), if C ∗ (E) is simple and purely infinite (see 2.6), it is determined up to isomorphism by its K-theory. 2.3: Examples a: For n ≥ 1 the graph On is given by: On0 = {?}, On1 = {e1 , . . . , en }, s(ek ) = r(ek ) = ?. Note that C ∗ (On ) ' On , where On is the Cuntz algebra, the C*-algebra generated by n isometries with orthogonal ranges which sum to 1. b: For n ≥ 0 the graph Pn is given by: Pn0 = {0, 1, . . . , n}, Pn1 = {e0 , . . . , en−1 }, s(ek ) = k, r(ek ) = k + 1. Note that C ∗ (Pn ) ' Mn+1 (C). c: For n ≥ 1 the graph Zn is given by: Zn0 = {1, . . . , n}, Zn1 = {e1 , . . . , en },  k + 1 if k < n s(ek ) = k, r(ek ) = 1 if k = n ∗ Note that C (Zn ) ' Mn (C(T)) (see [Ev2, 5.5], [aHR, 3.4]). d: Let A ∈ Mn (Z+ ) with no zero row or column. We associate a graph EA 0 = {1, . . . , n} and to A so that A is the incidence matrix of EA . Set EA 1 define EA so that there are Aij edges from i to j. Then C ∗ (EA ) ' OA (see [CK, 2.16]). e: The graph Z is given by: Z 0 = Z, Z 1 = {ek : k ∈ Z}, s(ek ) = k, r(ek ) = k + 1. Note that C ∗ (Z) ' K, where K is the C*-algebra of compact operators on a separable infinite dimensional Hilbert space. f: The graph Q is given by: Q0 = {0, 1}, Q1 = {e, f }, s(e) = 0, r(e) = 1, s(f ) = 0 = r(f ). It is easy to show that C ∗ (Q) is isomorphic to the Toeplitz algebra. g: The graph R is given by: R0 = Z, R1 = {ek , fk : k ∈ Z}, s(ek ) = k, r(ek ) = k + 1, s(fk ) = k = r(fk ). C ∗ (R) is a purely infinite C*-algebra expressible as an inductive limit of type I C*-algebras (see [KPR, 3.10]) and as a crossed product of O2 by Z (see §2.8). h: The graph S is given by: S 0 = {0, 1, 2 . . . }, S 1 = {ek , fk : k ≥ 0}, s(ek ) = r(fk ) = k, r(ek ) = s(fk ) = k + 1. It can be shown that C ∗ (S) ' O∞ ⊗ K (see §2.5).

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i: Let Fn be the free group on n generators g1 , . . . , gn and let Tn be the Cayley graph of Fn with respect to the given generating set: Tn0 = Fn , Tn1 = Fn × {1, . . . , n}, s(g, i) = g, r(g, i) = ggi . Note that Tn is a tree and that C ∗ (Tn ) is strongly Morita equivalent to C0 (∂Tn ) where ∂Tn is the boundary of the tree (see [KP, 4.3]). 2.4: If there are no sinks then C ∗ (E) ' C ∗ (GE ) where GE is the path groupoid of E (see [KPRR]). The path groupoid GE is based on Renault’s construction of 0 the Cuntz groupoid (see [Rn1, §III.2]). The unit space GE is identified with the ∞ infinite path space E . We follow the sign convention of [Rn1] (which differs from that of [KPR], [KPRR]): GE = {(x, k, y) : x, y ∈ E ∞ , k ∈ Z, xi+k = yi for sufficiently large i}. The various structure maps of GE are given by: r(x, k, y) = x, s(x, k, y) = y, (x, k, y)−1 = (y, −k, x), and (x, k, y)(y, l, z) = (x, k + l, z) (where we have identified x with the triple (x, 0, x)). Note that (x, k, y) ∈ GE iff there are p, q ∈ E ∗ and z ∈ E ∞ with r(p) = r(q) = s(z) such that x = pz, y = qz and k = |p| − |q|. One may define the natural topology on GE as follows: given (x, k, y) ∈ GE one obtains a local basis by taking p, q as above and setting Z(p, q) = {(pt, k, qt) : t ∈ E ∞ , s(t) = r(p) = r(q)}. Henceforth, unless otherwise specified, we assume that there are no sinks in the graphs considered and identify C ∗ (E) = C ∗ (GE ). Note that a graph with sinks may be extended in a natural way (by adding an infinite path to each sink) so that the C*-algebra of the resulting graph contains the C*-algebra of the original graph as a full corner (and thus is strongly Morita equivalent to it). The gauge automorphism group α : T → Aut C ∗ (E) leaves the dense algebra Cc (GE ) invariant and there is a cocycle a ∈ Z 1 (GE , Z) given by a(x, k, y) = k which determines it on this subalgebra (and hence on C ∗ (E)): for f ∈ Cc (GE ) ⊂ C ∗ (E) and t ∈ T one has (αt f )(x, k, y) = tk f (x, k, y) = ht, a(x, k, y)if (x, k, y). This is the usual action of a locally compact abelian group on a groupoid C*-algebra determined by a 1-cocycle with values in its dual, see [Rn1, II.5.1]. 2.5: The K-theory of C ∗ (E) may be computed as in [C4, 3.1] using the PimsnerVoiculescu exact sequence (see [PV, 2.4], [Bl, 10.2.1]). There is an exact sequence: M M ∆ 0 → K1 (C ∗ (E)) → Z −→ Z → K0 (C ∗ (E)) → 0 E0

E0

0 where E 0 Z is regarded as the free abelian group generated by {δv , : v ∈ E }, ∆ = ∆E is defined by X ∆E δ v = δ v − δr(e)

L

s(e)=v

L ∗ and the map E 0 Z → K0 (C (E)) is given by δv 7→ [Pv ] (see [PR, 4.2.4], [Pk, 3.1,3.2] — note that if E is a finite graph with incidence matrix A then with the obvious identifications ∆E = 1 − At ). For the graph S described in example 2.3h, a straightforward computation shows that K0 (C ∗ (S)) ' Z and K1 (C ∗ (S)) ' 0. It follows from the characterizations below (see §2.6) that C ∗ (S) is simple and purely infinite and from the Kirchberg-Phillips classification theorem (see [Ki], [Ph]) that

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C ∗ (S) is strongly Morita equivalent to O∞ . Hence, C ∗ (S) ' O∞ ⊗ K. 2.6: The stucture of C ∗ (E) is in some sense determined by the presence of loops in the graph E. For example, if there are no loops then C ∗ (E) is an af algebra (here E may have sinks [Dr, G1, KPR]). A graph E is said to satisfy condition (L) (see [KPR]) if for every loop x = (x1 , . . . , xn ) there is i = 1, . . . , n and e ∈ E 1 such that e 6= xi and s(e) = s(xi ). This is the natural analog of condition (I) considered by Cuntz and Krieger (see [CK]). By [KPR, 3.4], E satisfies condition (L) if and only if GE is essentially free (the points with trivial isotropy are dense in the unit space). It follows that if E satisfies condition (L), then any nondegenerate *representation induces an isomorphism with C ∗ (E) (see [KPR, 3.7]). Recall that a C*-algebra is purely infinite if every hereditary subalgebra contains an infinite projection. By [KPR, 3.9] (cf. [A-D, LS]), C ∗ (E) is purely infinite if and only if E satisfies condition (L) and every vertex connects to a loop, i.e. for every v ∈ E 0 there is a path x and a loop y, such that s(x) = v and r(x) = s(y). A graph E is said to satisfy condition (K) if for every simple loop x, there is a distinct simple loop y such that s(x) = s(y). This is the natural analog of condition (II) considered by Cuntz (see [C2]). If E satisfies condition (K), then GE is essentially principal (i.e. the points with trivial isotropy are dense in any closed invariant subset of the unit space, see [Rn1, II.4.3]). A subset H of E 0 is said to be hereditary if for each edge e, r(e) ∈ H if s(e) ∈ H and saturated if for each vertex v, if r(e) ∈ H for all e ∈ s−1 (v) then v ∈ H. If E satisfies condition (K), then the ideals of C ∗ (E) may be put in one-to-one correspondence with the saturated hereditary subsets of E 0 (see [KPRR, 6.6], cf. [Br, 3.3]). The ideal structure of a C*-algebra associated to a graph which may fail to satisfy condition (K) is discussed in [aHR, BPR]. A graph E is said to be cofinal if for every vertex v and infinite path x = (x1 , x2 , . . . ) there is a path y so that s(y) = v and r(y) = s(xi ) for some i. Note that if the graph is cofinal, conditions (K) and (L) are equivalent. By [KPRR, 6.8] C ∗ (E) is simple iff E is cofinal and satisfies condition (L). In that case there is a dichotomy: if there is a loop in E, C ∗ (E) is purely infinite otherwise C ∗ (E) is af (see [KPR, 3.11]). In independent work Goldstein has shown (see [G1, 3.19]) that if the graph E is countably infinite and transitive (for every u, v ∈ E 0 there is a path x such that s(x) = u and s(y) = v), then C ∗ (E) is simple, stable and purely infinite. 2.7: Given two graphs E and F , it is amusing to see how C ∗ (E × F ) is related to C ∗ (E) and C ∗ (F ). We define the product graph E × F as follows: (E × F )0 = E 0 × F 0 and (E × F )1 = E 1 × F 1 . Evidently, (E × F )n = E n × F n and (E × F )∞ = E ∞ × F ∞ and there is a groupoid embedding: GE×F → GE × GF given by [(w, y), k, (x, z)] 7→ [(w, k, x), (y, k, z)] where w, x ∈ E ∞ and y, z ∈ F ∞ . Let G be a compact abelian group; by analogy with the usual notion of product in the category of G-spaces, one defines the notion of product in the category of C*algebras with fixed G-actions as in [OPT, §2]. Given C*-algebras A, B with actions µ : G → Aut A and ν : G → Aut B, define A ⊗G B to be the fixed point algebra (A⊗B)λ under the action λ : G → Aut A⊗B defined by λg (a⊗b) = µg (a)⊗νg−1 (b). Note that the restrictions of µg ⊗1 and 1⊗νg to A⊗G B agree; this gives the G-action on A ⊗G B. Recall that every graph C*-algebra comes equipped with a canonical action by T, the gauge action.

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Proposition: Given graphs E and F , one has: C ∗ (E × F ) ' C ∗ (E) ⊗T C ∗ (F ); moreover, this isomorphism is T-equivariant. In particular it follows that for integers m, n ≥ 1 one has: Omn ' Om ⊗T On and, indeed, for non-negative integer matrices A, B with no zero row or column one has OA⊗B ' OA ⊗T OB . Proof : Define a *-homomorphism ϕ : C ∗ (E × F ) → C ∗ (E) ⊗ C ∗ (F ) by ϕ(P(u,v) ) = Pu ⊗ Pv and ϕ(S(e,f ) ) = Se ⊗ Sf . Then, ϕ is clearly injective; we must show that ϕ(C ∗ (E × F )) = (C ∗ (E) ⊗ C ∗ (F ))β where β : T → Aut C ∗ (E) ⊗ C ∗ (F ) is the automorphism group given by βt (a ⊗ b) = αt (a) ⊗ αt−1 (b). First observe that C ∗ (E) ⊗ C ∗ (F ) ' C ∗ (GE × GF ) and that, under this identification, ϕ is induced by the groupoid embedding above. The image of GE×F is a clopen subgroupoid which is equal to the kernel of the cocycle b ∈ Z 1 (GE × GF , Z) defined by b[(w, k, x), (y, l, z)] = k − l; note that, under the identification C ∗ (E) ⊗ C ∗ (F ) ' C ∗ (GE × GF ), β is determined by b (see [Rn1, II.5.1]): βt (f [(w, k, x), (y, l, z)]) =

ht, b[(w, k, x), (y, l, z)]if [(w, k, x), (y, l, z)]

= tk−l f [(w, k, x), (y, l, z)] for f ∈ Cc (GE × GF ). Hence, C ∗ (E × F ) ' C ∗ (GE×F ) ' C ∗ (GE × GF )β = C ∗ (E) ⊗T C ∗ (F ). Note further that under this isomorphism the gauge action on C ∗ (E × F ) is identified with the action on C ∗ (E) ⊗T C ∗ (F ) given above. 2 2.8: Fix a countable discrete group G; an action of G on a graph E consists of compatible actions on E 0 and E 1 , that is, the range and source maps are equivariant: r(ge) = gr(e) and s(ge) = gs(e) for all g ∈ G and e ∈ E 1 . By universality there is an induced action λ : G → Aut C ∗ (E) defined as follows: λg (Pv ) = Pgv and λg (Se ) = Sge for all v ∈ E 0 , e ∈ E 1 , g ∈ G. Note that the quotient E/G is a graph as well (which is locally finite if E is). Say that an action of G on E is free if G acts freely on E 0 (and E 1 ). Theorem (see [KP, 1.1]): Suppose G acts freely on the graph E. Then one has: C ∗ (E) oλ G ' C ∗ (E/G) ⊗ K(`2 (G)) where λ denotes the induced action on C ∗ (E). Moreover, if G is abelian, there is b on C ∗ (E/G) such that an action γ of G b C ∗ (E) ' C ∗ (E/G) oγ G, and such that under this isomorphism λ is identified with γˆ , the action dual to γ. The proof involves groupoid techniques and the notion of skew product (or (right) derived graph [GT, §2.1.1]) which enables one to reconstruct a free action from the quotient graph and certain patching data. Given a graph E, a group G and a function c : E 1 → G (sometimes referred to as a labelling), the skew product graph G ×c E (or E(c)) is defined as follows: (G ×c E)i = G × E i for i = 0, 1, r(g, e) = (gc(e), r(e)) and s(g, e) = (g, s(e)).

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Then the natural action of G (acting on the left) is free with quotient isomorphic to E (morever every free action is of this form — see [GT, Th. 2.2.2]). If G is abelian b → Aut C ∗ (E) defined by then the function c induces an action γ : G γχ (Pv ) = Pv and γχ (Se ) = hχ, c(e)iSe b Moreover, one has for all v ∈ E 0 , e ∈ E 1 , χ ∈ G. b ' C ∗ (G ×c E). C ∗ (E) oγ G The gauge automorphism group is an example of this with c : E → Z given by c(e) = 1 for all e ∈ E 1 . It follows that C ∗ (E) oα T ' C ∗ (Z ×c E) = C ∗ (Z × E) where Z is the graph given in example 2.3e. Since there are no loops in Z × E it follows that the crossed product is af. Hence, by duality or the above theorem, C ∗ (E) is stably isomorphic to a crossed product of an af algebra by Z. See [W, CE, Ev1, G2, G3] for other instances of actions induced by labellings (see also example 2.3g). Given a group G with generators g1 , . . . , gn , its Cayley graph is the skew product G ×c On where c : On → G is given by c(ek ) = gk (see example 2.3i). We obtain as a corollary of the above theorem that the crossed product of a group acting on the C*-algebra of its Cayley graph is isomorphic to On ⊗ K(`2 (G)). In [KQR] it is shown more generally that if G acts freely on the graph E, there is a coaction of G on C ∗ (E/G) such that C ∗ (E) is isomorphic to the crossed product of C ∗ (E/G) by this coaction in such a way that λ is identified with the dual action. The first assertion of the theorem then follows by duality (see [Ka1, Th. 8]). 2.9: A graph is said to be connected if when one ignores the orientation of the edges there is a path connecting any two vertices and a tree if, in addition, there is exactly one. If T is a tree, then the path groupoid GT is an equivalence relation on T ∞ with a “nice” quotient ∂T = T ∞ /GT (called the boundary of T ) in the sense that the quotient map T ∞ → ∂T is a local homeomorphism. It follows that C ∗ (T ) is strongly Morita equivalent to the abelian C*-algebra C0 (∂T ) (see [KP, 4.3]). If e is a tree and its fundamental E is a connected graph, then its universal cover E e is free and it induces group ΠE is free (see [Se, §I.3.3], [Hi]). The action of ΠE on E e an action on the boundary ∂ E. Thus, one has: Theorem (see [KP, 1.2]): Let E be a connected graph; then C ∗ (E) is strongly e o ΠE where E e is the universal cover of E, ΠE is its Morita equivalent to C0 (∂ E) e is the induced boundary action. fundamental group and the action of ΠE on ∂ E Moreover, ΠE is a free group. References [A-D] [aHR] [AR]

C. Anantharaman-Delaroche, Purely infinite C*-algebras arising from dynamical systems, Preprint, Universit´ e d’Orl´ eans, 1996. A. an Huef and I. Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergodic Theory Dyn. Sys. 17 (1997) 611-624. V. Arzumanian and J. Renault, Examples of pseudogroups and their C*-algebras, in Operator Algebras and Quantum Field Theory (Accademia Nazionale dei Lincei, Roma, July 1-6 1996), S. Doplicher, R. Longo, J. Roberts and L. Zsido, Eds., 93-104, International Press 1997.

10

[BPR] [Bl] [Br] [C1] [C2] [C3] [C4] [C5]

[CE] [CK] [D1] [D2] [D3] [D4] [DPZ] [DR1] [DR2] [DR3] [Dr] [El] [ETW] [EW] [Ev1] [Ev2] [EK] [EL] [FW] [G1] [G2] [G3] [GHJ] [GT] [Hi]

A. KUMJIAN

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A. KUMJIAN

[Pd] [Ph] [Pm]

[PV] [Pn]

[Pt1] [Pt2] [PtS] [Rh]

[Rn1] [Rn2] [RSc] [RSt1] [RSt2] [Rr1] [Rr2] [Rr3]

[Rr4] [Sa] [Se] [Sp] [Su] [SZ] [W]

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Department of Mathematics, University of Nevada, Reno NV 89557, USA E-mail address: [email protected]