Moves on k -graphs preserving Morita equivalence
Caleb Eckhardt, Kit Fieldhouse, Daniel Gent, Elizabeth Gillaspy, Ian Gonzales, David Pask
MMOVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE CALEB ECKHARDT, KIT FIELDHOUSE, DANIEL GENT, ELIZABETH GILLASPY,IAN GONZALES, AND DAVID PASK
Abstract.
We initiate the program of extending to higher-rank graphs ( k -graphs) the geometricclassification of directed graph C ∗ -algebras, as completed in the 2016 paper of Eilers, Restorff,Ruiz, and Sørensen [ERRS16]. To be precise, we identify four “moves,” or modifications, onecan perform on a k -graph Λ, which leave invariant the Morita equivalence class of its C ∗ -algebra C ∗ (Λ). These moves – insplitting, delay, sink deletion, and reduction – are inspired by the movesfor directed graphs described by Sørensen [Sø13] and Bates-Pask [BP04]. Because of this, ourperspective on k -graphs focuses on the underlying directed graph. We consequently include twonew results, Theorem 2.3 and Lemma 2.9, about the relationship between a k -graph and itsunderlying directed graph. Introduction
Recent years have seen a number of breakthroughs in the classification of C ∗ -algebras by K -theoretic invariants. For separable simple unital C ∗ -algebras A which have finite nuclear dimensionand satisfy the Universal Coefficient Theorem of [RS87], the Elliott invariant (consisting of theordered K -theory of A , its trace simplex, and the pairing between traces and K ( A )) is a classifyinginvariant [TWW17, EGLN15, GLN15]: two such C ∗ -algebras A, B are isomorphic if and only if theirElliott invariants are isomorphic. Work has already begun [EGLN17, GL18] on expanding theseresults to the non-unital setting.The Cuntz–Krieger algebras O A [CK80] associated to irreducible matrices A were one of the earlyclasses of C ∗ -algebras for which K -theory was shown to be a classifying invariant [CK80,Fra84,Rø95].When A is not irreducible, O A is not simple, leaving these C ∗ -algebras outside the scope of the Elliottclassification program. However, the proof of the K -theoretic classification of simple Cuntz–Kriegeralgebras draws heavily on the dynamical characterization of Cuntz–Krieger algebras as arising fromone-sided shifts of finite type [CK80]. As this dynamical characterization holds in the non-simplecase as well, Cuntz–Krieger algebras were a natural setting for a first foray into classification of non-simple C ∗ -algebras. This program was brought to fruition by Eilers, Restorff, Ruiz, and Sørensenin [ERRS16].Interpreting A as the adjacency matrix of a directed graph E A , we have a canonical isomorphism O A ∼ = C ∗ ( E A ). Using this perspective, as well as techniques from symbolic dynamics [Hua96,Boy02,BH], Eilers et al. obtained both a K -theoretic and a graph-theoretic classification of unital graph C ∗ -algebras. To be precise, [ERRS16] identifies 6 “moves” on directed graphs E with finitely manyvertices which preserve the stable isomorphism class of C ∗ ( E ) . The authors then use filtered K -theory to show that, for such graphs E, F , an isomorphism C ∗ ( E ) ⊗ K ∼ = C ∗ ( F ) ⊗ K can only existif we can pass from E to F by a finite sequence of these 6 moves and their inverses. Eilers et al. alsoshow in [ERRS16] that isomorphism of two unital graph C ∗ -algebras C ∗ ( E ) , C ∗ ( F ) is equivalent tothe existence of an order-preserving isomorphism of the filtered K -theory of C ∗ ( E ) and C ∗ ( F ).The K -theory of a graph C ∗ -algebra [Cun81, BHRS02] dictates that if C ∗ ( E ) is simple, it iseither approximately finite-dimensional or purely infinite. Kumjian and Pask developed the theoryof higher-rank graphs, or k -graphs, in [KP00] to provide a broader range of combinatorial examplesof C ∗ -algebras. Formally, a k -graph Λ is a countable category with a functor d : Λ → N k satisfying A graph E has finitely many vertices iff C ∗ ( E ) is unital. a r X i v : . [ m a t h . OA ] J un C. ECKHARDT, K. FIELDHOUSE, D. GENT, E. GILLASPY, I. GONZALES, AND D. PASK a factorization property (see Definition 2.1 below). However, k -graphs are also closely linked tobuildings [RS99, KV15] and to higher-rank shifts of finite type via textile systems [JM99]. Thegraph-theoretic inspiration for higher-rank graphs was made precise by Hazlewood et al. [HRSW13],who detailed in [HRSW13, Theorems 4.4 and 4.5] the correspondence between higher-rank graphson the one hand, and on the other hand, edge-colored directed graphs with an equivalence relationon their category of paths. In this perspective, the factorization property of a k -graph is encoded inthe set of “commuting squares,” or length-2 paths ab ∼ cd which are equivalent in the edge-coloreddirected graph.The paper at hand constitutes a first step towards extending the geometric classification of graph C ∗ -algebras to the setting of higher-rank graphs. Taking inspiration from [Dri99, BP04, CG06, Sø13]as well as from [ERRS16], we identify four moves (sink deletion, in-splitting, reduction, and delay)on row-finite, source-free k -graphs Λ which preserve the Morita equivalence class of C ∗ (Λ). Thesemoves for k -graphs were inspired by their analogues for directed graphs, and therefore involve addingor removing edges and vertices in Λ. Performing such a move on a k -graph affects the factorizationproperty, though, as length-2 paths may become longer or shorter. Thus, geometric classification inthe k -graph setting faces a new technical hurdle: one must identify how to adjust the factorizationproperty after each move, so that the resulting object is still a k -graph.As discussed in the introduction to [BP04], the moves of in-splitting and delay originate insymbolic dynamics. For shifts of finite type, the natural relations of conjugacy and flow equiv-alence [PS75] are generated by matrix operations which, when translated into the graph setting,correspond to the moves of insplitting, outsplitting and delay. (See [LM95, Sections 2.3 and 2.4]for more details.) For directed graphs, the analogues (S) of sink deletion and (R) of reductionwere first isolated by Sørensen [Sø13], drawing on the very general framework given in [CG06] formodifying a directed graph without changing its Morita equivalence class. The main result of [Sø13](Theorem 4.3) establishes that, for directed graphs E, F with finitely many vertices such that C ∗ ( E )and C ∗ ( F ) are simple, any stable isomorphism C ∗ ( E ) ⊗ K ∼ = C ∗ ( F ) ⊗ K must arise from a finitesequence of insplittings, outsplittings, Cuntz splice, the moves (S) and (R) , and their inverses. Asmentioned above, a series of papers by Eilers, Sørensen, and others followed, which culminated inthe complete classification of unital graph C ∗ -algebras in [ERRS16].We now outline the structure of the paper. The picture of higher-rank graphs as arising fromedge-colored directed graphs underlies our work in this paper, and so we take some pains in Section2 to assure the reader of the equivalence between our approach to k -graphs and the more commoncategory-theoretic perspective. To obtain our Morita equivalence results, we rely heavily on ageneralization of the gauge-invariant uniqueness theorem for k -graphs [KP00], and on Allen’s results[All08] about corners in higher-rank graphs, so we also review these notions in Section 2.Each of Sections 3 through 6 is dedicated to one of our four Morita equivalence preserving moveson k -graphs. For each move, we first ensure that its output is a k -graph, and then we show thatthe resulting k -graph C ∗ -algebra is Morita equivalent to our original C ∗ -algebra. We begin within-splitting in Section 3. We first describe conditions under which we can “in-split” a k -graph at avertex v – that is, create two copies of v and divide the edges with range v among the two copies – insuch a way that the resulting object is still a k -graph (Theorem 3.12). Theorem 3.13 then establishesthat insplitting produces a C ∗ -algebra which is isomorphic to our original one, not merely Moritaequivalent. Section 4 studies the move of “delaying” an edge by breaking it into two edges. In orderto delay an edge in a k -graph, the k -graph’s factorization rule also forces us to delay many of theedges of the same color. In Theorem 4.2, we show that this move results in a k -graph. Moreover,its C ∗ -algebra is Morita equivalent to that of our original k -graph (Theorem 4.3). In Section 5, weshow in Theorem 5.6 that if a vertex is a sink – that is, it emits no edges of a given color – then afterdeleting the sink and all incident edges, we are still left with a k -graph. The fact that this move doesnot change the Morita equivalence class of the k -graph C ∗ -algebra is established in Theorem 5.7.Finally, we turn to reduction in Section 6, where we identify when contraction of a “complete edge”(see Definition 6.1) in a k -graph produces a k -graph (Theorem 6.5). In this case, the C ∗ -algebra of OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 3 the resulting k -graph is always Morita equivalent to the original k -graph C ∗ -algebra, by Theorem6.6.Throughout the paper, we include examples showcasing the moves, and indicating the necessityof our hypotheses. Acknowledgments:
The authors thank Kenton Ke, Emily Morison, Jethro Thorne, and RyanWood for helpful comments. This research project was supported by NSF grant DMS-1800749 toE.G., and by the University of Montana’s Small Grants Program.2.
Notation
Fix an integer k ≥
1. As our main objects of study in this paper are k -graphs (higher-rankgraphs), we begin by recalling their definition. First, however, we specify that throughout thispaper we regard 0 as an element of N , and we view N k as a category, with composition of morphismsgiven by addition. Consequently, N k has one object (namely (0 , . . . , n = (cid:80) ki =1 n i e i ∈ N k , we write | n | = (cid:80) i n i . Definition 2.1. [KP00, Definitions 1.1] Let Λ be a countable category and d : Λ → N k a functor.If (Λ , d ) satisfies the factorization rule – that is, for every morphism λ ∈ Λ and n, m ∈ N k such that d ( λ ) = n + m , there exist unique µ, ν ∈ Λ such that d ( µ ) = m , d ( ν ) = n , and λ = µν – then (Λ , d )is a k -graph.We write Λ = { λ ∈ Λ : | d ( λ ) | = 1 } and Λ = d − (0). If e ∈ Λ , we say e is an edge of Λ, and Λ is the set of vertices of Λ.Observe that the factorization rule guarantees, for each λ ∈ Λ, the existence of unique v, w ∈ Λ such that vλw = λ ; we will write r ( λ ) for v and s ( λ ) for w . Similarly, we write v Λ = { λ ∈ Λ : r ( λ ) = v } and v Λ n = { λ ∈ v Λ : d ( λ ) = n } for any n ∈ N k . The sets Λ w, Λ n w are defined analogously.Our reason for the convention that the source of a morphism in Λ lies on its right, and its rangelies to the left, arises from the Cuntz–Krieger relations used to define k -graph C ∗ -algebras; seeDefinition 2.5 and Remark 2.6 below.We now briefly describe how to model k -graphs using k -colored graphs as we use this frameworkextensively for our constructions. Following [HRSW13] we let G = ( G , G , r, s ) denote a directedgraph with G its set of vertices and G its set of edges; r, s : G → G are the range and sourcemap respectively. For an integer n ≥ G n denote the paths of length n in G . By a slight abuseof notation, if δ ∈ G n we will write | δ | := n .We now color the graph G by assigning to each edge one of the standard basis vectors, e i , of N k and let G e i be the set of edges assigned to e i , so that G = (cid:83) ki =1 G e i . The path category , G ∗ = (cid:83) n ∈ N G n , may now be equipped with a degree functor d : G ∗ → N k , given on the vertices by d ( v ) = 0 for all v ∈ G , and on the edges by d ( f ) = e i if f was assigned the basis vector e i . On longerpaths, d is extended to be additive: d ( f n · · · f ) = (cid:80) ni =1 d ( f i ). (Our reason for this enumeration ofthe edges in a path is that in a k -graph, by Definition 2.1, we have s ( f i ) = r ( f i − ) whenever f i f i − is a well-defined product of morphisms in a k -graph. Consistency with this definition requires thatthe right-most edge in a path, f , denotes the path’s initial edge and the left-most edge, f n , its finaledge.) As usual, the range and source maps r, s : G → G extend to well-defined maps from G ∗ to G , which we continue to denote by r and s .Theorem 4.5 of [HRSW13] establishes that, if G is a k -colored graph as described above, thenΛ = G ∗ / ∼ is a k -graph for any ( r, s, d )-preserving equivalence relation ∼ on G ∗ which also satisfies(KG0) If λ ∈ G ∗ is a path such that λ = λ λ , then [ λ ] = [ p p ] whenever p ∈ [ λ ] and p ∈ [ λ ].(KG1) If f, g ∈ G are edges, then f ∼ g ⇔ f = g .(KG2) Completeness:
For every µ = µ µ ∈ G such that d ( µ ) = e i , d ( µ ) = e j , there exists aunique ν = ν ν ∈ G such that d ( ν ) = e j , d ( ν ) = e i and µ ∼ ν . C. ECKHARDT, K. FIELDHOUSE, D. GENT, E. GILLASPY, I. GONZALES, AND D. PASK (KG3)
Associativity:
For any e i - e j - e (cid:96) path abc ∈ G with i, j, (cid:96) all distinct, the e (cid:96) - e j - e i paths hjg, nrq constructed via the following two routes are equal.Route 1: Let ab ∼ de , so abc ∼ dec .Let ec ∼ f g , so abc ∼ df g .Let df ∼ hj , so abc ∼ hjg .Route 2: Let bc ∼ km , so abc ∼ akm .Let ak ∼ np , so abc ∼ npm .Let pm ∼ rq , so abc ∼ nrq . • •• •• •• • df j = r h = nbe ac mg = q kp Figure 1. (KG3)In fact, [HRSW13, Theorem 4.4] shows that every k -graph arises in this way. That is, given a k -graph Λ, we obtain a directed graph G by setting G = Λ , G = Λ . (This justifies our decisionto call Λ the vertices of Λ and Λ the edges of Λ.) Transferring the degree map d : Λ → N k to G makes G a k -colored graph; we obtain an equivalence relation on G ∗ by setting λ ∼ µ if the paths λ, µ represent the same morphism in Λ. The factorization rule in Λ then implies that ∼ satisfies(KG0) - (KG3).In this paper, we fully exploit the equivalence between k -colored directed graphs with equivalencerelations on the one hand, and k -graphs on the other hand. Our general strategy will be to define amove M on a k -graph Λ in terms of its impact on the 1-skeleton G and the equivalence relation ∼ which give rise to Λ. This produces a new colored graph G M with a new equivalence relation ∼ M ,which we then show satisfies (KG0) - (KG3) so that the quotient G M / ∼ M is a new k -graph Λ M .For λ ∈ G ∗ we notate its equivalence class under ∼ as [ λ ] ∈ Λ. For n ∈ N k we writeΛ n = { [ λ ] ∈ Λ : d ([ λ ]) = n } . For our purposes in this paper, we will also need an alternative characterization of the equivalencerelations on G ∗ which give rise to k -graphs. We begin by observing that an inductive application ofthe factorization rule of Definition 2.1 reveals that if Λ is a k -graph, then for any morphism λ ∈ Λ andordered n -tuple ( m , . . . , m n ) of elements of N k such that | m i | = 1 for all i and m + · · · + m n = d ( λ ),there is a unique set of edges λ , . . . , λ n ∈ Λ such that λ = λ n · · · λ where d ( λ i ) = m i . Definition 2.2.
For a finite path λ in an edge-colored directed graph G , let λ i denote the i th edgeof λ (counting from the source of λ ). The color order of λ is the | λ | -tuple ( d ( λ ) , d ( λ ) , . . . , d ( λ | λ | )).This leads us to the following condition on an equivalence relation ∼ on G ∗ :(KG4) For each λ ∈ G ∗ and each permutation of the color order of λ , there is a unique path µ ∈ [ λ ]with the permuted color order. Theorem 2.3.
Let G be an edge-colored directed graph and suppose ∼ is an ( r, s, d ) -preservingequivalence relation on G ∗ satisfying (KG0). The relation ∼ satisfies (KG1), (KG2), and (KG3)(and hence G ∗ / ∼ is a k -graph) if and only if ∼ satisfies (KG4).Proof. First, assume (KG0) and (KG4) hold for ∼ and consider an e i - e j - e (cid:96) path λ ∈ G . Convert λ into two e (cid:96) - e j - e i paths via the routes described in (KG3) and label them µ and ν . Since µ ∼ λ and ν ∼ λ by construction, the fact that ∼ is an equivalence relation implies that µ ∼ ν . Condition(KG4) and the fact that µ, ν have the same color order now gives µ = ν . Thus (KG3) holds.Similarly, if λ ∈ G , then there exists a unique µ ∈ [ λ ] of each permuted color order. Thus (KG2)holds. Finally for e, f ∈ G we have e ∼ f = ⇒ d ( e ) = d ( f ) = ⇒ e = f , since each color order hasa unique associated path. Also e = f = ⇒ e ∼ f . Thus (KG1) holds. OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 5 Now assume ∼ satisfies (KG0), (KG1), (KG2), and (KG3). We know from [HRSW13, Theorem4.5] that Λ := G ∗ / ∼ is a k -graph. Thus, fix δ ∈ G ∗ , and choose a sequence of basis vectors ( m j ) | δ | j =1 , with m j ∈ { e i } ki =1 for all j , such that d ( δ ) = (cid:80) | δ | j =1 m j . An inductive application of the factorizationrule of Definition 2.1 implies the existence of a unique path γ = γ | δ | · · · γ γ ∈ [ δ ] where d ( γ j ) = m j for every j . Since our ordering of the basis vectors ( m , . . . , m | d ( λ ) | ) was arbitrary, it follows that ∼ satisfies (KG4). (cid:3) Notation 2.4. A k -graph Λ is row-finite if for all v ∈ Λ and all 1 ≤ i ≤ k , we have |{ λ ∈ Λ e i : r ( λ ) = v }| < ∞ . We say v ∈ Λ is a source if there is i such that r − ( v ) ∩ Λ e i = ∅ .In this paper we will focus exclusively on row-finite source-free k -graphs. Definition 2.5. [KP00, Definition 1.5], [KPS12, Definition 7.4] Let Λ be a row-finite, source-free k -graph Λ. A Cuntz–Krieger Λ -family is a collection of projections { P v : v ∈ Λ } and partialisometries { T f : f ∈ Λ } satisfying the Cuntz–Krieger relations: (CK1) The projections P v are mutually orthogonal.(CK2) If a, b, f, g ∈ Λ satisfy af ∼ gb , then T a T f = T g T b . (CK3) For any f ∈ Λ we have T ∗ f T f = P s ( f ) . (CK4) For any v ∈ Λ and any 1 ≤ i ≤ k, we have P v = (cid:88) f : r ( f )= v,d ( f )= e i T f T ∗ f . There is a universal C ∗ -algebra for these generators and relations, which is denoted C ∗ (Λ) = C ∗ ( { p v , t f } ). For any Cuntz–Krieger Λ-family { P v , T f } , we consequently have a surjective ∗ -homomorphism π : C ∗ (Λ) → C ∗ ( { P v , T e } ), such that π ( p v ) = P v and π ( t f ) = T f for all v ∈ Λ , f ∈ Λ . Remark 2.6.
Observe that if { T f , P v } is a Cuntz–Krieger Λ-family, then (CK3) implies that T f P s ( f ) = T f . Similarly, by (CK4) and the fact that a sum of projections is a projection iff thoseprojections are orthogonal, P r ( f ) T f = T f . Thus, viewing edges in Λ as pointing from right to leftensures the compatibility of concatenation of edges in Λ with the multiplication in C ∗ (Λ).If Λ = G ∗ / ∼ , and µ, ν ∈ G ∗ represent the same equivalence class in Λ, then Condition (CK2),together with conditions (KG0)–(KG2), guarantees that t µ n · · · t µ = t ν n · · · t ν t ν . Thus, for [ µ ] ∈ Λ, we define t µ := t µ n · · · t µ . [KP00, Lemma 3.1] then implies that { t µ t ∗ ν : [ µ ] , [ ν ] ∈ Λ } densely spans C ∗ (Λ) . Remark 2.7.
We have opted to describe C ∗ (Λ) purely in terms of the partial isometries associatedto the vertices and edges, rather than the more common description using all of the partial isometries { t λ : λ ∈ Λ } , because all of our “moves” on k -graphs occur at the level of the edges.A crucial ingredient in our proofs that all of our moves preserve the Morita equivalence classof C ∗ (Λ) is the gauge-invariant uniqueness theorem . To state this theorem, observe first that theuniversality of C ∗ (Λ) implies the existence of a canonical action α of T k on C ∗ (Λ) which satisfies α z ( t e ) = z d ( e ) t e and α z ( p v ) = p v for all z ∈ T k , e ∈ Λ and v ∈ Λ . Theorem 2.8. [KP00, Theorem 3.4] Fix a row-finite source-free k -graph Λ and a ∗ -homomorphism π : C ∗ (Λ) → B . If B admits an action β of T k such that π ◦ α z = β z ◦ π for all z ∈ T k , and for all v ∈ Λ we have π ( p v ) (cid:54) = 0 , then π is injective. Many of the actions β that will appear in our applications of the gauge-invariant uniquenesstheorem take the form described in the following Lemma. The proof is a standard argument, usingthe universal property of C ∗ (Λ) to establish that β z is an automorphism for all z , and using an (cid:15)/ β is strongly continuous, so we omit the details. C. ECKHARDT, K. FIELDHOUSE, D. GENT, E. GILLASPY, I. GONZALES, AND D. PASK
Lemma 2.9.
Let (Λ , d ) be a row-finite source-free k -graph. Given a functor R : Λ → Z k , thefunction β : T k → Aut( C ∗ (Λ)) which satisfies β z ( t µ t ∗ ν ) = z R ( µ ) − R ( ν ) t µ t ∗ ν for all µ, ν ∈ Λ and z ∈ T k , is an action of T k on C ∗ (Λ) . In particular, we can apply the above Lemma whenever we have a function R : Λ → Z k suchthat, if we extend R to a function on G n by the formula R ( λ n · · · λ ) := R ( λ n ) + · · · + R ( λ ) ,R becomes a well-defined function on Λ.In addition to Theorem 2.8 and Lemma 2.9, our proofs that delay and reduction preserve Moritaequivalence will rely on Allen’s description [All08] of corners in k -graph C ∗ -algebras. To state Allen’sresult, we need the following definition. Definition 2.10.
Let (Λ , d ) be a row-finite k -graph. The saturation Σ( X ) of a set X ⊆ Λ ofvertices is the smallest set S ⊆ Λ which contains X and satisfies(1) (Heredity) If v ∈ S and r ( λ ) = v then s ( λ ) ∈ S ;(2) (Saturation) If s ( v Λ n ) ⊆ S for some n ∈ N k then v ∈ S .The following Theorem results from combining Remarks 3.2(2), Corollary 3.7, and Proposition4.2 from [All08]. Theorem 2.11. [All08] Let (Λ , d ) be a k -graph and X ⊆ Λ . Define P X = (cid:88) v ∈ X p v ∈ M ( C ∗ (Λ)) . If Σ( X ) = Λ , then P X C ∗ (Λ) P X is Morita equivalent to C ∗ (Λ) . In-splitting
The move of in-splitting a k -graph at a vertex v which we describe in this section should beviewed as the analogue of the out-splitting for directed graphs which was introduced by Bates andPask in [BP04]. This is because the Cuntz–Krieger conditions used by Bates and Pask to describethe C ∗ -algebra of a directed graph differ from the standard Cuntz–Krieger conditions for k -graphs.In the former, the source projection t ∗ e t e of each partial isometry t e , for e ∈ Λ , is required to equal p r ( e ) , whereas our Definition 2.5 requires t ∗ e t e = p s ( e ) . The following definition indicates the care that must be taken in in-splitting for higher-rankgraphs. The pairing condition of Definition 3.1 is necessary even for 2-graphs (cf. Examples 3.3below), but is vacuous for directed graphs. Although in- and out-splitting for directed graphs(cf. [BP04, Sø13]) allow one to “split” a vertex into any finite number of new vertices, the delicacyof the pairing condition has led us to “split” a vertex into only two new vertices.
Definition 3.1.
Let (Λ , d ) be a source-free k -graph with 1-skeleton G = (Λ , Λ , r, s ) and pathcategory G ∗ . Fix v ∈ Λ . Partition r − ( v ) ∩ Λ into two non-empty sets E and E satisfying the pairing condition : if a, f ∈ r − ( v ) ∩ Λ and there exist edges g, b ∈ Λ such that ag ∼ f b, then f and a are contained in the same set.We will use the partition E ∪ E of r − ( v ) ∩ Λ when we in-split Λ at v in Definition 3.5 below.First, however, we pause to examine some consequences of the pairing condition. Remark 3.2. If a (cid:54) = f are edges of the same color, then the relation ∼ underlying the k -graph Λ willnever satisfy ag ∼ f b . Thus, the pairing condition places no restrictions on edges of the same color.It follows that our definition of insplitting (Definition 3.5 below) is consistent with the definition ofinsplitting [BP04, Section 5] for directed graphs. OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 7 However, for k >
2, the pairing condition means that not all k -graphs can be in-split at allvertices. Satisfying the pairing condition requires that if f b ∼ ag then f, a are in the same set. Thismay force one of the sets E i to be empty, which is not allowed under Definition 3.1. Examples 3.3. ,(1) The property of having a valid partition E ∪ E of r − ( v ) ∩ Λ at a given vertex v dependsnot only on the 1-skeleton G of Λ, but also on the equivalence relation ∼ giving Λ = G ∗ / ∼ .For example, let Λ be a 2-graph with one vertex, Λ e = { a, b } and Λ e = { e, f } .(a) If we define ae ∼ ea, af ∼ f a, be ∼ eb, bf ∼ f b , repeatedly applying the pairingcondition gives E = { a, b, e, f } and so no valid partition is possible. Thus Λ cannot bein-split.(b) On the other hand, if we set ae ∼ ea, af ∼ eb, be ∼ f a, bf ∼ f b , we can take E = { a, e } and E = { b, f } . Thus in this case Λ can be in-split.(2) It may be possible to find two different valid partitions at a given vertex. Let Γ be a 2-graphwith one vertex, Γ e = { a, b, c, d } and Γ e = { e, f, g, h } , and the equivalence relation ae ∼ ea, af ∼ eb, ag ∼ ec, ah ∼ ed, be ∼ f a, bf ∼ f b, bg ∼ f c, bh ∼ f d,ce ∼ ga, cf ∼ gb, cg ∼ gc, ch ∼ gd, de ∼ hd, df ∼ hc, dg ∼ hb, dh ∼ ha. Then E = { a, c, e, g } , E = { b, f, d, h } and E = { a, e } , E = { b, c, d, f, g, h } are two parti-tions satisfying the pairing condition. Lemma 3.4.
For j ∈ { , } , E j has an edge of every color.Proof. Note that there exists e ∈ E j and s ( e ) is not a source. Thus for 1 ≤ i ≤ k there exists f ∈ r − ( s ( e )) ∩ Λ e i , and hence there exists a unique λ = λ λ ∈ G such that d ( λ ) = d ( e ), d ( λ ) = e i , and λ ∼ ef . Therefore, by the definition of E j , we have λ ∈ E j . (cid:3) Definition 3.5.
Let (Λ , d ) be a source-free k -graph. Fix v ∈ Λ and a partition E ∪E of r − ( v ) ∩ Λ satisfying Definition 3.1. We define the associated directed k -colored graph G I = (Λ I , Λ I , r I , s I ) withdegree map d I byΛ I = (Λ \ { v } ) ∪ { v , v } Λ I = (Λ \ s − ( v )) ∪ { f , f | f ∈ Λ , s ( f ) = v } , with d I ( g ) = d ( g ) for g ∈ Λ \ s − ( v ) and d I ( f i ) = d ( f ) . The range and source maps in G I are defined as follows:For f ∈ Λ such that s ( f ) (cid:54) = v, s I ( f ) = s ( f ) , for f ∈ Λ such that r ( f ) (cid:54) = v, r I ( f ) = r ( f ) and r I ( f i ) = r ( f ) , for f ∈ Λ such that s ( f ) = v, s I ( f i ) = v i for i = 1 , f ∈ Λ such that r ( f ) = v and f ∈ E i , r I ( f ) = v i . Examples 3.6. ,(1) The graph G shown below admits a unique equivalence relation ∼ such that G ∗ / ∼ is a2-graph Λ, because there is always at most one red-blue path (and the same number ofblue-red paths) between any two vertices. We may in-split at the vertex v with E = { a, e } and E = { b, f } . We duplicate x ∈ s − ( v ) to x , x with sources v and v respectively, and r ( x i ) = r ( x ) for each i . C. ECKHARDT, K. FIELDHOUSE, D. GENT, E. GILLASPY, I. GONZALES, AND D. PASK G • • •• v •• • •• pa qe xf b G I • • •• v •• v •• • pa qe x f x b Figure 2.
First example of in-splitting(2) We now give an example of an in-splitting where the vertex at which the splitting occurs hasa loop. The graph G in Figure 2 gives rise to multiple 2-graphs; we fix the 2-graph structureon G given by the equivalence relation ae ∼ ea, ce ∼ f a, gc ∼ bf, bg ∼ gb. Thus, the sets E = { c, f } , E = { b, g } satisfy the pairing condition, and we can in-split at v . G • v e a fc gb G I • v v e a fc g b g b Figure 3.
In-splitting at a vertex v which has loops. Remark 3.7.
While the vertex v at which we in-split the graph G from Figure 2 is a sink, andhence could also be handled by the methods of Section 5 below, one could easily modify Λ to besink-free (at the cost of a more messy 1-skeleton diagram) without changing the essential structureof the in-splitting at v .In order to describe the factorization on G I which will make it a k -graph, we first introduce somenotation. Definition 3.8.
Define a function par : G ∗ I → G ∗ by par ( w ) = w for all w ∈ Λ \{ v } and par ( v i ) = v for i = 1 , ,par ( f i ) = f, for f i ∈ { f , f | f ∈ Λ , s ( f ) = v } par ( f ) = f, for f ∈ Λ I \ { f , f | f ∈ Λ , s ( f ) = v } par ( λ ) = par ( λ ) · · · par ( λ n ) , for λ = λ · · · λ n ∈ G ∗ I . The effect of the function par is to remove the superscript on any edge (or path) of G I , returningits “parent” in G (or G ∗ ). OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 9 Definition 3.9.
We define an equivalence relation on G ∗ I by λ ∼ I µ if and only if par ( λ ) ∼ par ( µ ), r I ( λ ) = r I ( µ ), and s I ( λ ) = s I ( µ ). Define Λ I := G ∗ I / ∼ I ; we say that Λ I is the result of in-splitting Λ at v . Examples 3.10. ,(1) Consider again the directed colored graph of Example 3.6(1). Observe that x a ∼ I qp in G I since both paths have the same range and source in G I and [ par ( x a )] = [ xa ] = [ qp ] =[ par ( qp )] in Λ = G ∗ / ∼ .(2) In the directed colored graph of Example 3.6(2), we have b g ∼ I g b as both paths havethe same source and range and [ par ( b g )] = [ bg ] = [ gb ] = [ par ( g b )] in Λ. Observe thatalthough G admitted multiple factorizations, G I admits only this one. Remark 3.11. ,(1) For any λ ∈ G ∗ I , if s ( λ ) = s ( µ ) and par ( λ ) = par ( µ ) , we have λ = µ. To see this, observe thatby definition an edge e ∈ Λ satisfies e = par ( µ ) for at most two edges µ ∈ Λ I . If par ( µ ) = par ( ν ) and ν (cid:54) = µ , then without loss of generality we may assume s ( µ ) = v , s ( ν ) = v . Consequently, for η ∈ Λ I , at most one of the paths µη, νη is in G ∗ I , according to whether η ∈ E or η ∈ E . This implies our assertion.(2) Similarly, for any path λ ∈ G ∗ , we have λ = par ( µ ) for at least one path µ ∈ G ∗ I . Theorem 3.12. If (Λ , d ) is a source-free k -graph, then the result (Λ I , d I ) of in-splitting Λ at avertex v is also a source-free k -graph.Proof. Let (Λ , d ) be a source-free k -graph and let (Λ I , d I ) be produced by in-splitting at some v ∈ Λ . First note that Λ I satisfies (KG0) by our definition of par and the fact that Λ has thefactorization property. Lemma 3.4 and our hypothesis that Λ be source-free guarantee that allvertices in Λ I receive edges of all colors, so Λ I is source-free. Consider some path λ ∈ G ∗ I withcolor order ( m , . . . , m n ). Note that par ( λ ) also has color order ( m , . . . , m n ), and since Λ is a k -graph, for any permutation ( c , . . . , c n ) of ( m , . . . , m n ), there exists a unique µ (cid:48) ∈ Λ that hascolor order ( c , . . . , c n ) and µ (cid:48) ∈ [ par ( λ )]. By Remark 3.11, there exists a unique path µ ∈ Λ I suchthat par ( µ ) = µ (cid:48) and s ( µ ) = s ( λ ). By construction, µ has color order ( c , . . . , c n ) and µ ∈ [ λ ] I .Thus, [ λ ] I contains a unique path for each permutation of the color order of λ , and so (KG4) issatisfied. Therefore, by Theorem 2.3, Λ I is a k -graph. (cid:3) Theorem 3.13.
Let (Λ , d ) be a row-finite, source-free k-graph, and let (Λ I , d I ) be the in-split graphof Λ at the vertex v for the partition E ∪ E of r − ( v ) ∩ Λ . We have C ∗ (Λ I ) ∼ = C ∗ (Λ) .Proof. Let { s λ : λ ∈ Λ I ∪ Λ I } be the canonical Cuntz-Krieger Λ I -family which generates C ∗ (Λ I ).For λ ∈ Λ ∪ Λ , define T λ = (cid:88) par ( e )= λ s e . We first prove that { T λ : λ ∈ Λ ∪ Λ } is a Cuntz-Krieger Λ-family in C ∗ (Λ I ). Note that the set { T λ : λ ∈ Λ } is a collection of non-zero mutually orthogonal projections since each T λ is a sum ofprojections satisfying the same properties. Therefore { T λ : λ ∈ Λ } satisfies (CK1). Now, choose ab, cd ∈ G such that [ ab ] = [ cd ]. As in Remark 3.11, observe that the sum defining T a containsat most two elements, and the only way it will contain two elements is if s ( a ) = v . In that case, if ab ∈ G , then either b ∈ E or b ∈ E , so if f ∈ G I satisfies par ( f ) = b then r ( f ) ∈ { v , v } , and sothere is only one path ef ∈ G I whose parent is ab . Making the same argument for the paths in G I with parent cd and using the factorization rule in Λ I , we obtain T a T b = (cid:88) par ( e )= a s e (cid:88) par ( f )= b s f = (cid:88) par ( ef )= ab s e s f = (cid:88) par ( gh )= cd s g s h = T c T d . Thus { T λ : λ ∈ Λ ∪ Λ } satisfies (CK2). Now take f ∈ Λ . If s ( f ) (cid:54) = v we have T ∗ f T f = s ∗ f s f = s s ( f ) = T s ( f ) . If s ( f ) = v we have { g : par ( g ) = f } = { f , f } , so the fact that v = s ( f ) (cid:54) = s ( f ) = v impliesthat T ∗ f T f ( s f + s f ) ∗ ( s f + s f ) = s ∗ f s f + s ∗ f s f = s v + s v = T v . Thus { T λ : λ ∈ Λ ∪ Λ } satisfies (CK3). Finally fix a generator e i ∈ N k and fix w ∈ Λ . Wefirst observe that if two distinct edges in Λ I have the same parent, they must have different sources(namely v and v ) and orthogonal range projections, and therefore, for λ ∈ Λ , (cid:88) par ( e )= λ s e (cid:88) par ( e )= λ s ∗ e = (cid:88) par ( e )= λ s e s ∗ e . It follows that (cid:88) d ( e )= e i r ( e )= w T e T ∗ e = (cid:88) d ( e )= e i r ( e )= w (cid:88) par ( f )= e s f (cid:88) par ( f )= e s ∗ f = (cid:88) d ( e )= n i r ( e )= w (cid:88) par ( f )= e s f s ∗ f = (cid:88) d ( f )= e i r ( par ( f ))= w s f s ∗ f = (cid:88) par ( x )= w s x = T w . Therefore { T λ : λ ∈ Λ ∪ Λ } satisfies (CK4), and hence is a Cuntz–Krieger Λ family in C ∗ (Λ I ).Thus, by the universal property of C ∗ (Λ), there exists a ∗ -homomorphism π : C ∗ (Λ) → C ∗ (Λ I ) suchthat π ( t λ ) = T λ , where { t λ : λ ∈ Λ ∪ Λ } are the canonical generators of C ∗ (Λ). We will show that π is an isomorphism.Fix w ∈ Λ I and note that if par ( w ) (cid:54) = v then s w = T w ∈ π ( C ∗ (Λ)). Conversely if par ( w ) = v then w = v j for some j ∈ { , } . Thus, for a fixed generator e i ∈ N k we have (cid:88) r ( e )= vd ( e )= e i e ∈E j T e T ∗ e = (cid:88) r ( e )= vd ( e )= e i e ∈E j (cid:88) par ( e (cid:48) )= e s e (cid:48) s ∗ e (cid:48) = (cid:88) r ( e (cid:48) )= v j d ( e (cid:48) )= e i s e (cid:48) s ∗ e (cid:48) = s v j ∈ π ( C ∗ (Λ)) . Thus, all of the vertex projections of C ∗ (Λ I ) are in Im( π ), and therefore π ( C ∗ (Λ)) contains all ofthe generators of C ∗ (Λ I ). Hence π is surjective.Consider the canonical gauge actions α of T k on C ∗ (Λ I ) and β of T k on C ∗ (Λ). Observe that forall z ∈ T k , the fact that par is degree-preserving implies that α z ( T λ ) = (cid:88) par ( µ )= λ α z ( s µ ) = z d ( λ ) (cid:88) par ( µ )= λ s µ = z d ( λ ) T λ . Therefore, π ( β z ( t λ )) = π ( z d ( λ ) t λ ) = z d ( λ ) T λ = α z ( T λ ) = α z ( π ( t λ )) , so π intertwines the canonical gauge actions. The gauge invariant uniqueness theorem now impliesthat π is injective, and so C ∗ (Λ) ∼ = C ∗ (Λ I ) as claimed. (cid:3) Delay
Our goal in this section is to generalize to k -graphs the operation of delaying a graph at an edge –that is, breaking an edge in two by adding a vertex in the “middle” of the edge. The importance ofthis construction can be traced back to Parry and Sullivan’s analysis [PS75] of flow equivalence forshifts of finite type; Drinen realized [Dri99] that in the setting of directed graphs, these edge delayscorrespond to the expansion matrices used by Parry and Sullivan to complete the charaterizationof flow equivalence for shifts of finite type. Bates and Pask later generalized the “delay” operationin [BP04] and showed that the C ∗ -algebra of a delayed graph is Morita equivalent to the C ∗ -algebraof the original graph. OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 11 In the setting of higher-rank graphs, the “delay” operation becomes more intricate. So that theresulting object satisfies the factorization rule, after delaying one edge and adding a new vertex, newedges of other colors (incident with the new vertex) must be added. This procedure is described inDefinition 4.1 below, and Theorem 4.2 establishes that the resulting object Λ D is indeed a k -graph.Theorem 4.3 then proves that C ∗ (Λ D ) is Morita equivalent to the C ∗ -algebra of the original k -graphΛ. Definition 4.1.
Let (Λ , d ) be a k -graph and G = (Λ , Λ , r, s ) its underlying directed graph. Fix f ∈ Λ ; without loss of generality, assume d ( f ) = e . We first recursively define the set E e of allpossible elements of Λ e which will be affected by delaying f , in that elements of E e are oppositeto f in some commuting square in Λ. Namely, we set A = { f } ∪ { g ∈ Λ e : ag ∼ f b or ga ∼ bf where a, b ∈ Λ e i for 2 ≤ i ≤ k } ,A m = { e ∈ Λ e : ag ∼ eb or ga ∼ be where a, b ∈ Λ e i for 2 ≤ i ≤ k, g ∈ A m − } , E e = ∞ (cid:91) j =1 A j ⊆ Λ e . In the pictures below, the dashed edges would all lie in E e . • • • •• • • • f α β γ or • • • •• • • • ξ η ζ f Using E e we identify those commuting squares of degree ( e + e i ), i (cid:54) = 1 in Λ which contain an edgefrom E e . These squares will form the set E e i : E e i = { [ ga ] ∈ Λ : g ∈ E e , a ∈ Λ e i } . In the pictures above, if the solid black edges have degree e i , we have α, β, γ, ξ, η, ζ ∈ E e i . By delaying f , these squares will be turned into rectangles.To be precise, in the delayed graph, we will “delay” every edge in E e , replacing it with two edges: • • • •• • • •• • • • f f or • • • •• • • •• • • • f f E e D = { g , g : g ∈ E e } and add an edge for every square that has been turned into a rectangle. • • • •• • • •• • • • f f e α e β e γ or • • • •• • • •• • • • f e ξ e η e ζ f E e i D = { e α : α ∈ E e i } . Then define the k -colored graph G D = (Λ D , Λ D , r D , s D ) byΛ D = Λ ∪ { v g } g ∈E e , Λ e D = (Λ e \ E e ) ∪ E e D , with s D ( g ) = s ( g ) , s D ( g ) = v g , r D ( g ) = v g , r D ( g ) = r ( g );Λ e i D = Λ e i ∪ E e i D , with s D ( e α ) = v g such that bg represents α and d ( g ) = e ,r D ( e α ) = v h such that ha represents α and d ( h ) = e . In words, to construct G D from G , we add one vertex per delayed edge; each delayed edge becomestwo edges in G D ; and we add one edge for each square that was stretched into a rectangle by delayingthe edges in E . If α ∈ E e i we set d ( e α ) = e i , and all other edges inherit their degree from Λ.Let ι : G D → G be the partially defined inclusion map with domain (Λ D ∪ Λ D ) \ ( { k (cid:83) i =1 E e i D } ∪ { v e : e ∈ E e } ). Then, for edges g ∈ Λ D \ (cid:83) ki =1 E e i D , we can define s D ( g ) = s ( ι ( g )) , r D ( g ) = r ( ι ( g )) , d D ( g ) = d ( ι ( g )) . Now let G ∗ D be the path category for G D and define the equivalence relation ∼ D on bi-color paths µ = µ µ ∈ G D according to the following rules.Case 1: Assume µ , µ / ∈ (cid:83) ki =1 E e i D . Then we set [ µ ] D = ι − ([ ι ( µ )]).Case 2: Suppose µ j lies in E e D , so that µ j ∈ { g , g } for some edge g ∈ E e . If j = 1 and µ = g ,then r ( µ ) = s ( µ ) = ι − ( r ( g )) ∈ ι − (Λ ), and the edges in G D with source in ι − (Λ ) and degree e i for i (cid:54) = 1 are in ι − (Λ ). Therefore µ ∈ ι − (Λ e i ), and ι ( µ ) g is a bi-color path in G , so ι ( µ ) g ∼ ha for edges h ∈ E e , a ∈ Λ e i . There is then an edge e [ µ g ] ∈ Λ e i D with source s ( µ ) = v g and range v h = s ( h ); we define µ µ = µ g ∼ D h e [ µ g ] . • •• • ag hb −→ • •• •• • ag h e [ bg ] g h b Figure 4.
A commuting square in G and its “children” in G D , when h, g ∈ E e .If j = 1 and µ = g , the only edges in G D with source r ( g ) = v g and degree e i for i (cid:54) = 1 are ofthe form e [ bg ] = e [ ha ] for some commuting square bg ∼ ha in Λ. In this case, we will have h ∈ E e ,and r ( h ) = v h = r ( e [ bg ] ), so we set e [ bg ] g ∼ D h a .A similar argument shows that if j = 2, the path µ µ will be of the form h a or h e [ ha ] , whosefactorizations we have already described.Case 3: Assume µ is of the form e β e α for α ∈ E e i D , and β ∈ E e j D with i (cid:54) = j . Now s D ( e β ) = r D ( e α ) = v g for some g ∈ E e , and consequently α, β ∈ Λ are linked as shown on the left of Figure5. Since Λ is a k -graph, the 3-color path outlining βα generates a 3-cube in Λ, which is depicted onthe right of Figure 5. OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 13
16 C. ECKHARDT, K. FIELDHOUSE, D. GENT, E. GILLASPY, I. GONZALES, AND D. PASK • •• • ag hb ! • •• •• • ag h e [ bg ] g h b Figure 4.
A commuting square in G and its “children” in G D , when h, g e .If j = 1 and µ = g , the only edges in G D with source r ( g ) = v g and degree e i for i = 1 are of the form e [ bg ] = e [ ha ] for some commuting square bg ⇠ ha in ⇤. Inthis case, we will have h e , and r ( h ) = v h = r ( e [ bg ] ), so we set e [ bg ] g ⇠ D h a .A similar argument shows that if j = 2, the path µ µ will be of the form h a or h e [ ha ] , whose factorizations we have already described.Case 3: Assume µ is of the form e e ↵ for ↵ e i D , and e j D with i = j . Now s D ( e ) = r D ( e ↵ ) = v g for some g e , and consequently ↵, ⇤ are linked asshown on the left of Figure 5. Since ⇤ is a k -graph, the 3-color path outlining ↵ generates a 3-cube in ⇤, which is depicted on the right of Figure 5. • •• • •• g ↵ ⇠ • •• •• •• • ↵ Figure 5.
The commuting squares of edges from S ki =2 E e i D .Let and denote the faces of this cube which lie, respectively, opposite and ↵ .Since g e , all of the vertical edges of this cube are in E e , and so e j , e i . Moreover, the path e e is composable in ⇤ D , and has the same source and range as e e ↵ . Set e e ↵ ⇠ D e e .Observe that there are no two-color paths in G D of the form ge ↵ or e ↵ g for g ◆ (⇤ ) and ↵ e i , since r D ( g ) , s D ( g ) ◆ (⇤ ) but r D ( e ↵ ) , s D ( e ↵ ) v e : e e } . Extend ⇠ D to be an equivalence relation on G ⇤ D which satisfies (KG0) and (KG1);observe that ⇠ D satisfies (KG2) by construction. Define ⇤ D = G ⇤ D / ⇠ D . We call ⇤ D the graph of ⇤ delayed at the edge e . Theorem 4.2. If ⇤ is a row-finite source-free k -graph, then ⇤ D is also a source-free k -graph. Figure 5.
The commuting squares of edges from (cid:83) ki =2 E e i D .Let δ and γ denote the faces of this cube which lie, respectively, opposite β and α . Since g ∈ E e ,all of the vertical edges of this cube are in E e , and so δ ∈ E e j , γ ∈ E e i . Moreover, the path e γ e δ iscomposable in Λ D , and has the same source and range as e β e α . Set e β e α ∼ D e γ e δ .Observe that there are no two-color paths in G D of the form ge α or e α g for g ∈ ι − (Λ ) and α ∈ E e i , since r D ( g ) , s D ( g ) ∈ ι − (Λ ) but r D ( e α ) , s D ( e α ) ∈ { v e : e ∈ E e } . Extend ∼ D to be an equivalence relation on G ∗ D which satisfies (KG0) and (KG1); observe that ∼ D satisfies (KG2) by construction. Define Λ D = G ∗ D / ∼ D . We call Λ D the graph of Λ delayed atthe edge e . Theorem 4.2. If Λ is a row-finite source-free k -graph, then Λ D is also a source-free k -graph.Proof. Let (Λ , d ) be a k -graph, and let (Λ D , d D ) be the graph of Λ delayed at the edge e ∈ Λ e .Since ∼ D satisfies (KG0), (KG1), and (KG2) by construction, it suffices to show that ∼ D satisfies(KG3). Let µ = µ µ µ ∈ G D be a tri-colored path. Consider the following cases.Case 1: Assume µ j / ∈ k (cid:83) i =1 E e i D for all j ∈ { , , } . Then ι ( µ ) is a 3-colored path in Λ. Since wedefined [ µ ] D = ι − ([ ι ( µ )]), the fact that Λ satisfies (KG3) – hence, that ι ( µ ) uniquely determines a3-cube in Λ – implies that µ also gives rise to a well-defined 3-cube in Λ D .Case 2: Assume µ j ∈ E e D for one j ∈ { , , } . Then µ has one of the forms of tri-colored pathson the right-hand side of Figure 6. This follows from Definition 4.1, specifically the restrictions forwhen an edge in ι − (Λ ) can precede or follow an edge in E e D , and when an edge in E e i D can precedeor follow an edge in E e D .For example, suppose µ = g for some g ∈ E e . Then s D ( µ ) = r D ( µ ) = v g , so µ must beof the form e γ for some γ ∈ E e i with γ = [ gq ]. Then r D ( µ ) = s D ( µ ) = ι − ( r ( g )), and since d ( µ ) (cid:54)∈ { e , e i } we must have µ ∈ ι − (Λ e j ) for some j (cid:54) = i . But then, ι − ( µ ) gq ∈ G is a3-color path, which defines a unique 3-cube in Λ (as depicted on the left of Figure 6). It is nowstraightforward to check that the two routes for factoring µ in Λ D (as in (KG3)) arise as “children”of this cube in G D , so the fact that Λ is a k -graph implies that the two routes for factoring µ in Λ D lead to the same result. A similar analysis of the other possibilities for having one edge µ j ∈ E e D reveals that whenever this occurs, the factorization of µ in Λ D satisfies (KG3). v v v v v v v v aeq r fbg hs c td −→ v v v v v e v f v g v h v v v v ae q r f g b h e α e e γ e β f g e δ h s c td Figure 6.
A commuting cube in G and its “children” in G D , when e, f, g, h ∈ E e D We now observe that if a tri-colored path without an edge from E e D contains an edge from (cid:83) ki =2 E e i D , it must consist entirely of edges in (cid:83) ki =2 E e i D . To see this, suppose that a tri-colored path contains e γ for a commuting square γ ∈ Λ, but that µ contains no edges in E e D . Since s ( γ ) , r ( γ ) ∈ Λ D \ ι − (Λ ),the edge(s) preceding and following e γ must be of the form e α for some α ∈ E e i . Repeating theargument for e α if necessary shows that µ consists entirely of edges in (cid:83) ki =2 E e i D .Thus, the only remaining case isCase 3: Assume µ j ∈ k (cid:83) i =2 E e i D for all j ∈ { , , } , and without loss of generality assume µ = e α e β e γ is a blue-red-green path. Because of the definition of s D , r D for edges of the form e λ in G D , µ ∈ G ∗ D arises from a sequence of commuting squares α, β, γ in Λ which share edges in E e . Figure 7 belowdepicts (from left to right) the squares γ, β, α ∈ Λ. The color in each square λ reflects the color ofits horizontal edges, as these determine the degree of the edge e λ ∈ G D .
18 C. ECKHARDT, K. FIELDHOUSE, D. GENT, E. GILLASPY, I. GONZALES, AND D. PASK must be of the form e ↵ for some ↵ e i . Repeating the argument for e ↵ if necessaryshows that µ consists entirely of edges in S ki =2 E e i D .Thus, the only remaining case isCase 3: Assume µ j k S i =2 E e i D for all j , , } , and without loss of generalityassume µ = e ↵ e e is a blue-red-green path. Because of the definition of s D , r D foredges of the form e in G D , µ G ⇤ D arises from a sequence of commuting squares ↵, , in ⇤ which share edges in E e . Figure 7 below depicts (from left to right) thesquares , , ↵ ⇤. The color in each square reflects the color of its horizontaledges, as these determine the degree of the edge e G D . • • • •• • • • Figure 7.
Factorization squares in ⇤ that will be delayed to produce µ Because ⇤ is a k -graph, the rectangle in Figure 7, which is reproduced in the topline of Figure 8, determines a unique 4-dimensional cube in ⇤. Thus, as we followRoute 1 of (KG3) and the instructions given in Case 3 of Definition 4.1 to factor e ↵ e e = e ↵ e ⌘ e = e e ✏ e = e e e , the squares – and indeed all of the intermediate squares – must lie on the 4-dimensional cube determined by ↵ . To be precise, is the collection of green-red-blue squares on the left of the bottom row of Figure 8. Similarly, when we factor e ↵ e e via Route 2 of (KG3), we obtain the green-red-blue squares on the right of thebottom row of Figure 8. Because these squares lie on the same 4-cube as , andin the same position (compare the position of ⌫ on both), they must equal . Itfollows that applying either Route 1 or Route 2 to e ↵ e e gives us the same 3-coloredpath in G D . Figure 7.
Factorization squares in Λ that will be delayed to produce µ Because Λ is a k -graph, the rectangle in Figure 7, which is reproduced in the top line of Figure8, determines a unique 4-dimensional cube in Λ. Thus, as we follow Route 1 of (KG3) and theinstructions given in Case 3 of Definition 4.1 to factor e α e β e γ = e α e η e κ = e δ e (cid:15) e κ = e δ e φ e λ , the squares δφλ – and indeed all of the intermediate squares – must lie on the 4-dimensional cubedetermined by αβγ . To be precise, δφλ is the collection of green-red-blue squares on the left of thebottom row of Figure 8. Similarly, when we factor e α e β e γ via Route 2 of (KG3), we obtain thegreen-red-blue squares on the right of the bottom row of Figure 8. Because these squares lie on thesame 4-cube as δφλ , and in the same position (compare the position of ν on both), they must equal δφλ . It follows that applying either Route 1 or Route 2 to e α e β e γ gives us the same 3-colored pathin G D . OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 15 • • • •• • • • ν ν ν ν (cid:46) (cid:38)• •• • •• •• • • ν ν ν ν • •• • •• •• • • ν ν ν ν ↓ ↓• • •• • •• •• • • ν ν ν ν • • •• • •• • •• • • ν ν ν ν ↓ ↓•• • •• • ••• • •• • • ν ν ν ν = •• • •• • ••• • •• • • ν ν ν ν Figure 8.
Associativity in Λ D via factorization squares in ΛHaving confirmed that the factorization of an arbitrary tri-colored path in G D satisfies (KG3),we conclude that Λ D is a k -graph.It remains to check that Λ D is source-free and row-finite whenever Λ is. In constructing Λ D , allnewly-created vertices v g have an edge g of color 1 emanating from them. Moreover, since r ( g )is not a source in Λ, there is an edge b i ∈ Λ e i r ( g ) for each i ≥
2. Then, [ b i g ] ∈ E e i and hence e [ b i g ] ∈ Λ e i D v g for all i ≥
2. In other words, all of the new vertices v g emit at least one edge of eachcolor.Similarly, every vertex v ∈ ι − (Λ ) emits an edge of each color, because the same is true in Λ;if v emits an edge g ∈ E e then s D ( g ) = ι − ( s ( g )) = v , and all other edges emitted by ι ( v ) are in ι (Λ D ) and hence also occur in Λ D .Furthermore, the number of edges with range v in Λ D is the same as the number of edges withrange ι ( v ) ∈ Λ, if v (cid:54) = v g . In this setting, an edge in v Λ D \ ι − ( v Λ D ) is necessarily of the form g forsome g ∈ v E e , so | r − D ( v ) | = | v E e | + | ι − ( v Λ D ) | = | r − ( ι ( v )) | . If v = v g , then r − D ( v ) is still finite as long as Λ is row-finite: | r − D ( v g ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { g } ∪ k (cid:91) i =2 v g E e i D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) { α ∈ Λ : α = [ gb ] for some b ∈ Λ } (cid:12)(cid:12) = 1 + r − ( s ( g )) < ∞ . We conclude that Λ D is a row-finite, source-free k -graph whenever Λ is. (cid:3) Theorem 4.3.
Let (Λ , d ) be a row-finite, source-free k-graph and let (Λ D , d D ) be the graph of Λ delayed at an edge f . Then C ∗ (Λ D ) is Morita equivalent to C ∗ (Λ) .Proof. Let { t λ : λ ∈ Λ D ∪ Λ D } be the canonical Cuntz–Krieger Λ D -family generating C ∗ (Λ D ).Define S v = t v , for v ∈ Λ ,S h = (cid:26) t ι − ( h ) if h / ∈ E e t h t h if h ∈ E e , for h ∈ Λ . We claim that { S λ : λ ∈ Λ ∪ Λ } is a Cuntz–Krieger Λ-family in C ∗ (Λ D ). Note that since { t v : v ∈ Λ D } are mutually orthogonal projections, so are { S v : v ∈ Λ } . Therefore { S λ : λ ∈ Λ ∪ Λ } satisfies (CK1). Now take arbitrary a, b, g, h ∈ Λ such that ah ∼ gb . Assuming a, b, g, h / ∈ E e , Case1 of Definition 4.1 implies that S a S h = t ι − ( a ) t ι − ( h ) = t ι − ( g ) t ι − ( b ) = S g S b . Conversely, suppose either a, b ∈ E e , or g, h ∈ E e . Without loss of generality assume g, h ∈ E e .Then α := [ ah ] ∈ E e i D satisfies e α h ∼ D g b and ah ∼ D g e α . Hence S a S h = t a t h t h = t g t e α t h = t g t g t b = S g S b . Therefore { S λ : λ ∈ Λ ∪ Λ } satisfies (CK2).For (CK3), let h ∈ Λ \ E e ; observe that S h = t h , and hence S ∗ h S h = t ∗ h t h = t s D ( h ) = S s ( h ) . If h ∈ E e , we similarly have S ∗ h S h = t ∗ h t ∗ h t h t h = t s D ( h ) = S s ( h ) . Therefore { S λ : λ ∈ Λ ∪ Λ } satisfies (CK3).Finally, take an arbitrary v ∈ Λ . Observe that if h ∈ E e , then h is the only edge in Λ D suchthat d D ( h ) = e and r D ( h ) = v h . Thus t v h = t h t ∗ h and consequently (cid:88) h ∈ r − ( v ) ∩ Λ e S h S ∗ h = (cid:88) h ∈ r − ( v ) ∩ Λ e h/ ∈E e t ι − ( h ) t ∗ ι − ( h ) + (cid:88) h ∈ r − ( v ) ∩ Λ e h ∈E e t h t h t ∗ h t ∗ h = (cid:88) h ∈ r − ( v ) ∩ Λ e h/ ∈E e t ι − ( h ) t ∗ ι − ( h ) + (cid:88) h ∈ r − ( v ) ∩ Λ e h ∈E e t h t ∗ h = (cid:88) e ∈ r − D ( v ) ∩ Λ e D t e t ∗ e = t v = S v . Now, if 2 ≤ i ≤ k , the fact that v ∈ Λ means that ι − ( v )Λ e i D = ι − ( v Λ e i ): there are no edges of theform h j or e α with range v and degree e i . Consequently, (cid:88) h ∈ Λ ei S h S ∗ h = (cid:88) h ∈ v Λ ei t h t ∗ h = t v = S v . Therefore { S λ : λ ∈ Λ ∪ Λ } satisfies (CK4), and hence is a Cuntz–Krieger Λ-family in C ∗ (Λ D ).By the universal property of C ∗ (Λ), there exists a homomorphism π : C ∗ (Λ) → C ∗ (Λ D ).To see that that π is injective, we use Theorem 2.8 and Lemma 2.9. Define β : T k → Aut( C ∗ (Λ D ))by setting, for all z ∈ T k , β z ( t h ) = z d ( h ) t h , for h (cid:54) = g ,β z ( t h ) = t h , for h = g ,β z ( t v ) = t v , for v ∈ Λ , and extending β to be linear and multiplicative. By taking E = { g : g ∈ E e } and applyingLemma 2.9, we see that β is an action of T k on C ∗ (Λ D ). Let α be the canonical gauge action on OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 17 C ∗ (Λ) = C ∗ ( { s λ : λ ∈ Λ ∪ Λ } ) and note that for e / ∈ E e we have π [ α z ( s e )] = π [ z d ( e ) s e ] = z d ( e ) S e = z d ( e ) t e = β z ( t e ) = β z ( T e ) = β z ( π [ s e ])and for e ∈ E e , π [ α z ( s e )] = π [ z d ( e ) s e ] = z d ( e ) S e = z d ( e ) t e t e = β z ( t e t e ) = β z ( T e ) = β z ( π [ s e ]) . It is straightforward to check that α and β commute on the vertex projections. Therefore, β commutes with the canonical gauge action, so Theorem 2.8 implies that π is injective.To see that Im( π ) ∼ = C ∗ (Λ) is Morita equivalent to C ∗ (Λ D ), we invoke Theorem 2.11. Set X = ι − (Λ ) ⊆ Λ D ; we will show that the saturation Σ( X ) of X is Λ D . If g ∈ E e , then r ( g ) ∈ ι (Λ D )and g ∈ ι − ( r ( g ))Λ D . Therefore if H is hereditary and contains X , we must have s D ( g ) = v g ∈ H for all g ∈ E e . Consequently, H = Λ D . Since Λ D is evidently saturated, we have Σ( X ) = Λ D asclaimed. Theorem 2.11 therefore implies that P X C ∗ (Λ D ) P X ∼ = ME C ∗ (Λ D ) . We will now complete the proof that C ∗ (Λ) ∼ = ME C ∗ (Λ D ) by showing that P X C ∗ (Λ D ) P X = Im( π ) ∼ = C ∗ (Λ) . The generators { S λ : λ ∈ Λ ∪ Λ } of Im( π ) all satisfy P X S h P X = S h , so Im( π ) ⊆ P X C ∗ (Λ D ) P X .For the other inclusion, note that P X C ∗ (Λ D ) P X = span { t λ t ∗ µ : λ, µ ∈ G ∗ D , s D ( λ ) = s D ( µ ) , r ( λ ) , r ( µ ) ∈ X } . Given λ ∈ G ∗ D with r D ( λ ) ∈ X , create λ (cid:48) ∈ [ λ ] D by first replacing any path of the form g e [ ah ] in λ with its equivalent ah , and then replacing paths of the form e [ gb ] h with their equivalent g b . Notethat since r ( λ ) = r D ( λ ) ∈ X we cannot have λ | λ | ∈ E e i D for any i (cid:54) = 1 . Because of this, λ (cid:48) will containno edges in (cid:83) ki =2 E e i D .Thus, in λ (cid:48) , any occurrence of an edge of the form g will be preceded by g unless s D ( λ (cid:48) ) (cid:54)∈ X (in which case, if s D ( λ (cid:48) ) = v g , λ (cid:48) = g ). Consequently, if s D ( λ (cid:48) ) ∈ X , then t λ (cid:48)| λ | · · · t λ (cid:48) t λ (cid:48) = t λ (cid:48) = t λ is a product of operators of the form S h , S ∗ k for h, k ∈ Λ .Similarly, given µ ∈ G ∗ D with r D ( µ ) ∈ X and s D ( µ ) = s D ( λ ), create µ (cid:48) ∈ [ µ ] D by the procedureabove, so that µ (cid:48) contains no edges in (cid:83) ki =2 E e i D and any edge in µ (cid:48) of the form g (with the possibleexception of µ (cid:48) ) is preceded by g . It follows that for any µ, λ with r D ( µ ) , r D ( λ ) ∈ X and s D ( µ ) = s D ( λ ) ∈ X we have t λ t ∗ µ ∈ Im( π ).If t λ t ∗ µ ∈ P X C ∗ (Λ D ) P X and s D ( λ ) (cid:54)∈ X , write v g = s D ( λ ). As observed earlier, in this case wemust have λ (cid:48) = µ (cid:48) = g . Moreover, since v g receives precisely one edge of degree e (namely g ) wehave p v g = t g t ∗ g . It follows that t λ t ∗ µ = t λ (cid:48) t ∗ µ (cid:48) = t λ (cid:48) g t ∗ µ (cid:48) g . Observe that neither λ (cid:48) g nor µ (cid:48) g contains any edge in (cid:83) ki =2 E e i D , and every occurrence of an edgeof the form h in either path is preceded by h . Thus, in this case as well we can write t λ t ∗ µ as aproduct of operators of the form S h , S ∗ k .Since Im( π ) is norm-closed, it follows that every element in span { t µ t ∗ λ : λ, µ ∈ Λ D , r ( λ ) = r ( µ ) ∈ X } = P X C ∗ (Λ D ) P X lies in Im( π ). Thus, C ∗ (Λ) ∼ = Im( π ) = P X C ∗ (Λ D ) P X ∼ = ME C ∗ (Λ D ) , as claimed. (cid:3) Sink Deletion
In this section, we analyze the effect on C ∗ (Λ) of deleting a sink – a vertex which emits noedges of a certain color – from Λ. This should be viewed as the analogue of Move (S) , removinga regular source, for directed graphs, as the conventions used to define a Cuntz–Krieger familyin [ERRS16,Sø13] differ from the standard conventions for higher-rank graph C ∗ -algebras. We showin Theorem 5.6 that the result of deleting a sink from a k -graph is still a k -graph, and Theorem 5.7shows that the resulting C ∗ -algebra is Morita equivalent to the original k -graph C ∗ -algebra. Definition 5.1.
Let Λ be a k -graph. We say v ∈ Λ is an e i sink if s − ( v ) ∩ Λ e i = ∅ for some1 ≤ i ≤ k . We say v is a sink if it is an e i sink for some 1 ≤ i ≤ k . Definition 5.2.
Let (Λ , d ) be a k -graph. Let G = (Λ , Λ , r, s ), and G ∗ be its 1-skeleton andcategory of paths respectively. Let v ∈ Λ be a sink. We write w ≤ v if there exists a λ ∈ G ∗ suchthat s ( λ ) = v and r ( λ ) = w . Define the directed colored graph G S = (Λ S , Λ S , r S , s S ) byΛ S := { w : w (cid:54)≤ v } , Λ S := Λ \ { f ∈ Λ : r ( f ) ≤ v } ;we set r S = r, s S = s, d S = d . Let ι : G ∗ S → G ∗ be the inclusion map, and define an equivalencerelation on G ∗ S by µ ∼ λ when [ ι ( µ )] = [ ι ( λ )] ∈ Λ. Define Λ S = G ∗ / ∼ and call Λ S the k - graph of Λ with the sink v deleted. Example 5.3.
The graphs G and G S after deleting the blue sink v , where blue is the dashed color. G v •• • •• • G S •• •• • Example 5.4.
The following example highlights the fact that performing a sink-deletion may intro-duce new sinks. When Λ has a finite vertex set, performing successive sink deletions will eventuallyproduce a sink-free k -graph. G • w v G S • w Figure 9.
Sink deletion at v creating a new sink at w . Lemma 5.5. If (Λ , d ) is a k-graph with v ∈ Λ an e i sink, then { x ∈ Λ : x ≤ v } consists of e i sinks.Proof. If w ≤ v and w is not an e i sink, then there exists a y ∈ Λ and f ∈ Λ e i such that s ( f ) = w and r ( f ) = y . Thus there exists a path f λ ∈ s − ( v ) ∩ r − ( y ) . Furthermore, since Λ is a k -graphthere exists a path µg ∼ f λ with g ∈ Λ e i . That is, v is not an e i sink. (cid:3) Theorem 5.6. If (Λ , d ) is a source-free k-graph with v ∈ Λ a sink then (Λ S , d S ) , the graph of Λ with the sink v ∈ Λ deleted, is a source-free k-graph. OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 19 Proof.
Take λ ∈ G ∗ S . Since r ( ι ( λ )) (cid:54)≤ v , if µ = µ · · · µ n ∼ ι ( λ ), then r ( µ ) (cid:54)≤ v . In fact, we have s ( µ ) = s ( ι ( λ )) (cid:54)≤ v , and r ( µ i ) / ∈ V for all 1 ≤ i ≤ n . To see this, simply recall that if s ( η ) ≤ v then r ( η ) ≤ v as well. Consequently, µ ∈ ι ( G ∗ S ) and ι − ( µ ) ∼ S λ . Thus [ λ ] S = [ ι ( λ )], which satisfies(KG0) and (KG4) because Λ is a k -graph. Since λ ∈ G ∗ S was arbitrary, it follows that Λ S is a k -graph.To see that Λ S is source-free, note that whenever an edge e ∈ G was deleted in the process offorming Λ S , so too was the vertex r ( e ) ∈ G . Therefore, no sources were created in the formationof Λ S , so the k -graph Λ S is source-free. (cid:3) Theorem 5.7. If (Λ , d ) is a source free row finite k-graph with v ∈ Λ a sink and (Λ S , d S ) thek-graph of Λ with v deleted, then C ∗ (Λ) is Morita equivalent to C ∗ (Λ S ) .Proof. Let { s λ : λ ∈ Λ ∪ Λ } be the canonical Cuntz-Krieger Λ-family in C ∗ (Λ). Then for every λ ∈ Λ S ∪ Λ S define T λ = s ι ( λ ) . We first prove that { T λ : λ ∈ Λ S ∪ Λ S } is a Cuntz-Krieger Λ S -family in C ∗ (Λ). Note that { s x : x ∈ Λ } are non-zero and mutually orthogonal, and thus so are { T x : x ∈ Λ S } . Therefore { T λ : λ ∈ Λ S ∪ Λ S } satisfies (CK1). Since { s λ : λ ∈ Λ ∪ Λ } is a Cuntz-Krieger Λ-family in C ∗ (Λ), the factthat f g ∼ S hj iff ι ( f g ) = ι ( f ) ι ( g ) ∼ ι ( h ) ι ( j ) = ι ( hj ) tells us that if for f g ∼ S hj , then T f T g = s ι ( f ) s ι ( g ) = s ι ( h ) s ι ( k ) = T h T k , and therefore { T λ : λ ∈ Λ S ∪ Λ S } satisfies (CK2). Also, for f ∈ Λ we have T ∗ f T f = s ∗ ι ( f ) s ι ( f ) = s ι ( s ( f )) = T s ( f ) , and therefore { T λ : λ ∈ Λ S ∪ Λ S } satisfies (CK3). Finally note that for every f ∈ Λ , if r ( f ) (cid:54)≤ v then s ( f ) (cid:54)≤ v . Thus for every x ∈ Λ S , since x was not deleted, r − ( ι ( x )) = ι ( r − S ( x )). So, for everybasis vector e i of N k we have T x = s ι ( x ) = (cid:88) d ( ι ( λ ))= e i r ( ι ( λ ))= x s ι ( λ ) s ∗ ι ( λ ) = (cid:88) d S ( λ )= e i r S ( λ )= x T λ T ∗ λ . Thus (CK4) is satisfied, so { T λ : λ ∈ Λ S ∪ Λ S } is a Cuntz-Krieger Λ S family in C ∗ (Λ). By theuniversal property of C ∗ (Λ S ), then, there exists a ∗ -homomorphism π : C ∗ (Λ S ) → C ∗ (Λ) suchthat π ( t λ ) = T λ for any λ ∈ Λ S ∪ Λ S . Observe that π commutes with the canonical gauge actionson C ∗ (Λ) and C ∗ (Λ S ); moreover, π ( t x ) (cid:54) = 0 for any x ∈ Λ S . Consequently, the gauge invariantuniqueness theorem (Theorem 2.8) tells us that π is injective.We now invoke Theorem 2.11 to show that Im( π ) ∼ = ME C ∗ (Λ). Consider X = ι (Λ S ) ⊆ Λ , and set p = (cid:80) x ∈ Λ S p ι ( x ) . We claim that Σ( X ) = Λ , so that Theorem 2.11 implies that pC ∗ (Λ) p ∼ = ME C ∗ (Λ).To see this, recall from Lemma 5.5 that every vertex in Λ \ X is an e i sink. Moreover, the fact thatΛ is source-free implies that if w ∈ Λ then w Λ e i is nonempty. Since s ( w Λ e i ) ⊆ X , it follows thatevery w ∈ Λ lies in Σ( X ), as claimed.We now show that pC ∗ (Λ) p ∼ = Im( π ). To that end, observe that pC ∗ (Λ) p = span { s λ s ∗ µ : r ( λ ) , r ( µ ) ∈ X = ι (Λ S ) } = span { s λ s ∗ µ : r ( λ ) , r ( µ ) (cid:54)≤ v } . Moreover, if r ( λ ) (cid:54)≤ v then we must have s ( λ ) (cid:54)≤ v . It follows that if s λ s ∗ µ ∈ pC ∗ (Λ) p , then s λ , s µ ∈ Im( π ). Similarly, every generator s λ of Im( π ) lies in pC ∗ (Λ) p . We conclude that, asdesired, C ∗ (Λ S ) ∼ = Im( π ) = pC ∗ (Λ) p ∼ = ME C ∗ (Λ) . (cid:3) Reduction
In the geometric classification of unital graph C ∗ -algebras, the “delay” operation does not appear.Instead, we find its quasi-inverse reduction in the final list [ERRS16] of moves on graphs which encodeall Morita equivalences between graph C ∗ -algebras. Indeed, reduction – rather than delay – was acentral ingredient in [Sø13], and it is more easily recognized as a special case of the general resultof [CG06].For directed graphs, any delay can be undone by a reduction. As we will see in the followingpages, however, reduction for higher-rank graphs is not evidently an inverse to the “delay” movediscussed in Section 4. For this reason we have elected to include a detailed treatment of both moves.For row-finite directed graphs, reduction contracts an edge e to its source vertex v , and can occurwhenever s − ( v ) = { e } and all edges with range v emanate from the same vertex x (cid:54) = v . In thesetting of higher-rank graphs, we can only reduce complete edges (see Notation 6.1 below) whichemanate from a vertex v such that r − ( v ) is also a complete edge. Under these restrictions, however,reduction of a complete edge in Λ results in a new k -graph Λ R such that C ∗ (Λ) ∼ = ME C ∗ (Λ R ). (SeeTheorems 6.5 and 6.6 below.) Notation 6.1.
Let (Λ , d ) be a k -graph. We say a collection of edges, E ⊆ Λ , is a complete edge ifit has the following three properties:(1) E contains precisely one edge of each color;(2) s ( e ) = s ( f ) and r ( e ) = r ( f ) for every e, f ∈ E ;(3) if e ∈ E and a, b, f ∈ Λ satisfy ea ∼ f b or ae ∼ bf , then f ∈ E . Example 6.2.
The third condition in Notation 6.1 depends on the factorization rules. For example,consider the edge-colored directed graph below. v we f f e f e If we define f e ∼ e f and f e ∼ e f , then each set { e i , f i } is a complete edge, for i = 1 , , f e ∼ e f and f e ∼ e f , then there are no complete edges. Definition 6.3.
Let (Λ , d ) be a k -graph and G = (Λ , Λ , r, s ) its 1-skeleton. Fix v ∈ Λ such thatboth Λ v and v Λ are complete, and such that v (cid:54) = r (Λ v ) =: w . Define the directed colored graph G R = (Λ R , Λ R , r R , s R ) by Λ R = Λ \ { v } , Λ R = Λ \ Λ v,s R ( e ) = s ( e ) ,r R ( e ) = (cid:26) r ( e ) if r ( e ) (cid:54) = vw if r ( e ) = v. As the vertices and edges of G R are subsets of the vertices and edges of G , we write ι : Λ R ∪ Λ R → Λ ∪ Λ for the inclusion map. OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 21 Let G ∗ R be the path category of G R ; we will define a parent function par : G ∗ R → G ∗ . To thatend, fix an edge f ∈ Λ v and define par ( x ) = ι ( x ), for x ∈ Λ R ,par ( e ) = (cid:26) ι ( e ) if r ( ι ( e )) (cid:54) = vf ι ( e ) if r ( ι ( e )) = v , for e ∈ Λ R ,par ( λ ) = par ( λ | λ | ) · · · par ( λ ), for λ = λ | λ | · · · λ λ ∈ G ∗ R . Then define the degree map d R on G ∗ R such that d R ( e ) = d ( ι ( e )). Define an equivalence relation, ∼ R , on G ∗ R by µ ∼ R λ if par ( µ ) ∼ par ( λ ). Let Λ R = G ∗ R / ∼ R ; we call Λ R the graph of Λ reduced at v ∈ Λ . Example 6.4.
Figures 10 and 11 show the result of reduction at a vertex v in two different k -graphs.In both cases, we only picture the underlying 1-skeleton, as we have no choice in the factorization.Λ w v Λ R w Figure 10.
First example of reductionΓ • v w Γ R • w Figure 11.
Second example of reduction.
Theorem 6.5. If (Λ , d ) is a row-finite, source-free k -graph then (Λ R , d R ) , the graph of Λ reducedat v ∈ Λ , is a row-finite source-free k -graph.Proof. To see that ∼ R satisfies (KG0), suppose that λ = λ λ ∈ G ∗ R and that µ , µ ∈ G ∗ R satisfy par ( λ i ) ∼ par ( µ i ). Then the definition of the parent function, and the fact that ∼ satisfies (KG0),implies that par ( µ µ ) = par ( µ ) par ( µ ) ∼ par ( λ ) par ( λ ) = par ( λ ) . It follows that µ µ ∼ R λ , which establishes (KG0).Now, take an arbitrary λ ∈ G ∗ R and suppose ι ( λ ) = par ( λ ). It follows that ι ( λ ) never passesthrough v ; the fact that both v Λ and Λ v are complete edges therefore implies that no path in[ ι ( λ )] passes through v . As Λ is a k -graph, [ par ( λ )] satisfies (KG4). Since [ λ ] R = ι − ([ par ( λ )]) and ι is both injective, and onto { µ ∈ G ∗ : v not on µ } ⊇ [ par ( λ )], it follows that [ λ ] R also satisfies(KG4).If λ = λ | λ | · · · λ λ and ι ( λ ) (cid:54) = par ( λ ), then there exists at least one index 1 ≤ i ≤ | λ | suchthat r R ( λ i ) = w , where w is the range of the complete edge which was deleted to form Λ R . Forease of notation, in what follows we will assume that there is only one such index i , but the sameargument will work if there are several. Let the color order of λ be ( m , . . . , m | λ | ); then the color order of par ( λ ) is ( m , . . . , m i , d ( f ) , . . . , m | λ | ). Now [ par ( λ )] satisfies (KG4), so in particular, foreach permutation ( n , . . . , n | λ | ) of ( m , . . . , m | λ | ), there exists a unique path µ (cid:48) ∈ [ par ( λ )] such that d ( µ (cid:48) j ) = n j , j ≤ id ( f ) , j = i + 1 n j − , j > i + 1 . . Since µ (cid:48) and par ( λ ) both have an edge of degree d ( f ) in the ( i + 1)st position, and ∼ satisfies (KG0)and (KG1), we must have µ (cid:48) i +1 = ( par ( λ )) i +1 = f . Thus, µ (cid:48) ∈ Im( par ). Setting µ = par − ( µ (cid:48) ), wehave µ ∼ R λ . The fact that our permutation ( n , . . . , n | λ | ) was arbitrary implies that [ λ ] R includesa path of every color order; the fact that par is injective implies that such a path is unique. Thus,[ λ ] R satisfies (KG4), so Theorem 2.3 tells us that Λ R is a k -graph.To see that Λ R is row-finite, it suffices to observe that | x Λ e i R | = | par ( x )Λ e i | < ∞ for all x ∈ Λ R and for all 1 ≤ i ≤ k . The fact that Λ R is source-free follows from the analogous fact that 0 (cid:54) = | Λ e i par ( x ) | = | Λ e i R x | for all x and i . (cid:3) Theorem 6.6. If (Λ , d ) is a row-finite source-free k -graph, with (Λ R , d R ) the graph of Λ reduced at v ∈ Λ , then C ∗ (Λ) is Morita equivalent to C ∗ (Λ R ) .Proof. Let { s λ : λ ∈ Λ ∪ Λ } be the canonical Cuntz–Krieger Λ-family generating C ∗ (Λ). Define T λ = s par ( λ ) , for λ ∈ Λ R ∪ Λ R . We first prove that { T λ : λ ∈ Λ R ∪ Λ R } is a Cuntz–Krieger Λ R -family in C ∗ (Λ). Note that { s x : x ∈ Λ } are non-zero and mutually orthogonal, and thus so are { T x : x ∈ Λ R } . Thus { T λ : λ ∈ Λ R ∪ Λ R } satisfies (CK1). Further if ab, cd ∈ G ∗ R such that ab ∼ R cd , then [ par ( a ) par ( b )] =[ par ( ab )] = [ par ( cd )] = [ par ( c ) par ( d )]. By (KG0), we therefore have T a T b = s par ( a ) s par ( b ) = s par ( c ) s par ( d ) = T c T d , and therefore { T λ : λ ∈ Λ ∪ Λ } satisfies (CK2). Now fix e ∈ Λ R . If r ( ι ( e )) (cid:54) = v , we have ι ( e ) = par ( e ) and hence T ∗ e T e = s ∗ par ( e ) s par ( e ) = s s ( par ( e )) = s par ( s ( e )) = T s ( e ) . If r ( ι ( e )) = v , then we similarly have T ∗ e T e = ( s f s e ) ∗ ( s f s e ) = s ∗ e s ∗ f s f s e = s ∗ e s v s e = s s ( e ) = T s ( e ) . Therefore { T λ : λ ∈ Λ R ∪ Λ R } satisfies (CK3).Finally, to see that { T λ : λ ∈ Λ R ∪ Λ R } satisfies (CK4), we begin by considering (CK4) for x ∈ Λ R such that x (cid:54) = w (= r (Λ v )). Note that for any basis vector e i ∈ N k , (cid:88) r R ( e )= xd R ( e )= e i T e T ∗ e = (cid:88) r R ( e )= xd R ( e )= e i s e s ∗ e = (cid:88) r ( e )= xd ( e )= e i s e s ∗ e = s x = s par ( x ) = T x . Thus, (CK4) holds for such vertices x .In order to complete the proof that (CK4) holds, we first need a better understanding of theequivalence relation ∼ for paths which pass through v ∈ Λ . Since v Λ and Λ v are completeedges by hypothesis, each set contains precisely one edge of each color. Thus, if we write g i for theedge in v Λ with d ( g i ) = e i and h i for the edge in Λ v such that d ( h i ) = e i , then s v = s g i s ∗ g i forany i . Moreover, by our hypothesis that v Λ and Λ v are both complete edges, we have, for any1 ≤ i, j ≤ k , h j g i ∼ h i g j = ⇒ s h j s g i = s h i s g j = ⇒ s h j s g i s ∗ g i = s h i s g j s ∗ g i = ⇒ s h j = s h i s g j s ∗ g i . OVES ON k -GRAPHS PRESERVING MORITA EQUIVALENCE 23 Thus since f = h j for a unique j , s f s ∗ f = ( s h i s g j s ∗ g i )( s h i s g j s ∗ g i ) ∗ = s h i s g j s ∗ g i s g i s ∗ g j s ∗ h i = s h i s ∗ h i for all i . It follows that (cid:88) r R ( e )= wd R ( e )= e i T e T ∗ e = T g i T ∗ g i + (cid:88) r R ( e )= we (cid:54) = g i d R ( e )= e i T e T ∗ e = ( s f s g i )( s f s g i ) ∗ + (cid:88) e ∈ w Λ ei s ( e ) (cid:54) = v s e s ∗ e = s h i s ∗ h i + (cid:88) e ∈ w Λ ei s ( e ) (cid:54) = v s e s ∗ e = (cid:88) e ∈ w Λ ei s e s ∗ e = s w = T w . Therefore { T λ : λ ∈ Λ R ∪ Λ R } satisfies (CK4), and thus is a Cuntz–Krieger Λ R -family in C ∗ (Λ). Bythe universal property of C ∗ (Λ R ) , there exists a homomorphism π : C ∗ (Λ R ) → C ∗ (Λ) such that, if C ∗ (Λ R ) = C ∗ ( { t λ : λ ∈ Λ R ∪ Λ R } ), we have π ( t λ ) = T λ for all λ ∈ Λ R ∪ Λ R . The computationabove shows that our choice of edge f does not affect the validity of the construction.We now use the gauge-invariant uniqueness theorem and Lemma 2.9 to prove that π is injective.If G is the 1-skeleton of Λ, we define R : G ∗ → Z k by R ( e ) = d ( e ), for e ∈ Λ s.t. s ( e ) (cid:54) = v,R ( e ) = d ( e ) − d ( f ), for e ∈ Λ s.t. s ( e ) = v,R ( λ ) = | λ | (cid:88) i =1 R ( λ i ), for λ = λ | λ | · · · λ λ ∈ G ∗ R ( x ) = 0, for x ∈ Λ . To show that R induces a well defined function on Λ, suppose that µ ∼ ν and consider R ( µ ) , R ( ν ).If s ( µ i ) = v then the fact that Λ v is a complete edge implies that s ( ν i ) = v . Therefore R ( µ ) = d ( µ ) − l · d ( f ) and R ( ν ) = d ( ν ) − l · d ( f ), where l ∈ N counts the number of edges in µ with source v . Since d ( µ ) = d ( ν ) we conclude R ( µ ) = R ( ν ). Thus, the function β : T k → Aut( C ∗ (Λ)) definedby β z ( s µ s ∗ ν ) = z R ( µ ) − R ( ν ) s µ s ∗ ν is an action by Lemma 2.9.Let α be the canonical gauge action on C ∗ (Λ R ). For any e ∈ Λ R , s ( ι ( e )) (cid:54) = v so R ( ι ( e )) = d ( ι ( e )) = d R ( e ). Moreover, if r ( ι ( e )) (cid:54) = v , then ι ( e ) = par ( e ) and π ( t e ) = s ι ( e ) , and so for any z ∈ T k , π ( α z ( t e )) = π ( z d R ( e ) t e ) = z d ( ι ( e )) T e = z d ( ι ( e )) s ι ( e ) = z R ( ι ( e )) s ι ( e ) = β z ( s par ( e ) ) = β z ( π ( t e )) . If r ( ι ( e )) = v, we have π ( α z ( t e )) = π ( z d R ( e ) t e ) = z d R ( e ) T e = z d ( ι ( e )) s f s ι ( e ) = z R ( f ) z R ( e ) s f s ι ( e ) = β z ( s f s ι ( e ) ) = β z ( T e ) = β z ( π ( t e )) . It is straightforward to check that α z and β z also commute on the vertex projections. Therefore, π intertwines β with the canonical gauge action on C ∗ (Λ R ) and thus, by the gauge invariant uniquenesstheorem, π is injective.We now use Theorem 2.11 to show that Im ( π ) ∼ = C ∗ (Λ R ) is Morita equivalent to C ∗ (Λ). Define X := Λ \{ v } and set P X = (cid:80) x ∈ X s x ∈ M ( C ∗ (Λ)). Our first goal is to show that P X C ∗ (Λ) P X =Im( π ).To see that Im( π ) ⊆ P X C ∗ (Λ) P X , recall that, if λ ∈ Λ R , then the vertices par ( s R ( λ )) = s ( par ( λ )) , par ( r R ( λ )) = r ( par ( λ )) both lie in X . It follows that T λ ∈ P X C ∗ (Λ) P X for all λ ∈ Λ R : T λ = s par ( λ ) = s r ( par ( λ )) s par ( λ ) s s ( par ( λ )) = P X s par ( λ ) P X . Thus, π ( C ∗ (Λ R )) ⊆ P X C ∗ (Λ) P X . For the other inclusion, note that P X C ∗ (Λ) P X = span { s λ s ∗ µ : µ, λ ∈ G ∗ , s ( µ ) = s ( λ ) , r ( λ ) , r ( µ ) ∈ X } . We will show that each such generator s λ s ∗ µ is in Im( π ). We begin with the following special case. Claim 1: s h i s ∗ h j ∈ Im( π ) for all edges h i , h j ∈ Λ v .To see this, recall that s h i = s h j s g i s ∗ g j , and so(6.1) s h i s ∗ h j = s h j s g i ( s h j s g j ) ∗ . Now, write e (cid:96) = d ( f ). If (cid:96) = j then s h j s g i = T ι − ( g i ) ∈ Im( π ) and s h j s g j = T ι − ( g j ) , so s h i s ∗ h j ∈ Im( π ). Thus, we suppose (cid:96) (cid:54) = j .By appealing to the results of Section 5, without loss of generality we may assume that w = r ( h j )is not an e (cid:96) sink. Thus, there is an edge e ∈ Λ e (cid:96) w . Since Λ v is a complete edge, we must have eh j ∼ hf for some edge h . It follows that s h j s g i = s w s h j s g i = s ∗ e s e s h j s g i = s ∗ e s h s f s g i . If r ( e ) = r ( h ) ∈ X , then e = par ( ι − ( e )) and hf g i = par ( ι − ( h ) ι − ( g i )). Consequently, in this casewe have s h j s g i = s ∗ e T ι − ( h ) T ι − ( g i ) = T ∗ ι − ( e ) T ι − ( h ) T ι − ( g i ) ∈ Im( π ) . If r ( e ) = r ( h ) = v , then writing s v = s ∗ f s f we have s ∗ e s hfg i = ( s f s e ) ∗ s fh s fg i = T ∗ ι − ( e ) T ι − ( h ) T ι − ( g i ) ∈ Im( π ) . Similarly, we compute that s h j s g j = T ∗ ι − ( e ) T ι − ( h ) T ι − ( g j ) ∈ Im( π ) . Equation (6.1) then impliesthat s h i s ∗ h j ∈ Im( π ) for all 1 ≤ i, j ≤ k , so Claim 1 holds.Next, we establish our second claim via a case-by-case analysis. Claim 2: If η ∈ G ∗ and s ( η ) (cid:54) = v , then s η ∈ Im( π ).To see this, note first that if v does not lie on η , then η = par ( ι − ( η )), so T ι − ( η ) = s η ∈ P X C ∗ (Λ) P X .If v lies on η , and η ∼ ν , then the fact that Λ v, v Λ are complete edges means that ν will alsopass through v . If ν ∼ η is such that every edge of ν with source v is f , then ν ∈ Im( par ) andtherefore s ν = s η ∈ Im( π ).For the last case, Suppose that v lies on η but that for every path ν with ν ∼ η , there is an edge ν i in ν with s ( ν i ) = v and ν i (cid:54) = f . Without loss of generality, suppose that i is the smallest such.By hypothesis, s ( η ) = s ( ν ) (cid:54) = v , so i (cid:54) = 1 . As established in the proof of Claim 1, s ν i = s ∗ e s h s f for edges e of degree d ( f ) and h of degree d ( ν i ), and we have s ν i s ν i − = s ∗ e s hfν i − ∈ Im( π ) . By construction, ν i − · · · ν does not pass through v , and so s ν i − ··· ν ∈ Im( π ). An inductive appli-cation of this argument now shows that s ν = s η lies in Im( π ) whenever s ( η ) (cid:54) = v . This completesthe proof of Claim 2.Finally, consider an arbitrary generator s λ s ∗ µ of P X C ∗ (Λ) P X , with λ = λ | λ | · · · λ , µ = µ | µ | · · · µ ∈ G ∗ . If s ( λ ) = s ( µ ) (cid:54) = v , then applying Claim 2 to λ and µ , we see that s λ s ∗ µ ∈ Im( π ). If v = s ( λ ) = s ( µ ) , then by Claim 1, s λ s ∗ µ ∈ Im( π ). Since r ( λ ) = r ( µ ) = w (cid:54) = v , it follows from Claim 2 that if η := λ | λ | · · · λ and ζ := µ | µ | · · · µ , then s η , s ζ ∈ Im( π ). 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