When is the Cuntz-Krieger algebra of a higher-rank graph approximately finite-dimensional?
aa r X i v : . [ m a t h . OA ] O c t WHEN IS THE CUNTZ-KRIEGER ALGEBRA OF A HIGHER-RANKGRAPH APPROXIMATELY FINITE-DIMENSIONAL?
D. GWION EVANS AND AIDAN SIMS
Abstract.
We investigate the question: when is a higher-rank graph C ∗ -algebraapproximately finite dimensional? We prove that the absence of an appropriate higher-rank analogue of a cycle is necessary. We show that it is not in general sufficient, butthat it is sufficient for higher-rank graphs with finitely many vertices. We give a detaileddescription of the structure of the C ∗ -algebra of a row-finite locally convex higher-rankgraph with finitely many vertices. Our results are also sufficient to establish that if the C ∗ -algebra of a higher-rank graph is AF, then its every ideal must be gauge-invariant.We prove that for a higher-rank graph C ∗ -algebra to be AF it is necessary and sufficientfor all the corners determined by vertex projections to be AF. We close with a numberof examples which illustrate why our question is so much more difficult for higher-rankgraphs than for ordinary graphs. Introduction
A directed graph E consists of countable sets E and E and maps r, s : E → E .We call elements of E vertices and elements of E edges and think of each e ∈ E asan arrow pointing from s ( e ) to r ( e ). When r − ( v ) is finite and nonempty for all v , thegraph C ∗ -algebra C ∗ ( E ) is the universal C ∗ -algebra generated by a family of mutuallyorthogonal projections { p v : v ∈ E } and a family of partial isometries { s e : e ∈ E } such that s ∗ e s e = p s ( e ) for all e ∈ E and p v = P r ( e )= v s e s ∗ e for all v ∈ E [18, 33].Despite the elementary nature of these relations, the class of graph C ∗ -algebras isquite rich. It includes, up to strong Morita equivalence, all AF algebras [16, 52], allKirchberg algebras whose K group is free abelian [51] and many other interesting C ∗ -algebras besides [25, 26]. We know this because we can read off a surprising amount ofthe structure of a graph C ∗ -algebra (for example its K -theory [35, 42], and its wholeprimitive ideal space [27]) directly from the graph. In particular, a graph C ∗ -algebra isAF if and only if the graph contains no directed cycles [32, Theorem 2.4]. Moreover, if E contains a directed cycle and C ∗ ( E ) is simple, then C ∗ ( E ) is purely infinite. So everysimple graph C ∗ -algebra is classifiable either by Elliott’s theorem or by the Kirchberg-Phillips theorem.In 2000, Kumjian and Pask introduced higher-rank graphs, or k -graphs, and their C ∗ -algebras [31] as a generalisation of graph algebras designed to model Robertson andSteger’s higher-rank Cuntz-Krieger algebras [45]. These have proved a very interesting Date : October 19, 2018.2010
Mathematics Subject Classification.
Primary 46L05.
Key words and phrases.
Graph C ∗ -algebra, C ∗ -algebra, AF algebra, higher-rank graph, Cuntz-Krieger algebra.This research was supported by the Australian Research Council and by an LMS travelling lecturergrant. source of examples in recent years [15, 36], but remain far less-well understood thantheir 1-dimensional counterparts, largely because their structure theory is much morecomplicated. In particular, a general structure result for simple k -graph algebras is stilllacking; even a satisfactory characterisation of simplicity itself is in full generality fairlyrecent [47]. The examples of [36] show that there are simple k -graph algebras which areneither AF nor purely infinite, indicating that the question is more complicated than fordirected graphs. Some fairly restrictive sufficient conditions have been identified whichensure that a simple k -graph C ∗ -algebra is AF [31, Lemma 5.4] or is purely infinite [49,Proposition 8.8], but there is a wide gap between the two.Deciding whether a given C ∗ -algebra is AF is an interesting and notoriously diffi-cult problem. The guiding principle seems to be that if, from the point of view of itsinvariants, it looks AF and it smells AF, then it is probably AF. This point of viewled to the discovery and analyses of non-AF fixed point subalgebras of group actionson non-standard presentations of AF algebras initiated by [2] and [30] and continuedby [19, 6] and others. Numerous powerful AF embeddability theorems (the canonicalexample is [38]; and more recently for example [28, 11, 50]) have also been uncovered.These results demonstrate that algebraic obstructions — beyond the obvious one of sta-ble finiteness — to approximate finite dimensionality of C ∗ -algebras are hard to comeby. On the other hand, proving that a given C ∗ -algebra is AF can be a highly nontriv-ial task (cf. [6, 9] and the series of penetrating analyses of actions of finite subgroupsof SL ( Z ) on the irrational rotation algebra initiated by [5, 8, 54] and culminating in[17]). Moreover, non-standard presentations of AF algebras have found applications inclassification theory [38], and also to long-standing questions such as the Powers-Sakaiconjecture [29].In this paper, we consider more closely the question of when a k -graph C ∗ -algebrais AF. The question is quite vexing, and we have not been able to give a completeanswer (see Example 4.2). However, we have been able to weaken the existing necessarycondition for the presence of an infinite projection, and also to show that for a k -graph C ∗ -algebra to be AF, it is necessary that the k -graph itself should contain no directedcycles; indeed, we identify a notion of a higher-dimensional cycle the presence of whichprecludes approximate finite dimensionality of the associated C ∗ -algebra. Our resultsare sufficiently strong to completely characterise when a unital k -graph C ∗ -algebra isAF, and to completely describe the structure of unital k -graph C ∗ -algebras associated torow-finite k -graphs. We also provide some examples confirming some earlier conjecturesof the first author. Specifically, we construct a 2-graph Λ which contains no cycles andin which every infinite path is aperiodic, but such that C ∗ (Λ) is finite but not AF,and we construct an example of a 2-graph which does not satisfy [20, Condition (S)]but does satisfy [20, Condition (Γ)] and whose C ∗ -algebra is AF. We close with anintriguing example of a 2-graph Λ II whose infinite-path space contains a dense set ofperiodic points, but whose C ∗ -algebra is simple, unital and AF-embeddable, and sharesmany invariants with the 2 ∞ UHF algebra. If, as seems likely, the C ∗ -algebra of Λ II is strongly Morita equivalent to the 2 ∞ UHF algebra, it will follow that the structuretheory of simple k -graph algebras is much more complex than for graph algebras.We remark that a proof that C ∗ (Λ II ) is indeed AF would provide another interestingnon-standard presentation of an AF algebra. It would open up the possibility that F k -GRAPH C ∗ -ALGEBRAS 3 known constructions for k -graph C ∗ -algebras might provide new insights into questionsabout AF algebras. Acknowledgements.
We thank David Evans for suggesting the title of the paper asa research question. We also thank Bruce Blackadar, Alex Kumjian, Efren Ruiz andMark Tomforde for helpful discussions, and Andrew Toms and Wilhelm Winter forhelpful email correspondence. Finally, Aidan thanks Gwion for his warm hospitality inRome and again in Aberystwyth. 2.
Background
We introduce some background relating to k -graphs and their C ∗ -algebras. See [31,40, 41] for details.2.1. Higher-rank graphs.
Fix an integer k >
0. We regard N k as a semigroup underpointwise addition with identity element denoted 0. When convenient, we also thinkof it as a category with one object. We denote the generators of N k by e , . . . e k , andfor n ∈ N k and i ≤ k we write n i for the i th coordinate of n ; so n = ( n , n , . . . , n k ) = P ki =1 n i e i . For m, n ∈ N k , we write m ≤ n if m i ≤ n i for all i , and we write m ∨ n forthe coordinatewise maximum of m and n , and m ∧ n for the coordinatewise minimumof m and n . Observe that m ∧ n ≤ m, n ≤ m ∨ n , and that m ′ := m − ( m ∧ n ) and n ′ := n − ( m ∧ n ) is the unique pair such that m − n = m ′ − n ′ and m ′ ∧ n ′ = 0. For n ∈ N k , we write | n | for the length | n | = P ki =1 n i of n .As introduced in [31], a graph of rank k or a k -graph is a countable small categoryΛ equipped with a functor d : Λ → N k , called the degree functor , which satisfies the factorisation property : for all m, n ∈ N k and all λ ∈ Λ with d ( λ ) = m + n , there existunique µ, ν ∈ Λ such that d ( µ ) = m , d ( ν ) = n and λ = µν .We write Λ n for d − ( n ). If d ( λ ) = 0 then λ = id o for some object o of Λ. Hence r ( λ ) := id cod( λ ) and s ( λ ) := id dom( λ ) determine maps r, s : Λ → Λ which restrict to theidentity map on Λ (see [31]). We think of elements of Λ both as vertices and as pathsof degree 0, and we think of each λ ∈ Λ as a path from s ( λ ) to r ( λ ). If v ∈ Λ and λ ∈ Λ, then the composition vλ makes sense if and only if v = r ( λ ). With this in mind,given a subset E of Λ, and a vertex v ∈ Λ , we write vE for the set { λ ∈ E : r ( λ ) = v } .Similarly, Ev denotes { λ ∈ E : s ( λ ) = v } . In particular, for v ∈ Λ and n ∈ N k , wehave v Λ n = { λ ∈ Λ : d ( λ ) = n and r ( λ ) = v } . Moreover, given a subset H of Λ , we let EH denote the set { λ ∈ E : s ( λ ) ∈ H } and set HE = { λ ∈ E : r ( λ ) ∈ H } .We say that Λ is row-finite if v Λ n is finite for all v ∈ Λ and n ∈ N k . We say that Λhas no sources if v Λ n is nonempty for all v ∈ Λ and n ∈ N k . We say that Λ is locallyconvex if, whenever µ ∈ Λ e i and r ( µ )Λ e j = ∅ with i = j , we have s ( µ )Λ e j = ∅ also.For λ ∈ Λ and m ≤ n ≤ d ( λ ), we denote by λ ( m, n ) the unique element of Λ n − m suchthat λ = λ ′ λ ( m, n ) λ ′′ for some λ ′ , λ ′′ ∈ Λ with d ( λ ′ ) = m and d ( λ ′′ ) = d ( λ ) − n .For µ, ν ∈ Λ, a minimal common extension of µ and ν is a path λ such that d ( λ ) = d ( µ ) ∨ d ( ν ) and λ = µµ ′ = νν ′ for some µ ′ , ν ′ ∈ Λ. Equivalently, λ is a minimal commonextension of µ and ν if d ( λ ) = d ( µ ) ∨ d ( ν ) and λ (0 , d ( µ )) = µ and λ (0 , d ( ν )) = ν . Wewrite MCE( µ, ν ) for the set of all minimal common extensions of µ and ν , and we saythat Λ is finitely aligned if MCE( µ, ν ) is finite (possibly empty) for all µ, ν ∈ Λ. If Γ is asub- k -graph of Λ, then for µ, ν ∈ Γ we write MCE Γ ( µ, ν ) and MCE Λ ( µ, ν ) to emphasise D. GWION EVANS AND AIDAN SIMS in which k -graph we are computing the set of minimal common extensions. We haveMCE Γ ( µ, ν ) = MCE Λ ( µ, ν ) ∩ Γ × Γ.For λ ∈ Λ and E ⊆ r ( λ )Λ, the set of paths τ ∈ s ( λ )Λ such that λτ ∈ MCE( λ, µ ) forsome µ ∈ E is denoted Ext( λ ; E ). That is,Ext( λ ; E ) = [ µ ∈ E { τ ∈ s ( λ )Λ : λτ ∈ MCE( λ, µ ) } . By [22, Proposition 3.12], we have Ext( λµ ; E ) = Ext( µ ; Ext( λ ; E )) for all composable λ, µ and all E ⊆ r ( λ )Λ.Fix a vertex v ∈ Λ . A subset F ⊆ v Λ is called exhaustive if for every λ ∈ v Λ thereexists µ ∈ F such that MCE( λ, µ ) = ∅ . By [41, Lemma C.5], if E ⊂ r ( λ )Λ is exhaustive,then Ext( λ ; E ) ⊆ s ( λ )Λ is also exhaustive.2.2. Higher-rank graph C ∗ -algebras. Let Λ be a finitely aligned k -graph. A Cuntz-Krieger Λ-family is a subset { t λ : λ ∈ Λ } of a C ∗ -algebra B such that(CK1) { t v : v ∈ Λ } is a family of mutually orthogonal projections;(CK2) t µ t ν = t µν whenever s ( µ ) = r ( ν );(CK3) t ∗ µ t ν = P µα = νβ ∈ MCE( µ,ν ) t α t ∗ β for all µ, ν ∈ Λ; and(CK4) Q λ ∈ E ( t v − t λ t ∗ λ ) = 0 for all v ∈ Λ and finite exhaustive sets E ⊆ v Λ.The C ∗ -algebra C ∗ (Λ) of Λ is the universal C ∗ -algebra generated by a Cuntz-KriegerΛ-family; the universal family in C ∗ (Λ) is denoted { s λ : λ ∈ Λ } .The universal property of C ∗ (Λ) ensures that there exists a strongly continuous action γ of T k on C ∗ (Λ) satisfying γ z ( s λ ) = z d ( λ ) s λ for all z ∈ T k and λ ∈ Λ, where z d ( λ ) isdefined by the standard multi-index formula z d ( λ ) = z d ( λ ) z d ( λ ) . . . z d ( λ ) k k .The Cuntz-Krieger relations can be simplified significantly under additional hypothe-ses. For details of the following, see [41, Appendix B]. Suppose that Λ is row-finite andlocally convex. For n ∈ N k , defineΛ ≤ n := [ m ≤ n { λ ∈ Λ m : s ( λ )Λ e i = ∅ for all i ≤ k such that m i < n i } . Then (CK3) and (CK4) are equivalent to(CK3 ′ ) t ∗ µ t µ = t s ( µ ) for all µ ∈ Λ, and(CK4 ′ ) t v = P λ ∈ v Λ ≤ n t λ t ∗ λ for all v ∈ Λ and n ∈ N k .If Λ is has no sources, then Λ ≤ n = Λ n for all n , so if Λ is row-finite and has no sourcesthen (CK4 ′ ) is equivalent to(CK4 ′′ ) t v = P λ ∈ v Λ n t λ t ∗ λ for all v ∈ Λ and n ∈ N k .Note that (CK3) implies (CK3 ′ ) for all k -graphs Λ.Recall from [37] that a graph trace on a row-finite k -graph Λ with no sources is afunction g : Λ → R + such that g ( v ) = P λ ∈ v Λ n g ( s ( λ )) for all v ∈ Λ and n ∈ N k . Agraph trace g is called faithful if g ( v ) = 0 for all v ∈ Λ . Proposition 3.8 of [37] describeshow faithful graph traces on Λ correspond with faithful gauge-invariant semifinite traceson C ∗ (Λ). We call a graph trace g finite if P v ∈ Λ g ( v ) converges to some T ∈ R + , andwe say that a finite graph trace g is normalised if P v ∈ Λ g ( v ) = 1. F k -GRAPH C ∗ -ALGEBRAS 5 Lemma 2.1.
Let Λ be a row-finite k -graph with no sources. Each normalised finitefaithful graph trace g on Λ determines a faithful bounded gauge-invariant trace τ g on C ∗ (Λ) which is normalised in the sense that the limit over increasing finite subsets F of Λ of τ g (cid:16) P v ∈ F s v (cid:17) is equal to : specifically, τ g ( s µ s ∗ ν ) = δ µ,ν g ( s ( µ )) for all µ, ν ∈ Λ .Moreover, g τ g is a bijection between normalised finite faithful graph traces on Λ andnormalised faithful gauge-invariant traces on C ∗ (Λ) .Proof. By [37, Proposition 3.8], the map g τ g is a bijection between faithful (not neces-sarily finite or normalised) graph traces on Λ and faithful semifinite lower-semicontinuousgauge-invariant traces on C ∗ (Λ). So it suffices to show that τ g is finite if and only if g is finite, and that τ g is normalised if and only if g is normalised. For this, for each finite F ⊆ Λ let P F := P v ∈ F s v ∈ C ∗ (Λ). Then the P F form an approximate identity, and so τ g is finite if and only if lim F τ g ( P F ) = P v ∈ F g ( v ) converges. Moreover, each of g and τ g is normalised if and only if each of these sums converges to 1. (cid:3) Infinite paths and aperiodicity.
For each m ∈ ( N ∪ {∞} ) k , we define a k -graphΩ k,m by Ω k,m = { ( p, q ) ∈ N k × N k : p ≤ q ≤ m } , with r ( p, q ) = ( p, p ) , s ( p, q ) = ( q, q ) , and d ( p, q ) = q − p. It is standard to identify Ω k,m with { p ∈ N k : p ≤ m } by ( p, p ) p , and we shall silentlydo so henceforth.If Λ and Γ are k -graphs, then a k -graph morphism φ : Λ → Γ is a functor from Λ toΓ which preserves degree: d Γ ( φ ( λ )) = d Λ ( λ ) for all λ ∈ Λ.Given a k -graph Λ and m ∈ N k , each λ ∈ Λ m determines a k -graph morphism x λ : Ω k,m → Λ by x λ ( p, q ) := λ ( p, q ) for all ( p, q ) ∈ Ω k,m . Moreover, each k -graphmorphism x : Ω k,m → Λ determines an element x (0 , m ) of Λ m . Thus we identify thecollection of k -graph morphisms from Ω k,m to Λ with Λ m when m ∈ N k . Extending thisidea, given m ∈ ( N ∪ {∞} ) k \ N k , we regard k -graph morphisms x : Ω k,m → Λ as pathsof degree m in Λ and write d ( x ) := m and r ( x ) for x (0); we denote the set of all suchpaths by Λ m . When m = ( ∞ , ∞ , . . . , ∞ ), we denote Ω k,m by Ω k and we call a path x of degree m in Λ an infinite path. We denote by W Λ the collection S m ∈ ( N ∪{∞} ) k Λ m ofall paths in Λ; our conventions allow us to regard Λ as a subset of W Λ .For each n ∈ N k there is a shift map σ n : { x ∈ W Λ : n ≤ d ( x ) } → W Λ such that d ( σ n ( x )) = d ( x ) − n and σ n ( x )( p, q ) = x ( n + p, n + q ) for 0 ≤ p ≤ q ≤ d ( x ) − n .Given x ∈ W Λ and λ ∈ Λ r ( x ), there is a unique λx ∈ W Λ satisfying d ( λx ) = d ( λ ) + d ( x ), ( λx )(0 , d ( λ )) = λ and σ d ( λ ) ( λx ) = x . For x ∈ W Λ and n ≤ d ( x ), we then have x (0 , n ) σ n ( x ) = x .A boundary path in Λ is a path x : Ω k,m → Λ with the property that for all p ∈ Ω k,m and all finite exhaustive sets E ⊆ x ( p )Λ, there exists µ ∈ E such that x ( p, p + d ( µ )) = µ .We denote by ∂ Λ the collection of all boundary paths in Λ. Lemma 5.15 of [22] impliesthat for each v ∈ Λ , the set v∂ Λ := { x ∈ ∂ Λ : r ( x ) = v } is nonempty. Fix x ∈ ∂ Λ. If n ≤ d ( x ), then σ n ( x ) ∈ ∂ Λ, and if λ ∈ Λ r ( x ), then λx ∈ ∂ Λ [22, Lemma 5.13]. Recallalso from [40] that if Λ is row-finite and locally convex, then ∂ Λ coincides with the setΛ ≤∞ = { x ∈ W Λ : x ( n )Λ e i = ∅ whenever n ≤ d ( x ) and n i = d ( x ) i } . D. GWION EVANS AND AIDAN SIMS
Recall from [34] that a k -graph Λ is said to be aperiodic if for all µ, ν ∈ Λ suchthat s ( µ ) = s ( ν ) there exists τ ∈ s ( µ )Λ such that MCE( µτ, ντ ) = ∅ . By [34, Proposi-tion 3.6 and Theorem 4.1], the following are equivalent:(1) Λ is aperiodic;(2) for all distinct m, n ∈ N k and v ∈ Λ there exists x ∈ v∂ Λ such that either m ∨ n d ( x ) or σ m ( x ) = σ n ( x );(3) for all v ∈ Λ there exists x ∈ v∂ Λ such that for distinct m, n ≤ d ( x ), σ m ( x ) = σ n ( x );(4) for every nontrivial ideal I of C ∗ (Λ) there exists v ∈ Λ such that s v ∈ I .Here, and in the rest of the paper, an “ideal” of a C ∗ -algebra always means a closed2-sided ideal.2.4. Skeletons.
We will frequently wish to present a k -graph visually. To do this, wedraw its skeleton and, if necessary, list the associated factorisation rules. Given a k -graph Λ, the skeleton of Λ is the coloured directed graph E Λ with vertices E = Λ , edges E := S ki =1 Λ e i and with colouring map c : E → { , . . . , k } given by c ( α ) = i if and only if α ∈ Λ e i . In pictures in this paper, edges of degree e will be drawnas solid lines and those of degree e as dashed lines. If α, β ∈ E have distinct colours,say c ( α ) = i and c ( β ) = j , and if s ( α ) = r ( β ), then αβ ∈ Λ e i + e j and the factorisationproperty in Λ implies that there are unique edges β ′ , α ′ ∈ E such that c ( β ′ ) = c ( β )and c ( α ′ ) = c ( α ) and such that α ′ α ββ ′ is a commuting diagram in Λ. we call such a diagram a square and we denote by C thecollection of all such squares. We write αβ ∼ C β ′ α ′ , or just αβ ∼ β ′ α ′ . We call thelist of all such relations the factorisation rules for E Λ . It turns out that Λ is uniquelydetermined up to isomorphism by its skeleton and factorisation rules [23, 24]. Moreover,given a k -coloured directed graph E and a collection of factorisation rules of the form αβ ∼ β ′ α ′ where αβ and β ′ α ′ are bi-coloured paths of opposite colourings with the samerange and source, there exists a k -graph with this skeleton and set of factorisation rulesif and only if both of the following conditions are satisfied: (1) the relation ∼ is bijectivein the sense that for each ij -coloured path αβ , there is exactly one ji -coloured path β ′ α ′ such that αβ ∼ β ′ α ′ ; and (2) if αβ ∼ β α , α γ ∼ γ α and β γ ∼ γ β , and if βγ ∼ γ β , αγ ∼ γ α and α β ∼ β α , then α = α , β = β and γ = γ . Observethat (2) is vacuous unless α, β and γ are of three distinct colours, so if k = 2, thencondition (1) by itself characterises those lists of factorisation rules which determine2-graphs.If E Λ has the property that given any two vertices v, w and any two colours i, j ≤ k ,there is at most one path f g from w to v such that c ( f ) = i and c ( g ) = j , then thereis just one possible complete collection of squares possible for this skeleton. In thissituation, we just draw the skeleton to specify Λ, and do not bother to list the squares. F k -GRAPH C ∗ -ALGEBRAS 7 Cycles and generalised cycles
In this section we present a necessary condition on an arbitrary k -graph for its C ∗ -algebra to be AF.As with graph C ∗ -algebras, the necessary conditions for k -graph C ∗ -algebras to beAF which we have developed involve the presence of cycles of an appropriate sort in the k -graph. To formulate a result sufficiently general to deal with the examples which weintroduce later, we propose the notion of a generalised cycle . We have not been able toconstruct a non-AF k -graph C ∗ -algebra which could not be recognised as such by thepresence of a generalised cycle in the complement of some hereditary subgraph, but wehave no reason to believe that such an example does not exist. For the origins of thefollowing definition, see [20, Lemma 4.3] Definition 3.1.
Let Λ be a finitely aligned k -graph. A generalised cycle in Λ is a pair( µ, ν ) ∈ Λ × Λ such that µ = ν , s ( µ ) = s ( ν ), r ( µ ) = r ( ν ), and MCE( µτ, ν ) = ∅ for all τ ∈ s ( µ )Λ. Lemma 3.2.
Let Λ be a finitely aligned k -graph. Fix a pair ( µ, ν ) ∈ Λ × Λ such that µ = ν , s ( µ ) = s ( ν ) and r ( µ ) = r ( ν ) . Then the following are equivalent: (1) The pair ( µ, ν ) is a generalised cycle; (2) The set
Ext( µ, { ν } ) is exhaustive; and (3) { µx : x ∈ s ( µ ) ∂ Λ } ⊆ { νy : y ∈ s ( ν ) ∂ Λ } .Proof. Suppose that ( µ, ν ) is a generalised cycle. Fix λ ∈ s ( µ )Λ. Then MCE( µλ, ν ) = ∅ ,and hence Ext( µλ, { ν } ) = ∅ . By [22, Proposition 3.12], we haveExt( µλ, { ν } ) = Ext( λ ; Ext( µ, { ν } )) , and hence there exists α ∈ Ext( µ, { ν } ) such that MCE( λ, α ) = ∅ . Hence Ext( µ, { ν } ) isexhaustive. This proves (1) = ⇒ (2).Now suppose that Ext( µ, { ν } ) is exhaustive. Since Λ is finitely aligned, Ext( µ, { ν } ) isalso finite, and hence it is a finite exhaustive subset of s ( µ )Λ. Fix x ∈ s ( µ ) ∂ Λ. By defi-nition of ∂ Λ there exists α ∈ Ext( µ, { ν } ) such that x (0 , d ( α )) = α . Hence ( µx )(0 , d ( µ ) ∨ d ( ν )) = µα ∈ MCE( µ, ν ), and it follows that ( µx )(0 , d ( ν )) = ( µα )(0 , d ( ν )) = ν . Thus y := σ d ( ν ) ( µx ) satisfies y ∈ s ( ν ) ∂ Λ and µx = νy . This proves (2) = ⇒ (3).Finally suppose that { µx : x ∈ s ( µ ) ∂ Λ } ⊆ { νy : y ∈ s ( ν ) ∂ Λ } . Fix τ ∈ s ( µ )Λ. Since s ( τ ) ∂ Λ = ∅ [22, Lemma 5.15], we may fix z ∈ s ( τ ) ∂ Λ, and then x := τ z ∈ s ( µ ) ∂ Λalso [22, Lemma 5.13]. By hypothesis, we then have µx = νy for some y ∈ s ( ν ) ∂ Λ. Inparticular, ( µx )(0 , d ( µτ ) ∨ d ( ν )) ∈ MCE( µτ, ν ), and hence the latter is nonempty. Thisproves (3) = ⇒ (1). (cid:3) In the language of [22], condition (3) of Lemma 3.2 says that the cylinder sets Z ( µ )and Z ( ν ) are nested: Z ( µ ) ⊆ Z ( ν ).For the remainder of the paper, the term cycle , as distinct from generalised cycle , willcontinue to refer to a path λ ∈ Λ \ Λ such that r ( λ ) = s ( λ ). When — as in Section 5— we wish to emphasise that we mean a cycle in the traditional sense, rather than ageneralised cycle, we will also use the term conventional cycle. To see where the definition of a generalised cycle comes from, observe that if λ is aconventional cycle in a k -graph, then ( λ, r ( λ )) is a generalised cycle. There are plenty D. GWION EVANS AND AIDAN SIMS of examples of k -graphs containing generalised cycles but no cycles (see Example 6.1),but when k = 1, the two notions more or less coincide: Lemma 3.3.
Let Λ be a -graph. Suppose that ( µ, ν ) is a generalised cycle in Λ . Thenthere is a conventional cycle λ ∈ Λ \ Λ such that either µ = νλ or ν = µλ .Proof. Since Λ is a 1-graph, either d ( µ ) ≤ d ( ν ) or vice versa. We will assume that d ( µ ) ≤ d ( ν ) and show that ν = µλ for some conventional cycle λ ; if instead d ( ν ) ≤ d ( µ )then the same argument gives ν = µλ . If d ( µ ) = d ( ν ), then MCE( µ, ν ) = ∅ forces µ = ν which is impossible for a generalised cycle, so d ( µ ) < d ( ν ). Then τ := s ( µ ) ∈ s ( µ )Λsatisfies MCE( µτ, ν ) = ∅ . This forces ν = µλ for some λ . Now r ( λ ) = s ( µ ) and s ( λ ) = s ( ν ) = s ( µ ), so λ is a conventional cycle. (cid:3) The main result in this section is the following.
Theorem 3.4.
Let Λ be a finitely aligned k -graph. If C ∗ (Λ) is AF, then Λ contains nogeneralised cycles. The proof deals separately with two cases. To delineate the cases, we introduce thenotion of an entrance to a generalised cycle.
Definition 3.5.
Let Λ be a finitely aligned k -graph. An entrance to a generalised cycle( µ, ν ) is a path τ ∈ s ( ν )Λ such that MCE( ντ, µ ) = ∅ .If λ is a conventional cycle then an entrance to the conventional cycle λ means anentrance to the associated generalised cycle ( λ, r ( λ )); that is a path τ ∈ r ( λ )Λ such thatMCE( τ, λ ) = ∅ . Remark . A generalised cycle ( µ, ν ) has an entrance if and only if the reversed pair( ν, µ ) is not a generalised cycle.
Lemma 3.7.
Suppose that ( µ, ν ) is a generalised cycle. Then s µ s ∗ µ ≤ s ν s ∗ ν . Moreover, s µ s ∗ µ = s ν s ∗ ν if and only if the generalised cycle ( µ, ν ) has no entrance.Proof. Since Ext( µ, { ν } ) ⊂ s ( µ )Λ ( d ( µ ) ∨ d ( ν )) − d ( µ ) , for distinct α, β ∈ Ext( µ, { ν } ), we have s α s ∗ α s β s ∗ β = 0. In particular, applying (CK4),0 = Y α ∈ Ext( µ, { ν } ) ( s s ( µ ) − s α s ∗ α ) = s s ( µ ) − X α ∈ Ext( µ, { ν } ) s α s ∗ α . Hence s µ s ∗ µ = s µ s s ( µ ) s ∗ µ = X α ∈ Ext( µ, { ν } ) s µα s ∗ µα . For each α ∈ Ext( µ, { ν } ), we have µα = νβ for some β ∈ Λ, and hence s µα s ∗ µα = s ν ( s β s ∗ β ) s ∗ ν ≤ s ν s ∗ ν , giving s µ s ∗ µ ≤ s ν s ∗ ν .Suppose that the generalised cycle ( µ, ν ) has no entrance. Then ( ν, µ ) is also ageneralised cycle, and the preceding paragraph gives s ν s ∗ ν ≤ s µ s ∗ µ also.Now suppose that the generalised cycle ( µ, ν ) has an entrance τ ; so MCE( ντ, µ ) = ∅ .Then s ντ s ∗ ντ ≤ s ν s ∗ ν and s ντ s ∗ ντ s µ s ∗ µ = X λ ∈ MCE( ντ,µ ) s λ s ∗ λ = 0 . Hence s ν s ∗ ν − s µ s ∗ µ ≥ s ντ s ∗ ντ > (cid:3) F k -GRAPH C ∗ -ALGEBRAS 9 Corollary 3.8 ([20, Lemma 4.3]) . Let Λ be a finitely aligned k -graph which containsa generalised cycle with an entrance. Then C ∗ (Λ) contains an infinite projection. Inparticular C ∗ (Λ) is not AF.Proof. Let ( µ, ν ) be the generalised cycle with an entrance. By Lemma 3.7, we have s ν s ∗ ν > s µ s ∗ µ = s µ s ∗ ν s ν s ∗ µ ∼ s ν s ∗ µ s µ s ∗ ν = s ν s ∗ ν . Hence s ν s ∗ ν is an infinite projection. The last statement follows immediately. (cid:3) We must now show that when Λ contains a generalised cycle with no entrance, C ∗ (Λ)is not AF. The following result is the key step. The argument is essentially that of [7,Proposition 4.4.1], and we thank George Elliott for directing our attention to [7]. Proposition 3.9.
Let A be a unital C ∗ -algebra carrying a normalised trace T , and let β : T → Aut( A ) be a strongly continuous action. Let U be a unitary in A , and supposethat there exists n ∈ Z \ { } satisfying β z ( U ) = z n U for all z ∈ T . Then U does notbelong to the connected component of the identity in the unitary group U ( A ) .Proof. Let α : R → Aut( A ) be the action determined by α t ( a ) := β e πit ( a ). Let D ( δ ) := (cid:8) a ∈ A : lim t → t ( α t ( a ) − a ) exists (cid:9) , and let δ : D ( δ ) → A be the generator of α ; thatis δ ( a ) := lim t → t ( α t ( a ) − a ) for a ∈ D ( δ ). Note that U ∈ D ( δ ) since we have(3.1) δ ( U ) = lim t → t ( β e πit ( U ) − U ) = lim t → e nπit − t U = 2 nπiU. Let µ denote the normalised Haar measure on T . Define a map τ : A → C by τ ( a ) := R T T ( β z ( a )) dµ ( z ). We claim that τ is a normalised β -invariant (and hence α -invariant)trace on A . Given a ∈ A , for each z ∈ T we have T ( β z ( a ∗ a )) = T ( β z ( a ) ∗ β z ( a )) ≥ T is a trace. Hence τ ( a ∗ a ) ≥ τ is positive. It is clearly linear, and it satisfies τ (1) = 1because β fixes 1. For a, b ∈ A we calculate: τ ( ab ) = Z T T ( β z ( a ) β z ( b )) dµ ( z ) = Z T T ( β z ( b ) β z ( a )) dµ ( z ) = τ ( ba ) . So τ is a trace. Finally, to see that τ is β -invariant, note that for a ∈ A , we have τ ( β z ( a )) = R T T ( β w ( β z ( a ))) dµ ( w ) = R T T ( β zw ( a )) dµ ( w ) = R T β w ′ ( a ) dµ ( z − w ′ ) = τ ( a )by left-invariance of µ .It now follows from [39, p 281, lines 7–16] that for a unitary V ∈ D ( δ ) which is alsoin the connected component U ( A ) of the identity, we have τ ( V ∗ δ ( V )) = 0. However,using (3.1), we have τ ( U ∗ δ ( U )) = τ ( U ∗ nπiU ) = τ (2 nπi A ) = 2 nπi , and it follows that U
6∈ U ( A ). (cid:3) Proposition 3.10.
Let Λ be a finitely aligned k -graph, and let φ : Z k → Z be a homo-morphism. Suppose that there exists N ∈ Z \{ } and a partial isometry V ∈ span { s µ s ∗ ν : µ, ν ∈ Λ , φ ( d ( µ ) − d ( ν )) = N } such that V V ∗ = V ∗ V . Then C ∗ (Λ) is not AF.Proof. Let P = V ∗ V . Then V is a unitary in P C ∗ (Λ) P .For each i ≤ k , let φ i = φ ( e i ) so that φ ( n ) = P ki =1 φ i n i for all n ∈ Z k . Define ahomomorphism ι φ : T → T k by ι ( z ) i = z φ i for 1 ≤ i ≤ k , and define β : T → Aut( C ∗ (Λ))by β z := γ ι φ ( z ) for all z ∈ T . For µ, ν ∈ Λ we have β z ( s µ s ∗ ν ) = γ ι φ ( z ) ( s µ s ∗ ν ) = ι φ ( z ) d ( µ ) − d ( ν ) s µ s ∗ ν = z φ ( d ( µ ) − d ( ν )) s µ s ∗ ν . In particular, since V ∈ span { s µ s ∗ ν : φ ( d ( µ ) − d ( ν )) = N } , we have β z ( V ) = z N V forall z ∈ T so that β fixes P and hence restricts to an action on P C ∗ (Λ) P . Now supposethat C ∗ (Λ) is an AF algebra; we seek a contradiction. Since corners of AF algebras areAF [14, Exercise III.2], P C ∗ (Λ) P is a unital AF algebra, and hence carries a normalisedtrace. We may therefore apply Proposition 3.9 to see that V does not belong to theconnected component of the unitary group of P C ∗ (Λ) P . This is a contradiction sincethe unitary group of any unital AF algebra is connected. (cid:3) Proof of Theorem 3.4.
We prove the contrapositive statement. Let ( µ, ν ) be a gener-alised cycle in Λ. If ( µ, ν ) has an entrance, then Corollary 3.8 implies that C ∗ (Λ) is notAF. So suppose that ( µ, ν ) has no entrance.Since d ( µ ) = d ( ν ) there exists i such that d ( µ ) i = d ( ν ) i . Define φ : Z k → Z by φ ( n ) := n i , let N := d ( µ ) i − d ( ν ) i , and let V := s µ s ∗ ν . By Lemma 3.7, we have V V ∗ = V ∗ V , so Proposition 3.10 applied to V, N, φ implies that C ∗ (Λ) is not AF. (cid:3) Using the characterisation of gauge-invariant ideals in k -graph algebras of [49], andusing also that quotients of AF algebras are AF, we can extend the main theoremsomewhat, at the expense of a more technical statement. Example 6.3 indicates thatthe extended result is genuinely stronger. Corollary 3.11.
Let Λ be a finitely aligned k -graph. Suppose that there exists a saturatedhereditary subset H of Λ such that Λ \ Λ H contains a generalised cycle. Then C ∗ (Λ) is not AF.Moreover, given a saturated hereditary subset H of Λ , a pair ( µ, ν ) ∈ (Λ \ Λ H ) is ageneralised cycle in Λ \ Λ H if and only if d ( µ ) = d ( ν ) , s ( µ ) = s ( ν ) , r ( µ ) = r ( ν ) , and MCE Λ ( ν, µτ ) \ Λ H = ∅ for every τ ∈ Λ \ Λ H .Proof. For the first statement observe that by [49, Lemma 4.1], Λ \ Λ H is a finitelyaligned k -graph, and [49, Corollary 5.3] applied with B = FE(Λ \ Λ H ) \ E H impliesthat C ∗ (Λ \ Λ H ) is a quotient of C ∗ (Λ). If Λ \ Λ H contains a generalised cycle, thenTheorem 3.4 implies that C ∗ (Λ \ Λ H ) is not AF, and since quotients of AF algebras areAF, it follows that C ∗ (Λ) is not AF either.For the final statement, observe thatMCE Λ \ Λ H ( α, β ) = MCE Λ ( α, β ) \ Λ H. So by Remark 3.6, a generalised cycle in Λ \ Λ H is a pair of distinct paths ( µ, ν ) inΛ \ Λ H with the same range and source such that for every τ ∈ s ( µ )Λ \ Λ H , the setMCE Λ ( ν, µτ ) \ Λ H is nonempty as claimed. (cid:3) Theorem 3.4 combined with the results of [34] shows in particular that aperiodicityof every quotient graph is necessary for C ∗ (Λ) to be AF. We use this to show that if C ∗ (Λ) is AF, then its ideals are indexed by the saturated hereditary subsets of Λ . Proposition 3.12.
Let Λ be a finitely aligned k -graph such that C ∗ (Λ) is AF. Then forevery saturated hereditary subset H of Λ , and every pair η, ζ of distinct paths in Λ \ Λ H ,there exists τ ∈ s ( η )Λ \ Λ H such that MCE( ητ, ζ τ ) ⊂ Λ H . Moreover every ideal of C ∗ (Λ) is gauge-invariant. F k -GRAPH C ∗ -ALGEBRAS 11 Proof.
For the first statement of the Proposition, we prove the contrapositive. Supposethat there exist a saturated hereditary H ⊂ Λ and distinct paths η, ζ ∈ Λ \ Λ H such thatfor every τ ∈ s ( η )Λ \ Λ H , we have MCE( ητ, ζ τ ) ∩ (Λ \ Λ H ) = ∅ . Let Γ := Λ \ Λ H . Thepaths η, ζ ∈ Γ have the property that for every τ ∈ s ( η )Γ, we have MCE Γ ( ητ, ζ τ ) = ∅ .That is, Γ is not aperiodic in the sense of [34, Definition 3.1].By [34, Proposition 3.6 and Definition 3.5], there exist v ∈ Γ and distinct m, n ∈ N k such that m ∨ n ≤ d ( x ) and σ m ( x ) = σ n ( x ) for all x ∈ v∂ Γ. By [34, Lemma 4.3], therethen exist µ, ν, α ∈ Γ such that d ( µ ) = m , d ( ν ) = n , r ( µ ) = r ( ν ), s ( µ ) = s ( ν ) = r ( α ),and µαx = ναx for all x ∈ s ( α ) ∂ Γ. In particular { µαx : x ∈ s ( µα ) ∂ Λ } ⊆ { ναy : y ∈ s ( να ) ∂ Λ } . So (3) = ⇒ (1) of Lemma 3.2 implies that ( µα, να ) is a generalised cycle inΓ, and then Theorem 3.4 implies that C ∗ (Γ) is not AF.Corollary 5.3 of [49] implies that C ∗ (Γ) is isomorphic to the quotient of C ∗ (Λ) by theideal generated by { s v : v ∈ H } . Since quotients of AF algebras are AF, it follows that C ∗ (Λ) is also not AF.To prove the second statement, suppose that C ∗ (Λ) is indeed AF. The previousstatement combined with [34, Lemma 4.4] implies that for each saturated hereditary H ⊆ Λ , each v ∈ Λ \ H and each finite F ⊂ Λ v , there exists τ ∈ v Λ \ Λ H such thatMCE( µτ, ντ ) = ∅ for all distinct µ, ν ∈ F . We may now run the proof of [48, Theo-rem 6.3], leaving out Lemma 6.4 and the first two paragraphs of the proof of Lemma 6.7and using τ in place of the path x (0 , N ) in the remainder of the proof of Lemma 6.7,to see that the conclusion of Theorem 6.3 holds for any relative Cuntz-Krieger algebraassociated to Λ \ Λ H ; and then the argument of [49, Theorem 7.2] implies that everyideal of C ∗ (Λ) is gauge-invariant as claimed. (cid:3) Corners and skew-products
We begin this section with a characterisation of approximate finite-dimensionalityof C ∗ (Λ) in terms of the same property for corners of the form s v C ∗ (Λ) s v . We thendescribe a recipe for constructing examples of k -graphs whose C ∗ -algebras are AF. Proposition 4.1.
Let (Λ , d ) be a finitely aligned k -graph. Then C ∗ (Λ) is AF if andonly if the corners s v C ∗ (Λ) s v , v ∈ Λ are all AF.Proof. It is standard that corners of AF algebras are AF (see, for example, [14, Exer-cise III.2]), proving the “only if” implication.For the “if” direction, suppose that each s v C ∗ (Λ) s v is AF. For each finite F ⊂ Λ , let P F := P v ∈ F s v . We claim that each ideal I F := C ∗ (Λ) P F C ∗ (Λ) is AF. We proceed byinduction on | F | . If | F | = 1, say F = { v } , then I F ∼ Me s v C ∗ (Λ) s v is AF by hypothesis.Now suppose that I F is AF whenever | F | ≤ n and fix F ⊂ Λ with | F | = n + 1. Fix v ∈ F , and let G = F \ { v } . Then I G and I { v } are both AF by the inductive hypothesis,and hence I G / ( I { v } ∩ I G ) is also AF because quotients of AF algebras are AF. Since I F = I G + I { v } , there is an exact sequence I { v } → I F → I G / ( I { v } ∩ I G ) . Since extensions of AF algebras by AF algebras are also AF (see, for example [14,Theorem III.6.3]), it follows that I F is AF as claimed. Clearly G ⊆ F implies I G ⊆ I F . Thus C ∗ (Λ) = S F ⊂ Λ finite I F is AF because the classof AF algebras is closed under taking countable direct limits (this follows from an ε/ (cid:3) We now describe a class of examples of k -graphs whose C ∗ -algebras are AF. Theprimary motivation is the example Λ I discussed in Section 6.Recall from [31, Definition 1.9] that if f : N k → N l is a surjective homomorphism,and Λ is an l -graph, then there is a pullback k -graph f ∗ (Λ) equal as a set to { ( λ, n ) ∈ Λ × N k : d ( λ ) = f ( n ) } with pointwise operations and degree map d ( λ, n ) := n . Alsorecall that if c : Λ → Z k is a functor from a k -graph to Z k , then we can form theskew-product k -graph Λ × c Z k , which is equal as a set to Λ × Z k with structure maps r ( λ, m ) = ( r ( λ ) , m ), s ( λ, m ) = ( s ( λ ) , m + c ( λ )), and ( λ, m )( µ, m + c ( λ )) = ( λµ, m ). Example . Let E be a row-finite 1-graph, and denote the degree functor on E by | · | : E → N . Let c be a function from E to { , e , . . . , e k − } ⊆ Z k , and for λ = λ · · · λ n ∈ E n , let c ( λ ) := P ni =1 c ( λ i ). Define c ( v ) = 0 for v ∈ E . Define f : N k → N by f ( n ) = P ki =1 n i , and a functor c : f ∗ ( E ) → Z k by c ( λ, n ) j := ( c ( λ ) j − P i = j n i if j < k , | λ | if j = k .Let Λ be the skew-product k -graph Λ = f ∗ ( E ) × c Z k . Identifying ( E × N k ) × N k with E × N k × N k , we have f ∗ ( E ) × c Z k = { ( α, m, q ) ∈ E × N k × Z k : | α | = f ( m ) } . Observe that p ( λ, n, a ) := λ defines a functor from Λ to E . In particular, each v ∈ Λ has the form v = ( p ( v ) , , q ) for some q ∈ Z k , and then µ = ( p ( µ ) , d ( µ ) , q ) for all µ ∈ v Λ.We will show that C ∗ (Λ) is AF, with the corners of C ∗ (Λ) determined by vertexprojections isomorphic to corresponding corners of the AF core of C ∗ ( E ). Lemma 4.3.
Consider the situation of Example 4.2. Fix a vertex v = ( p ( v ) , , q ) ∈ Λ .Fix µ, ν ∈ v Λ , let m := d ( µ ) and n := d ( ν ) , and express s ( µ ) as ( w, , q ) ∈ E × { } × Z k = Λ . Then s ∗ µ s ν = P τ ∈ p ( s ( µ )) E | n | s ( τ,n,q + c ( µ,m )) s ∗ ( ητ,m,q + c ( ν,n )) if p ( µ ) = p ( ν ) η , P τ ∈ p ( s ( ν )) E | m | s ( ζτ,n,q + c ( µ,m )) s ∗ ( τ,m,q + c ( ν,n )) if p ( ν ) = p ( µ ) ζ , otherwise.Proof. Let N := | µ | + | ν | . We have s ∗ µ s ν = X λ ∈ v Λ m + n s ∗ µ s λ s ∗ λ s ν = X ξ ∈ wE N s ∗ ( p ( µ ) ,m,q ) s ( ξ,m + n,q ) s ∗ ( ξ,m + n,q ) s ( p ( ν ) ,n,q ) . Factorising each ξ ∈ wE N as ξ = ξ m ξ ′ m = ξ n ξ ′ n where | ξ m | = | m | and | ξ n | = | n | gives(4.1) s ∗ µ s ν = X ξ ∈ wE N ( s ∗ ( p ( µ ) ,m,q ) s ( ξ m ,m,q ) ) s ( ξ ′ m ,n,q + c ( ξ m ,m )) s ∗ ( ξ ′ n ,m,q + c ( ξ n ,n )) ( s ∗ ( ξ n ,n,q ) s ( p ( ν ) ,n,q ) ) . F k -GRAPH C ∗ -ALGEBRAS 13 The Cuntz-Krieger relations ensure that s ∗ ( p ( µ ) ,m,q ) s ( ξ m ,m,q ) = 0 unless ξ m = p ( µ ), andlikewise s ∗ ( ξ n ,n,q ) s ( p ( ν ) ,n,q ) = 0 unless ξ n = p ( ν ). So s ∗ µ s ν = 0 unless there exists ξ ∈ E N such that ξ = p ( µ ) ξ ′ m = p ( ν ) ξ ′ n which occurs if and only if either: (1) p ( µ ) = p ( ν ) η , and ξ ′ m = τ and ξ ′ n = ητ for some τ ∈ E | n | ; or (2) p ( ν ) = p ( µ ) ζ , and ξ ′ n = τ and ξ ′ m = ζ τ forsome τ ∈ E | m | . Using this to simplify (4.1), we obtain the desired formula for s ∗ µ s ν . (cid:3) Lemma 4.4.
Consider the situation of Example 4.2. Fix α, β ∈ E N . Define a, b ∈ N k by a j = ( c ( β ) j j < kN − | c ( β ) | j = k and b j = ( c ( α ) j + c j j < k , N − | c ( α ) | j = k .Then ( α, a ) , ( β, b ) ∈ f ∗ ( E ) . If s ( α ) = s ( β ) , then s ( α, a, q ) = s ( β, b, q ) for all q ∈ N k ,and if s ( α ) = s ( β ) , then s ( α, a, q ) = s ( β, b, q ) for all q ∈ N k .Proof. Clearly f ( a ) = f ( b ) = N = | α | = | β | , so ( α, a ) , ( β, b ) ∈ f ∗ ( E ). For j = k , wehave c ( α, a ) j = c ( α ) j − X i = j a i = b j − ( N − a j ) = N + a j + b j , and similarly c ( β, b ) j = N + b j + a j . Since c ( α, a ) k = N = c ( β, b ) k , we have c ( α, a ) = c ( β, b ) and the result follows. (cid:3) Corollary 4.5.
Consider the situation of Example 4.2. For v ∈ Λ and N ∈ N , let B N ( v ) := span { s µ s ∗ ν : µ, ν ∈ v Λ , | µ | = | ν | = N } . Then for each N ∈ N , the space B N ( v ) is a finite dimensional C ∗ -algebra with nonzero matrix units { w η,ζ : η, ζ ∈ p ( v ) E N , s ( η ) = s ( ζ ) } . Moreover s µ s ∗ ν = P ξ ∈ s ( µ )Λ e s µξ s ∗ νξ determines an inclusion B N ( v ) ⊆ B N +1 ( v ) , and s v C ∗ (Λ) s v = S ∞ N =1 B N ( v ) . For each v ∈ Λ , we have s v C ∗ (Λ) s v ∼ = t p ( v ) C ∗ ( E ) γ t p ( v ) . If Λ is cofinal, then C ∗ (Λ) is strongly Morita equivalent to s w C ∗ ( E ) γ s w for any w ∈ E .Proof. We first claim that if µ, ν, α, β ∈ v Λ with | µ | = | ν | = | α | = | β | = N , s ( µ ) = s ( ν ), s ( α ) = s ( β ), p ( µ ) = p ( α ) and p ( ν ) = p ( β ), then s µ s ∗ ν = s α s ∗ β .To see this, let m := d ( µ ) , n := d ( ν ) , a := d ( α ), and b := d ( β ), and use Lemma 4.3 tocalculate s µ s ∗ ν = s µ s ∗ ν X σ ∈ p ( v ) E N s ( σ,n + b,q ) s ∗ ( σ,n + b,q ) = X τ ∈ s ( p ( ν )) E N s ( p ( µ ) τ,m + b,q ) s ∗ ( p ( ν ) τ,n + b,q ) . (4.2)Likewise,(4.3) s α s ∗ β = X τ ∈ s ( p ( β )) E N s ( p ( α ) τ,a + n,q ) s ∗ ( p ( β ) τ,b + n,q ) . We have p ( µ ) = p ( α ) and p ( ν ) = p ( β ) by assumption. So it suffices to show that a + n = m + b . That s ( µ ) = s ( ν ) implies in particular that q + c ( µ ) = q + c ( ν ) and hence that for j = k , c ( p ( µ )) j − X i = j m i = c ( p ( ν )) − X i = j n i . Similarly, each c ( p ( α )) j − X i = j a i = c ( p ( β )) − X i = j b i . Since p ( µ ) = p ( α ) and p ( ν ) = p ( β ), we may subtract the two equations above to obtain X i = j ( m i − a i ) = X i = j ( n i − b i ) for all j = k. Moreover, P ki =1 a i = N = P ki =1 m i , and similarly for the n i and b i , so we obtain m j − a j = n j − b j for all j < k ; and then m k − a k = n k − b k also because | m | = | n | = | a | = | b | = N . So m − a = n − b , and rearranging we obtain a + n = m + b , proving theclaim.For η, ζ ∈ p ( v ) E N , Lemma 4.4 yields a, b ∈ N k with | a | = | b | = N and c ( η, a ) = c ( ζ , b ).We then have s ( η, a, q ) = s ( ζ , b, q ) if and only if s ( η ) = s ( ζ ). For each pair η, ζ ∈ p ( v ) E n such that s ( η ) = s ( ζ ), define w η,ζ := s ( η,a,q ) s ∗ ( ζ,b,q ) . The above claim shows that the w η,ζ depend only on η and ζ and not on our choice of a and b . The claim also implies that s µ s ∗ ν = w ( p ( µ ) ,p ( ν )) whenever µ, ν ∈ v Λ N with s ( µ ) = s ( ν ) (this forces s ( p ( µ )) = s ( p ( ν )).Hence B N = span { w η,ζ : η, ζ ∈ p ( v ) E N , s ( η ) = s ( ζ ) } . The w η,ζ are nonzero because s µ s ∗ ν = 0 in C ∗ (Λ) whenever s ( µ ) = s ( ν ) [31, Re-marks 1.6(iv)]. It remains to check that they are matrix units. We have w ∗ η,ζ = ( s ( η,a,q ) s ∗ ( ζ,b,q ) ) ∗ = s ( ζ,b,q ) s ∗ ( η,a,q ) = w ζ,η for any choice of a, b for which this makes sense. Lemma 4.3 implies that w η,ζ w α,β = 0if α = ζ .Suppose that α = ζ . Let a, b, m, n ∈ N k be the unique elements such that | a | = | b | = | m | = | n | = N and a j = c ( β ) j , b j = c ( α ) j , m j = c ( ζ ) j and n j = c ( η j ) for j < k . Soby Lemma 4.4 and Lemma 4.3, we have w α,β = s ( α,a,q ) s ∗ ( β,b,q ) and w η,ζ = s ( η,m,q ) s ∗ ( ζ,n,q ) .Since α = ζ , Lemma 4.3 and the composition formula in Λ gives w η,ζ w ∗ α,β = X τ ∈ s ( ζ ) E N s ( ητ,m + a,q ) s ∗ ( βτ,n + b,q ) = X τ ∈ s ( ζ ) E N s ( η,a,q ) s ( τ,m,q + c ( η,a )) s ∗ ( τ,b,q + c ( β,n )) s ∗ ( β,n,q ) = s ( η,a,q ) (cid:16) X τ ∈ s ( ζ ) E N s ( τ,m,q + c ( η,a )) s ∗ ( τ,b,q + c ( β,n )) (cid:17) s ∗ ( β,n,q ) (4.4)We claim that c ( η, a ) = c ( β, n ). We have c ( η, a ) k = N = c ( β, n ) k . Fix j < k . Then c ( η, a ) j = c ( η ) j − N + a j = c ( η ) j − N + m j + ( a j − m j )= c ( ζ ) j − N + n j + ( a j − b j ) by definition of m, n. F k -GRAPH C ∗ -ALGEBRAS 15 The symmetric calculation gives c ( β, n ) j = c ( α ) j − N + a j + ( n j − m j ). Since α = ζ ,we have b = m also, so c ( η, a ) j = c ( β, n ) j as claimed.We now have s ( η, a, q ) = s ( β, n, q ), so that w η,β = s ( η,a,q ) s ∗ ( β,n,q ) , and (4.4) becomes w η,ζ w α,β = s ( η,a,q ) (cid:16) X τ ∈ s ( ζ ) E N s ( τ,m,q + c ( η,a )) s ∗ ( τ,m,q + c ( η,a )) (cid:17) s ∗ ( β,n,q ) = s ( η,a,q ) (cid:16) X λ ∈ s ( η,a,q )Λ m s λ s ∗ λ (cid:17) s ∗ ( β,n,q ) , which is equal to s ( η,a,q ) s ∗ ( β,n,q ) by (CK4), and hence to w η,β . This proves that B N ( v ) isfinite-dimensional with matrix units as claimed.The indicated inclusion B N ( v ) ⊆ B N +1 ( v ) is an immediate consequence of the Cuntz-Krieger relations. To see that s v C ∗ (Λ) s v is the closure of the union of the B N , observethat s v C ∗ (Λ) s v is spanned by elements of the form s µ s ∗ ν where µ, ν ∈ v Λ and s ( µ ) = s ( ν ).Fix such a spanning element. Writing v = ( p ( v ) , , q ), we have µ = ( p ( µ ) , d ( µ ) , q ) and ν = ( p ( ν ) , d ( ν ) , q ) and then s ( µ ) = s ( ν ) = ⇒ c ( p ( µ ) , d ( µ )) = c ( p ( ν ) , d ( ν )) = ⇒ | p ( µ ) | = | p ( ν ) | = ⇒ s µ s ∗ ν ∈ B | p ( µ ) | ( v ) . So S ∞ N =1 B N ( v ) contains all the spanning elements of s v C ∗ (Λ) s v , whence its closure isequal to s v C ∗ (Λ) s v .It is routine to check that, for each N ∈ N , the set A N ( p ( v )) := { s α s ∗ β : α, β ∈ p ( v ) E N , s ( α ) = s ( β ) } is a set of nonzero matrix units for a finite-dimensional subalgebraof C ∗ ( E ) γ and that s p ( v ) C ∗ ( E ) γ s p ( v ) is the closure of the increasing union of the A N with inclusions s α s ∗ β P f ∈ s ( α ) E s αf s ∗ αf . So w α,β s α s ∗ β determines isomorphisms B N ( v ) → A N ( p ( v )) which respect the inclusion maps. So the inductive limits s v C ∗ (Λ) s v and s p ( v ) C ∗ ( E ) γ s p ( v ) are isomorphic also. For the final statement observe that the proofof [31, Proposition 4.8] shows that if Λ is cofinal then every s v is full in C ∗ (Λ), andhence C ∗ (Λ) is strongly Morita equivalent to s v C ∗ (Λ) s v for any v . (cid:3) Higher-rank graphs with finitely many vertices
In this section, we completely characterise the higher-rank graphs with finitely manyvertices whose C ∗ -algebras are AF. We then go on to prove that the standard dichotomyfor simple graph C ∗ -algebras persists for row-finite locally convex k -graphs with finitelymany vertices, and we describe the structure of non-simple unital finite higher-rankgraph C ∗ -algebras. Remark . A standard argument [32, Proposition 1.4] implies that if Λ is a finitelyaligned k -graph, then C ∗ (Λ) is unital if and only if Λ is finite. So one may regard theresults in this section as results about unital k -graph C ∗ -algebras.In the sequel we denote by K ( H ) the C ∗ -algebra of compact operators on a separableHilbert space H . When H has finite dimension n we identify K ( H ) with M n ( C ) in thecanonical way. Furthermore, given a countable set S , for each α, β ∈ S , θ α,β denotesthe canonical matrix unit in K ( ℓ ( S )). The following theorem extends [20, Lemma 4.2]. Theorem 5.2.
Let Λ be a finitely aligned k -graph such that Λ is finite. Then (1) C ∗ (Λ) is AF if and only if Λ contains no cycles, and (2) C ∗ (Λ) is finite-dimensional if and only if Λ contains no cycles and is row-finite,in which case there is an isomorphism M v ∈ Λ ,v Λ= { v } M Λ v ( C ) ∼ = C ∗ (Λ) which takes θ α,β to s α s ∗ β . Before proving the theorem, we establish two technical results. Recall from [48] that satiated collections E (see [48, Definition 4.1]) of finite exhaustive subsets of Λ indexthe relative Cuntz-Krieger algebras C ∗ (Λ; E ) which interpolate between the Toeplitzalgebra T C ∗ (Λ) and the Cuntz-Krieger algebra C ∗ (Λ) [48, Corollary 5.6]. Moreover, allquotients of C ∗ (Λ) by gauge-invariant ideals can be realised as relative Cuntz-Kriegeralgebras associated to complements of saturated hereditary subgraphs [49, Theorem 5.5]. Lemma 5.3.
Let Λ be a finitely aligned k -graph. Suppose that Λ is finite, and that Λ contains no cycles. Let E be a satiated subset of FE(Λ) . If v ∈ Λ satisfies v Λ = { v } ,then C ∗ (Λ; E ) s v C ∗ (Λ; E ) ∼ = K ( ℓ (Λ v )) .Proof. Since v Λ = { v } for µ, ν ∈ Λ v , we haveΛ min ( µ, ν ) = ( { ( v, v ) } if µ = ν ∅ otherwise.In particular, the relative Cuntz-Krieger relations imply that if µ, ν, α, β ∈ Λ v , then s µ s ∗ ν s α s ∗ β = δ ν,α s µ s ∗ β . Hence C ∗ (Λ; E ) s v C ∗ (Λ; E ) = span { s µ s ∗ ν : µ, ν ∈ Λ v } , and thatthere is an isomorphism of K ( ℓ (Λ v )) with C ∗ (Λ; E ) s v C ∗ (Λ; E ) which takes θ µ,ν to s µ s ∗ ν . (cid:3) Proposition 5.4.
Let Λ be a finitely aligned k -graph such that Λ is finite and such that Λ contains no cycles, and let E be a satiated subset of FE(Λ) . Then C ∗ (Λ; E ) is AF.Proof. We proceed by induction on | Λ | . If | Λ | = 1, then since Λ has no cycles, Λ = { v } where v is the unique element of Λ , and hence C ∗ (Λ) = C is certainly AF.Now suppose that for any finitely aligned k -graph Γ with no cycles and with fewervertices than Λ, and for any satiated subset E ′ of FE(Γ), the C ∗ -algebra C ∗ (Γ; E ′ ) is AF.Let { s λ : λ ∈ Λ } denote the universal generating relative Cuntz-Krieger (Λ; E )-familyin C ∗ (Λ; E ). Since Λ is finite, and since Λ contains no cycles, there exists v ∈ Λ suchthat v Λ = { v } , and then Lemma 5.3 implies that the ideal I = C ∗ (Λ; E ) s v C ∗ (Λ; E ) isAF. Let H := { v ∈ Λ : s v I } and let E ′ := (cid:8) E ∈ FE(Λ) \ E : Q λ ∈ E ( s r ( λ ) − s λ s ∗ λ ) ∈ I (cid:9) . If H = ∅ , then 1 C ∗ (Λ; E ) = P v ∈ Λ s v ∈ I , so C ∗ (Λ; E ) = I is AF and we are done.So suppose that H = ∅ . Let Γ := Λ \ Λ H . An application of the gauge-invariantuniqueness theorem [48, Theorem 6.1] for relative Cuntz-Krieger algebras shows that C ∗ (Γ; E ′ ) ∼ = C ∗ (Λ; E ) /I . Moreover Γ ⊂ Λ \ { v } , so Γ has fewer vertices than Λ. Theinductive hypothesis therefore implies that C ∗ (Λ; E ) /I is AF. Since I is AF and theclass of AF algebras is closed under extensions (see, for example, [14, Theorem III.6.3]),it follows that C ∗ (Λ) is itself AF. (cid:3) F k -GRAPH C ∗ -ALGEBRAS 17 Proof of Theorem 5.2. (1) If Λ contains a cycle, then Theorem 3.4 implies that C ∗ (Λ)is not AF; and if Λ contains no cycle, then C ∗ (Λ) is AF by Proposition 5.4 applied with E = FE(Λ).(2) First suppose that Λ is not row-finite. Then there exist v ∈ Λ and n ∈ N k such that v Λ n is infinite. Hence { s λ s ∗ λ : λ ∈ v Λ n } is an infinite family of mutuallyorthogonal nonzero projections in C ∗ (Λ), whence C ∗ (Λ) is not finite-dimensional. Nowsuppose that Λ is row-finite and contains no cycles. Let Λ denote the collection ofvertices v ∈ Λ such that v Λ = { v } . Since Λ contains no cycles, Λ n = ∅ whenever | n | ≥ | Λ | . Since Λ is finite and Λ is row-finite, Λ itself is finite. In particular, ΛΛ is finite. Fix w ∈ Λ . We claim that w ΛΛ is exhaustive. Indeed, fix λ ∈ w Λ. Asabove, the set { n ∈ N k : s ( λ )Λ n = ∅} is bounded; let n be a maximal element of thisset, and fix τ ∈ s ( λ )Λ n . By definition of n , we have s ( τ ) ∈ Λ , so λτ ∈ w ΛΛ trivially has a common extension with λ . By definition of Λ , as in Lemma 5.3 we have s ∗ µ s ν = δ µ,ν s s ( µ ) for µ, ν ∈ Λ . Hence [41, Proposition 3.5] implies that s w = X λ ∈ w ΛΛ s λ s ∗ λ Y λλ ′ ∈ w ΛΛ s λλ ′ s ∗ λλ ′ = X λ ∈ w ΛΛ s λ s ∗ λ . Hence C ∗ (Λ) = M v ∈ Λ span { s α s ∗ β : α, β ∈ Λ v } . Lemma 5.3 implies that each span { s α s ∗ β : α, β ∈ Λ v } ∼ = M Λ v ( C ). (cid:3) We show next that for row-finite locally convex k -graphs with finitely many vertices,the standard dichotomy for simple graph C ∗ -algebras persists: if Λ is a row-finite lo-cally convex k -graph with finitely many vertices and C ∗ (Λ) is simple then C ∗ (Λ) iseither finite-dimensional or purely infinite. It seems likely that a similar result holdsfor arbitrary k -graphs with finitely many vertices (though “finite dimensional” wouldbe replaced with “isomorphic to K ( H ) for some finite- or countably-infinite-dimensionalHilbert space”), but the arguments provided here would require substantial modifica-tion. We first need two technical results. Lemma 5.5.
Let Λ be a row-finite k -graph, and suppose that ρ ∈ Λ is a cycle with noentrance. For each m ∈ N k such that m ∧ d ( ρ ) = 0 , define a map P ρ : v Λ m → v Λ m by P ρ ( µ ) := ( ρµ )(0 , m ) . Then P ρ is bijective.Proof. Fix µ ∈ v Λ m . Since ρ has no entrance, Λ min ( ρ, µ ) = ∅ . Fix ( σ, τ ) ∈ Λ min ( ρ, µ ).Then in particular, τ ∈ s ( µ )Λ m . Now ( µτ )(0 , d ( ρ )) = ρ because ρ does not have anentrance. Hence µ = P ρ (( µτ )( d ( ρ ) , d ( ρ ) + m )). Since µ ∈ v Λ m was arbitrary, it followsthat P ρ is surjective. Since Λ is row-finite, v Λ n is finite, so that P ρ | v Λ n is surjectiveimplies that it is bijective. (cid:3) Lemma 5.6.
Let Λ be a row-finite k -graph, and suppose that ρ ∈ Λ is a cycle with noentrance. For each µ ∈ r ( ρ )Λ such that d ( µ ) ∧ d ( ρ ) = 0 and for each n ∈ N , there is aunique element of s ( µ )Λ nd ( ρ ) . Moreover, there exists p ∈ N such that the unique elementof s ( λ )Λ pd ( ρ ) is a cycle. Proof.
Fix µ ∈ r ( ρ )Λ such that d ( µ ) ∧ d ( ρ ) = 0, and let m = d ( µ ). Observe that foreach n ∈ N ,(5.1) P nρ ( µ ) = P ρ ( P n − ρ ( µ )) = ( ρP n − ρ ( µ ))(0 , m ) = · · · = ( ρ n µ )(0 , m ) . Fix n ∈ N . Let τ n := (cid:0) ρ n P − nρ ( µ ) (cid:1) ( m, m + nd ( ρ )). Since µ = P nρ ( P − nρ ( µ )) = (cid:0) ρ n P − nρ ( µ ) (cid:1) (0 , m ), we have µτ n = ρ n P nρ ( µ ), and in particular, τ n ∈ s ( µ )Λ nd ( ρ ) . Tosee that s ( µ )Λ nd ( ρ ) = { τ n } , let λ ∈ s ( µ )Λ nd ( ρ ) . Then ( µλ )(0 , nd ( ρ )) = ρ n because ρ hasno entrance. Let α := ( µλ )( nd ( ρ ) , m + nd ( ρ )), so ρ n α = µλ . Then P nρ ( α ) = µ by (5.1),so α = P − nρ ( µ ), and hence µλ = ρ n α = ρ n P − nρ ( µ ). Thus λ = τ n .Since Λ m is finite, there exist l, n ∈ N with l < n such that P lρ ( µ ) = P nρ ( µ ) . Let p := n − l . Then µ = P − nρ ( P nρ ( µ )) = P − nρ ( P lρ ( µ )) = P − ( n − l ) ρ ( µ ) = P pρ ( µ ) , and then by definition of the τ n , we have s ( τ p ) = s (cid:0)(cid:0) ρ p P − pρ ( µ ) (cid:1) ( m, m + pd ( ρ )) (cid:1) = s ( P − ( n − l ) ρ ( µ )) = s ( µ ) = r ( τ p ) . (cid:3) In the following proof and some later results, given a cycle τ in a k -graph Λ, we write τ ∞ for the unique element of W Λ such that d ( τ ∞ ) i is equal to ∞ when d ( τ ) i > d ( τ ) i = 0, and such that ( τ ∞ )( n · d ( τ ) , ( n + 1) · d ( τ )) = τ for all n ∈ N . Corollary 5.7.
Let Λ be a row-finite locally convex k -graph such that | Λ | is finite and C ∗ (Λ) is simple. If Λ contains no cycles, then Λ contains a unique source v , and C ∗ (Λ) ∼ = M Λ v ( C ) . Otherwise, C ∗ (Λ) is purely infinite.Proof. Suppose that Λ does not contain a cycle. Then Λ is finite by [20, Remark 4.1],and then [20, Lemma 4.2] shows that C ∗ (Λ) is equal to the direct sum over all sources w in Λ of M Λ w ( C ). Since C ∗ (Λ) is simple, there can be just one summand, and theresult follows.Now suppose that Λ contains a cycle. Since C ∗ (Λ) is simple, Λ is cofinal and has nolocal periodicity by [44, Theorem 3.4]. Hence if Λ contains a cycle with an entrance,then [49, Proposition 8.8] implies that C ∗ (Λ) is purely infinite. It therefore suffices toshow that Λ contains a cycle with an entrance.We suppose for contradiction that no cycle in Λ has an entrance. For each cycle λ ∈ Λlet I λ := { i ≤ k : d ( λ ) i = 0 } , and fix a cycle ρ in Λ such that I ρ is maximal with respectto set inclusion amongst the sets I λ . We claim that if µ ∈ r ( ρ )Λ and m < n ≤ d ( µ ),then µ ( m ) = µ ( n ). To see this, suppose for contradiction that µ ( m ) = µ ( n ). ByLemma 5.6 there exist p ∈ N \ { } and a cycle τ of degree pd ( ρ ) with r ( τ ) = µ ( m ).Since d ( µ ) ∧ d ( ρ ) = 0, we have ( n − m ) ∧ d ( ρ ) = 0, so τ µ ( m, n ) is a cycle with I τµ ( m,n ) = I τ ⊔ I µ ( m,n ) ) I τ = I ρ , contradicting our choice of ρ .Since Λ is finite, it follows that there exists µ ∈ r ( ρ )Λ such that d ( µ ) ∧ d ( ρ ) = 0and such that s ( µ )Λ e i = ∅ whenever e i ∧ d ( ρ ) = 0. Another application of Lemma 5.6implies that there exists p ∈ N k and a cycle τ ∈ s ( µ )Λ pd ( ρ ) . Since r ( τ )Λ e i = ∅ , the graphmorphism τ ∞ belongs to Λ ≤∞ , and since cycles in Λ have no entrance, s ( µ )Λ ≤∞ = { τ ∞ } .In particular, σ d ( τ ) ( x ) = x for all x ∈ s ( µ )Λ ≤∞ , which contradicts that Λ has no localperiodicity. (cid:3) F k -GRAPH C ∗ -ALGEBRAS 19 We conclude the section with the following description of the C ∗ -algebras of row-finitelocally convex k -graphs with finitely many vertices: each such C ∗ -algebra either containsan infinite projection or is strongly Morita equivalent (denoted ∼ Me ) to a direct sum ofmatrix algebras over the continuous functions on tori of dimension at most k (with theconvention that a dimension zero torus is a point). To prove the result, we need someterminology. Let Λ be a row-finite locally convex k -graph such that | Λ | is finite, andsuppose that C ∗ (Λ) does not contain an infinite projection. We will call paths µ suchthat r ( µ ) = s ( µ ) and r ( µ )Λ e i = ∅ whenever d ( µ ) i = 0 initial cycles , and we will say thata vertex v ∈ Λ is a vertex on the initial cycle µ if v ∈ ( µ ∞ ) := { µ ∞ ( n ) : n ≤ d ( µ ∞ ) } .We write IC(Λ) for the collection of initial cycles in Λ, and IC(Λ) for the collection ofvertices of Λ which lie on an initial cycle. Lemma 5.8.
Let Λ be a row-finite locally convex k -graph such that | Λ | is finite, andsuppose that C ∗ (Λ) does not contain an infinite projection. Let µ be an initial cycle of Λ . Let G µ := { m − n : m, n ≤ d ( µ ∞ ) , µ ∞ ( m ) = µ ∞ ( n ) } . Then G µ is a subgroup of Z k .Proof. It is clear that 0 ∈ G and that − G = G , so we just need to show that G is closedunder addition. Suppose that µ ∞ ( m ) = µ ∞ ( n ) and that µ ∞ ( p ) = µ ∞ ( q ), so that m − n and p − q are elements of G ; we must show that ( m − n ) + ( p − q ) ∈ G . We calculate: µ ∞ ( m + p ) = σ m + p ( µ ∞ )(0) = σ m ( σ p ( µ ∞ ))(0) = σ m ( σ q ( µ ∞ ))(0) = σ m + q ( µ ∞ )(0) . A symmetric argument shows that µ ∞ ( n + q ) = σ m + q ( µ ∞ )(0) also. Hence ( m − n ) +( p − q ) = ( m + p ) − ( n + q ) ∈ G as required. (cid:3) Proposition 5.9.
Let Λ be a row-finite locally convex k -graph such that | Λ | is finite,and suppose that C ∗ (Λ) does not contain an infinite projection. Then there exist n ≥ and l , . . . , l n ∈ { , . . . , k } such that C ∗ (Λ) ∼ Me L ni =1 C ( T l i ) .Proof. Since C ∗ (Λ) contains no infinite projection, Lemma 3.7 implies that no cycle inΛ has an entrance.For p ∈ N , let p := ( p, p, . . . , p ) ∈ N k . Let N := | Λ | . Fix λ ∈ Λ ≤ N . Since N = | Λ | ,there exist p < q ≤ N such that the vertices λ ( p ∧ d ( λ )) and λ ( q ∧ d ( λ )) coincide. By[40, Lemma 3.12 and Lemma 3.6], the path µ := λ ( p ∧ d ( λ ) , q ∧ d ( λ )) belongs to Λ ≤ p − q ,so r ( µ )Λ e i = ∅ whenever d ( µ ) i = 0. Since µ has no entrance, r ( µ )Λ n = { µ ∞ (0 , n ) } forall n ≤ d ( µ ∞ ).By the preceding paragraph, for every λ ∈ Λ ≤ N , we have s ( λ ) ∈ IC(Λ) . By theCuntz-Krieger relations, X λ ∈ Λ ≤ N s λ s s ( λ ) s ∗ λ = X v ∈ Λ X λ ∈ v Λ ≤ N s λ s ∗ λ = X v ∈ Λ p v = 1 C ∗ (Λ) , so P v ∈ IC(Λ) s v is a full projection in C ∗ (Λ). For each initial cycle µ , we write P µ for P v ∈ ( µ ∞ ) s v .Given two initial cycles µ, ν either ( µ ∞ ) = ( ν ∞ ) , or ( µ ∞ ) ∩ ( ν ∞ ) = ∅ . We write µ ∼ ν if ( µ ∞ ) = ( ν ∞ ) . Since cycles in Λ have no entrance, if µ, ν ∈ IC(Λ), with µ ν ,then v Λ w = ∅ for all v ∈ ( µ ∞ ) and w ∈ ( ν ∞ ) , and hence P µ C ∗ (Λ) P µ ⊥ P ν C ∗ (Λ) P ν . In particular, C ∗ (Λ) ∼ Me (cid:16) X v ∈ IC(Λ) s v (cid:17) C ∗ (Λ) (cid:16) X v ∈ IC(Λ) s v (cid:17) = X [ µ ] ∈ IC(Λ) / ∼ P µ C ∗ (Λ) P µ = M [ µ ] ∈ IC(Λ) / ∼ P µ C ∗ (Λ) P µ . It therefore suffices to show that for µ ∈ IC(Λ), we have P µ C ∗ (Λ) P µ ∼ Me C ( T l ) for some l ≤ k .For this, fix µ ∈ IC(Λ). For v ∈ ( µ ∞ ) , we have v = µ ∞ ( m ) for some m , andthen s v = s ∗ µ (0 ,m ) s µ (0 ,m ) = s ∗ µ (0 ,m ) s r ( µ ) s µ (0 ,m ) , so s r ( µ ) is full in P µ C ∗ (Λ) P µ . It thereforesuffices to show that s r ( µ ) C ∗ (Λ) s r ( µ ) ∼ = C ( T l ) for some l ≤ k . By [1, Corollary 3.7], s r ( µ ) C ∗ (Λ) s r ( µ ) is isomorphic to the universal C ∗ -algebra generated by elements { t α,β : r ( α ) = r ( β ) = r ( µ ) , s ( α ) = s ( β ) } such that(1) t ∗ α,β = t β,α ,(2) t α,β t η,ζ = P ( τ,ρ ) ∈ Λ min β,η t ατ,ζρ , and(3) for every finite exhaustive subset E of r ( µ )Λ, Q λ ∈ E ( t v,v − t λ,λ ) = 0.Since µ has no entrance, relation (3) holds if and only if each t α,α = t v,v , and then (2)implies that t v,v is a unit for the corner, and that each t α,β is a unitary. If α ∈ r ( µ )Λ, then α = µ ∞ (0 , d ( α )). For m, n ≤ d ( µ ∞ ) with µ ∞ ( m ) = ( µ ∞ )( n ), let α = µ ∞ (0 , m − m ∧ n )and β = µ ∞ (0 , n − m ∧ n ). Then (2) implies that t α,β − t µ ∞ (0 ,m ) ,µ ∞ (0 ,n ) = t α,β ( t r ( µ ) ,r ( µ ) − t µ ∞ (0 ,n ) ,µ ∞ (0 ,n ) ) , and since { µ ∞ (0 , n ) } is exhaustive in r ( µ )Λ, it follows that t α,β = t µ ∞ (0 ,m ) ,µ ∞ (0 ,n ) . Inparticular, if G µ is the group obtained from Lemma 5.8, then there is a well-definedfunction ( m − n ) u m − n := t µ ∞ (0 ,m ) ,µ ∞ (0 ,n ) from G µ to s r ( µ ) C ∗ (Λ) s r ( µ ) .For α, β, η, ζ , τ and ρ as in (2), we have d ( ατ ) − d ( ζ ρ ) = (cid:0) d ( α ) + ( d ( β ) ∨ d ( ζ )) − d ( β ) (cid:1) + (cid:0) d ( ζ ) + ( d ( β ) ∨ d ( ζ )) − d ( ζ ) (cid:1) = ( d ( α ) − d ( β )) + ( d ( η ) − d ( ζ )) , and hence for g, h ∈ G µ we have u g u h = u g + h .Hence s r ( µ ) C ∗ (Λ) s r ( µ ) is the universal C ∗ -algebra generated by a unitary representa-tion of G µ , namely C ∗ ( G µ ). Since G µ is a subgroup of Z k it is isomorphic to Z l for some l ≤ k , so C ∗ ( G µ ) ∼ = C ( T l ) as required. (cid:3) Examples
In this final section, we present some examples which illustrate our results. Webegin with an example that illustrates the need for the fairly technical definition of ageneralised cycle.Before we discuss it, recall that the C ∗ -algebra of a directed graph is AF if and onlyif the graph contains no cycle. There are two obvious generalisations of the notion ofa cycle to the setting of k -graphs: paths whose range and source coincide, or periodicinfinite paths. Examples have appeared previously in the literature to show that thereexist k -graphs containing no path whose range and source coincide whose C ∗ -algebrasare not AF (for example the pullback of Ω by the homomorphism ( p, q ) p + q ; see F k -GRAPH C ∗ -ALGEBRAS 21 [31, Example 1.7, Definition 1.9, and Corollary 2.5(iii)]) and that there exist k -graphs inwhich every infinite path is aperiodic and the C ∗ -algebra is not AF (see [36, Examples6.5 and 6.6]). However, to our knowledge, the following is the first known example ofa k -graph which contains no (conventional) cycle and in which every infinite path isaperiodic but such that the C ∗ -algebra is not AF. This confirms the conjecture statedat the opening of [20, Section 4.1]. Example . Let Λ be the 2-graph with skeleton v v v v · · · α β α α β β α α α β β β and factorisation rules α ij β i +1 k ∼ β ij +1(mod i ) α i +1 k +1(mod i +1) . Wright’s argument [55] showsthat Λ is aperiodic in the sense of [34], meaning that every vertex receives at leastone aperiodic infinite path. However, we claim that it has the stronger property thatevery infinite path in Λ is aperiodic. (For 1-graphs this is equivalent to requiring thatthe graph contains no cycles; it is also equivalent to the condition that the associatedgroupoid is principal.)To see this, fix x ∈ Λ ≤∞ , say r ( x ) = v i − , and factorise x as x = α ij β i +1 j α i +2 j β i +3 j . . . Then σ e ( x ) = β i +1 j α i +2 j β i +3 j α i +4 j . . . = α i +1 j − i +1) β i +2 j − i +2) α i +3 j − i +3) β i +4 j − i +4) . . . and σ e ( x ) = σ e ( β ij +1(mod i ) α i +1 j +1(mod i +1) β i +2 j +1(mod i +2) α i +3 j +1(mod i +3) β i +4 j +1(mod i +4) ) . . . = α i +1 j +1(mod i +1) β i +2 j +1(mod i +2) α i +3 j +1(mod i +3) β i +4 j +1(mod i +4) . . . Hence for m, n ∈ N , σ ( m,n ) ( x ) = α i + m + nj m + n +( n − m )(mod i + m + n ) β i + m + n +1 j m + n +1 +( n − m )(mod i + m + n +1) α i + m + n +2 j m + n +2 +( n − m )(mod i + m + n +2) β i + m + n +3 j m + n +3 +( n − m )(mod i + m + n +3) . . . So for p ∈ N , we can recover p from y := σ p ( x ) as follows: • if r ( x ) = v l and r ( σ p ( x )) = v l ′ , then p + p = l ′ − l . • for i ≥
0, we have y (( i, i ) , ( i + 1 , i )) = α r ( y )+2 ik ( i ) for some k ( i ) ∈ Z / ( r ( y ) + 2 i ) Z .Moreover, p − p ≡ k ( i ) − j r ( y )+2 i (mod r ( y ) + 2 i ) for all i . Hence the sequence d i := k ( i ) − ( j r ( y )+2 i ∈ Z ) is either constant or else increases by 2 at each step. We have − ( r ( y ) + 2 i ) < p − p < r ( y ) + 2 i for i >
0, so if ( d i ) is constant then p ≥ p and p − p = d i for i ≥
1, and if d i +1 = d i + 2 for all i , then p < p and p − p = d i − ( r ( y ) + 2 i ) for i ≥ • We now know p + p and p − p ; we then have p = ( p + p )+( p − p )2 , and then p = p + p − p .In particular, if σ p ( x ) = σ q ( x ), then p = q by the above, and it follows that x is notperiodic. Hence Λ has no periodic boundary paths. It also has no cycles. However,( α , β ) is a generalised cycle, so C ∗ (Λ) is not AF. Since g ( v n ) := 1 / ( n − C ∗ (Λ) carries a faithful trace,and hence is finite. Remark . The 2-graph of the preceding example contains a generalised cycle, so wewere able to use Theorem 3.4 to see that its C ∗ -algebra was not AF. We believe thatit is possible to construct a similar example which contains no generalised cycle and noperiodic paths whose C ∗ -algebra is simple and finite but not AF, using Proposition 3.10in place of Theorem 3.4.Our second example demonstrates a 2-graph which contains no generalised cycle, butso that a quotient graph does contain such a cycle. In particular Corollary 3.11 is agenuinely stronger result than Theorem 3.4. Example . Consider the unique 2-graph S with the skeleton illustrated in Figure 1.It is straightforward to check that this 2-graph contains no generalised cycles. However,...... ...... αβ ...... ...... ...... . . . . . .. . . . . .. . . . . .. . . . . . Figure 1.
The skeleton of the 2-graph S . F k -GRAPH C ∗ -ALGEBRAS 23 the collection H of vertices to the left of the middle (those contained in the grey rec-tangle) form a saturated hereditary subset of Λ , and the quotient graph does containa generalised cycle, namely ( α, β ).We now present two examples of 2-graphs with the same skeleton, one of them AF,the other not obviously so. The AF example is a 2-graph which satisfies Condition (Γ) of[20, Definition 4.6] but not Condition (S) [20, Definition 4.3], confirming a conjecture ofthe first author. The other example is intriguing, because it strongly suggests that thereare 2-graph C ∗ -algebras C ∗ (Λ) which are AF but whose canonical diagonal subalgebras(in the sense of Kumjian) as AF algebras are not conjugate to their maximal abeliansubalgebras span { s λ s ∗ λ : λ ∈ Λ } . (By contrast, whenever the C ∗ -algebra of a directedgraph E is AF, it has an AF decomposition for which the canonical diagonal subalgebrais precisely span { s λ s ∗ λ : λ ∈ E ∗ } , where E ∗ is the finite path space of E .) Example . Consider the skeleton v e e f f Let P I be the 2-graph with this skeleton and factorisation rules e i f j = f i e j . By [31,Corollary 3.5(iii)], C ∗ ( P I ) ∼ = O ⊗ C ( T ).Let c ( e ) = c ( f ) := (0 , c ( e ) := (1 ,
1) and c ( f ) := ( − , c extends to a functor on P I . The skew-product graph Λ I := P I ⋊ c Z has theskeleton illustrated in Figure 2. To keep notation compact, we write e i,jl for ( e l , ( i, j )) v ( − , − v ( − , v ( − , v ( − , ...... v ( − , − v ( − , v ( − , v ( − , ...... v ( − , − v ( − , v ( − , v ( − , ...... v (0 , − v (0 , v (0 , v (0 , ...... v (1 , − v (1 , v (1 , v (1 , ...... v (2 , − v (2 , v (2 , v (2 , ...... v (3 , − v (3 , v (3 , v (3 , ....... . . . . .. . . . . .. . . . . .. . . . . . Figure 2.
The common skeleton of the 2-graphs Λ I and Λ II . and f i,jl for ( f l , ( i, j )) for all l ∈ { , } and i, j ∈ Z . So locally, the labelling looks like v ( i,j ) v ( i − ,j +1) v ( i,j +1) v ( i +1 ,j +1) f i,j e i,j f i,j e i,j The factorisation rules are e i,j f i,j +12 = f i,j e i − ,j +12 , e i,j f i +1 ,j +11 = f i,j e i,j +11 ,e i,j f i,j +11 = f i,j e i − ,j +11 , e i,j f i +1 ,j +12 = f i,j e i,j +12 . We claim that C ∗ (Λ I ) is strongly Morita equivalent to the UHF algebra of type 2 ∞ . Tosee this, we invoke Corollary 4.5. Let B be the 1-graph with B = { v } and B = { a, b } whose C ∗ -algebra is canonically isomorphic to O . Then e → ( a, (1 , e → ( b, (1 , f → ( a, (0 , f → ( b, (0 , P I with the pullback f ∗ ( B ) under the homomorphism f : ( m, n ) → m + n from N to N . The 2-graph Λ I isisomorphic to the one obtained from Example 4.2 by setting c ( a ) = e and c ( b ) = 0in N . Since Λ I is cofinal, it follows from Corollary 4.5 that C ∗ (Λ I ) is strongly Moritaequivalent to s v C ∗ ( B ) γ s v . The fixed-point algebra C ∗ ( B ) γ is precisely the classicalcore of O , which is the 2 ∞ UHF algebra [13, 1.5].
Example . Consider the 2-graph P II with the same skeleton as P I but with factorisa-tion rules e i f j = f j e i . This is isomorphic to B × B , so C ∗ ( P II ) ∼ = O ⊗ O as in [31,Corollary 3.5(iv)]. The formula given for c in Example 6.4 also extends to a functor on P II , and we write Λ II for the corresponding skew-product graph. Then Λ II has the sameskeleton as Λ I , but factorisation rules e i,j f i,j +12 = f i,j e i,j +11 , e i,j f ( i +1) , ( j +1)1 = f i,j e i − ,j +12 ,e i,j f i,j +11 = f i,j e i − ,j +11 , e i,j f i +1 ,j +12 = f i,j e i,j +12 . For each i, j , let x i,j denote the unique infinite path x i,j : Ω → Λ II such that x i,j ( n, n + (1 , e i, ( j + | n | )1 and x i,j ( n, n + (0 , f i, ( j + | n | )2 for all n ∈ N . Then σ (1 , ( x i,j ) = x i, ( j +1) = σ (0 , ( x i,j ) for all i, j , and in particular everyvertex of Λ II receives a periodic infinite path. On the other hand, for each i, j there isa unique infinite path y i,j : Ω → Λ II defined by y i,j ( n, n + (1 , e ( i + n − n ) , ( j + | n | )2 and y i,j ( n, n + (0 , f ( i + n − n ) , ( j + | n | )1 for all n ∈ N . Since each y i,j is injective from Ω → Λ it is aperiodic. So Λ II satisfiesthe aperiodicity condition. It is also cofinal, so C ∗ (Λ II ) is simple by [31, Proposition 4.8].6.1. The C ∗ -algebra of Λ II . We will spend some time analysing C ∗ (Λ II ). We believethat it is isomorphic to C ∗ (Λ I ), but via an isomorphism which cannot easily be describedin terms of the presentation of each as a k -graph C ∗ -algebra. As supporting evidencefor this conjecture, setting p := s v (0 , , we prove: that pC ∗ (Λ II ) p has a unique tracialstate; that C ∗ (Λ II ) (and thus pC ∗ (Λ II ) p ) is AF-embeddable; that the K -theory of both pC ∗ (Λ II ) p and C ∗ (Λ II ) is ( Z [ ] , { } ) (as groups); that Murray-von Neumann equivalence F k -GRAPH C ∗ -ALGEBRAS 25 in pC ∗ (Λ II ) p of the canonical representatives of the generators of its K -group is equiv-alent to K equivalence characterised by equality under the trace; and that the orderon its K -group is the standard unperforated order. So all the evidence suggests that C ∗ (Λ II ) is strongly Morita equivalent to the 2 ∞ UHF algebra, and hence also to C ∗ (Λ I ).To indicate why this might be surprising, we close by showing that if pC ∗ (Λ II ) p is indeedthe 2 ∞ UHF algebra, then its diagonal subalgebra as an AF algebra is not conjugateto the canonical maximal abelian subalgebra span { s λ s ∗ λ : λ ∈ v (0 , Λ II } , even thoughthe two subalgebras are canonically isomorphic under an isomorphism which preserves K -classes in the enveloping algebras.Recall that a normalised trace on C ∗ (Λ II ) is a trace such that P v ∈ F τ ( s v ) convergesto 1 as F increases over finite subsets of Λ and that for a hereditary subset H of Λ ,we may identify C ∗ ( H Λ II ) with the subalgebra of C ∗ (Λ II ) generated by { s λ : λ ∈ H Λ II } . Lemma 6.6.
Let H := { v ( i,j ) : j ≥ , | i | ≤ j } be the hereditary subset of Λ generatedby v . Let T := P ∞ j =1 (2 j − − j . There is a normalised trace τ on C ∗ ( H Λ II ) given by τ ( s v ( i,j ) ) = T i − j , and τ ( s µ s ∗ ν ) = δ µ,ν τ ( s s ( µ ) ) . Moreover, this is the unique normalisedtrace on C ∗ ( H Λ II ) .Proof. The function g : v ( i,j ) → T − j determines a normalised finite faithful graphtrace on each of H Λ II and H Λ I . Lemma 2.1 implies that there are faithful normalisedtraces τ II g : C ∗ ( H Λ II ) → C and τ I g : C ∗ ( H Λ I ) → C satisfying τ g ( s µ s ∗ ν ) = δ µ,ν g ( s ( µ )) forall µ, ν . Since C ∗ ( H Λ I ) is strongly Morita equivalent to M ∞ , τ I g is the unique suchtrace on C ∗ ( H Λ I ), and hence g is the unique normalised finite graph trace on H Λ I . Itis then also the unique normalised finite graph trace on H Λ II , so another application ofLemma 2.1 implies that any trace on C ∗ ( H Λ II ), which is nonzero on each s v and zeroon each s µ s ∗ ν such that d ( µ ) = d ( ν ), must agree with τ II g .We claim that τ II g is the unique trace on C ∗ ( H Λ II ). To see this, fix a trace τ on C ∗ ( H Λ II ). By the above, it suffices to show that τ ( s v ) = 0 for all v ∈ H and that τ ( s µ s ∗ ν ) = 0 whenever d ( µ ) = d ( ν ). To see that τ ( s v ) = 0 for all v , fix v ∈ H . Since τ isnormalised, we have τ ( s w ) = 0 for some w . Since Λ is cofinal, [34, Remark A.3] impliesthat there exists n ∈ N such that v Λ s ( α ) = ∅ for all α ∈ w Λ n . Since s w = P α ∈ w Λ n s α s ∗ α ,there exists α ∈ w Λ n such that τ ( s α s ∗ α ) = 0. Fix ξ ∈ v Λ s ( α ). Then τ ( s v ) ≥ τ ( s ξ s ∗ ξ ) = τ ( s ξ s ∗ α s α s ∗ ξ ) = τ ( s α s ∗ ξ s ξ s ∗ α ) = τ ( s α s ∗ α ) = 0 . It remains to show that τ ( s µ s ∗ ν ) = 0 when d ( µ ) = d ( ν ). If s ( µ ) = s ( ν ), this is trivial,and if r ( µ ) = r ( ν ), then the trace property gives τ ( s µ s ∗ ν ) = τ ( s ∗ ν s µ ) = τ ( s ∗ ν s r ( ν ) s r ( µ ) s µ ) =0. So we may suppose that s ( µ ) = s ( ν ) and r ( µ ) = r ( ν ). Factorise µ = ηα and ν = ζ β where d ( η ) = d ( ζ ) = d ( µ ) ∧ d ( ν ). Then d ( α ) ∧ d ( β ) = 0 and τ ( s µ s ∗ ν ) = τ ( s η s α s ∗ β s ∗ ζ ) = τ ( s ∗ ζ s η s α s ∗ β ) = δ η,ζ τ ( s α s ∗ β ) . In particular, it suffices to show that τ ( s α s ∗ β ) = 0. If r ( α ) = r ( β ) then by theabove argument, we are done. If not then let K := | α | . Since r ( α ) = r ( β ) and s ( α ) = s ( β ), we have α = x i,j (0 , Ke h ) and β = x i,j (0 , Ke l ) for some i, j ∈ Z and h, l such that { h, l } = { , } . By the Cuntz-Krieger relations and the trace property, τ ( s α s ∗ β ) = τ ( s ∗ β s α ) = X α α ′ = β β ′ ∈ Λ ( K,K ) τ ( s α ′ s ∗ β ′ ) . Let α = x i, ( j + K ) (0 , d ( β )) and β = x i, ( j + K ) (0 , d ( α )), then MCE( α , β ) = { α α } = { β β } so that τ ( s α s ∗ β ) = τ ( s α s ∗ β ) . Repeating this, we obtain pairs α n , β n such that r ( α n ) = r ( β n ) = s ( α n − ) = s ( β n − )and d ( α n ) = d ( β n − ) and vice versa for all n , and such that τ ( s α n s ∗ β n ) = τ ( s α m s ∗ β m ) forall m, n . Now suppose that K = 0 so that α = β and let z := τ ( s α s ∗ β ); we must showthat z = 0. Let v n = r ( α n ) = r ( β n ) for all n . Since K = 0, we have v m = v n for distinct m, n . It follows thatspan { s α n s ∗ β n } = ∞ M n =0 s v n (cid:0) span { s α n s ∗ β n } (cid:1) s v n ⊆ ∞ M n =1 s v n C ∗ (Λ II ) s v n . Since the C ∗ -norm on a direct sum is the supremum norm, it follows that the series ∞ X n =0 n s α n s ∗ β n converges to some S ∈ C ∗ ( H Λ II ). By continuity of τ , we have τ ( S ) = ∞ X n =0 τ ( 1 n s α n s ∗ β n ) = ∞ X n =0 zn , and this forces z = 0 since the harmonic series does not converge. (cid:3) Corollary 6.7.
Let τ be the trace on C ∗ (Λ II ) constructed in Lemma 6.6. There is aunique tracial state τ on pC ∗ (Λ II ) p given by τ = τ ( p ) τ ( a ) . In particular pC ∗ (Λ II ) p and C ∗ (Λ II ) are stably finite. Recall that C ∗ (Λ II ) is the skew-product of B × B by the Z -valued functor c satisfying c ( a, v ) = c ( v, b ) = (0 , c ( b, v ) = (1 ,
1) and c ( v, a ) = ( − , l for the lengthfunctor l ( α, β ) := | α | + | β | from B × B to Z , and we write γ l for the correspondinginduced action satisfying γ lz ( s λ ) = z l ( λ ) s λ . Lemma 6.8.
The C ∗ -algebra C ∗ ( B × B ) γ l is isomorphic to N Z M ⋊ σ Z , where σ isthe (Bernoulli) shift automorphism that translates each tensor factor one position to theright.Proof. Let S and S be the canonical generators of the Cuntz algebra O , and let F be the AF core of O . We will prove that C ∗ ( B × B ) γ l is isomorphic to C ∗ ( F ⊗ F ∪{ U } ) ⊂ O , where U := S ∗ ⊗ S + S ∗ ⊗ S is unitary. The Lemma will follow since C ∗ ( F ⊗ F ∪ { U } ) ∼ = N Z M ⋊ σ Z by [12, Proposition 3.3].First, note that u := s ( v,a ) s ∗ ( a,v ) + s ( v,b ) s ∗ ( b,v ) is a unitary in C ∗ ( B × B ). We will showthat C ∗ ( B × B ) γ l = C ∗ ( C ∗ (Λ) γ ∪ { u } ) . For this, observe that C ∗ ( B × B ) γ l = span { s α s ∗ β : α, β ∈ B × B , l ( α ) = l ( β ) } . Since d ( α ) = d ( β ) = ⇒ l ( α ) = l ( β ), and since γ lz ( u ) = u for all z ∈ T , we have C ∗ ( C ∗ (Λ) γ ∪ { u } ) ⊆ C ∗ ( B × B ) γ l . F k -GRAPH C ∗ -ALGEBRAS 27 For the reverse inclusion, since u is unitary, it suffices to show that for each spanningelement s α s ∗ β of C ∗ ( B × B ) γ l , there exists n ∈ Z such that u n s α s ∗ β ∈ C ∗ ( B × B ) γ .For this, fix α = ( α , α ) , β = ( β , β ) ∈ B × B such that l ( α ) = l ( β ). For each z ∈ T we have γ z ( u n s α s ∗ β ) = (( z ( − , ) n u )( z ( n, − n ) s α s ∗ β ) = u n s α s ∗ β ∈ C ∗ ( B × B ) γ , as required,where n = | α | − | β | = | β | − | α | .By [31, Corollary 3.5(iv)], there is an isomorphism ψ : C ∗ ( B × B ) ∼ = O ⊗ O satisfying ψ ( s ( α,β ) ) = S α ⊗ S β (with the obvious identification of paths and multi-indices). Hence the restriction of ψ to C ∗ ( B × B ) γ l is the required isomorphism, since ψ ( u ) = U and ψ ( C ∗ ( B × B ) γ ) = F ⊗ F . (cid:3) Proposition 6.9.
The C ∗ -algebra C ∗ (Λ II ) is AF-embeddable.Proof. Since every automorphism of a UHF algebra is approximately inner, N Z M ⋊ σ Z is AF-embeddable by [53, 3.6 Theorem]. Thus Lemma 6.8 implies that C ∗ ( B × B ) γ l isAF-embeddable. For all λ, µ ∈ Λ II , c ( λ ) = c ( µ ) = ⇒ l ( λ ) = l ( µ ), from which it followsthat C ∗ ( B × B ) γ c ⊆ C ∗ ( B × B ) γ l ; thus C ∗ ( B × B ) γ c is also AF-embeddable. By[46, Proposition], C ∗ (Λ II ) is strongly Morita equivalent, and thus stably isomorphic, to C ∗ ( B × B ) γ c . Hence, C ∗ (Λ II ) is itself AF-embeddable. (cid:3) Remark . It is known that N Z M ⋊ σ Z is a simple, unital, (non AF) A T -algebra ofreal rank zero and has a unique tracial state [10]. Thus the same is true for C ∗ ( B × B ) γ l ,which is strongly Morita equivalent to C ∗ (( B × B ) × l Z ) by [46, Proposition]. Hence C ∗ (( B × B ) × l Z ) is another example of a simple, 2-graph C ∗ -algebra that is neitherAF nor purely infinite.To prove our K-theory results we will need the fact that the K -group of C ∗ (Λ II ) isgenerated by the classes of its vertex projections. The following Lemma proves this factholds in general. Lemma 6.11.
Let Λ be a row-finite -graph with no sources. Suppose that the degreeof each cycle in Λ has zero first coordinate or the degree of each cycle has zero secondcoordinate. Then K ( C ∗ (Λ)) is generated by { [ s v ] : v ∈ Λ } .Proof. As in [21, Definition 3.6], for i = 1 , M i denote the Λ × Λ integer matrix M i ( v, w ) = | v Λ e i w | . By our hypothesis, and without loss of generality, the coordinategraph of Λ corresponding to e (see [31, Examples 1.10.(i)]), which we denote by Λ ,contains no cycles and so it is AF by [31, Examples 1.7.(i)] and [32, Theorem 2.4]. It iswell known that the K -group of an AF algebra is trivial so that K (Λ ) is the trivialgroup. From [21, Remarks 3.19] (or the well-known formulae for the K-groups of directedgraph C ∗ -algebras) we get ker(1 − M t ) ∼ = K (Λ ) ∼ = { } . Therefore the block-columnmatrix (cid:18) M t − − M t (cid:19) of [21, Proposition 3.16] has trivial kernel, and the Lemma now follows from [21, Propo-sition 3.16]. (cid:3) Proposition 6.12.
The K -group K ( pC ∗ (Λ II ) p ) is generated by the classes { [ s λ s ∗ λ ] : λ ∈ v (0 , Λ } , and two such classes are equal if and only if the associated projectionsare Murray-von Neumann equivalent in pC ∗ (Λ II ) p . Let τ be the unique tracial state on pC ∗ (Λ II ) p defined in Corollary 6.7. Then τ induces an order-isomorphism K ( τ ) between the ordered K -group of pC ∗ (Λ II ) p and ( Z [ ] , Z [ ] ∩ R + ) , and hence an order-isomorphism between the ordered K -group of C ∗ (Λ II ) and ( Z [ ] , Z [ ] ∩ R + ) . In particular K ( τ ) carries the class of the identity to , and carries { [ s λ s ∗ λ ] : λ ∈ v , Λ II } to [0 , ∩ Z [ ] . Moreover, K ( C ∗ (Λ II )) , and hence also K ( pC ∗ (Λ II ) p ) , is trivial.Proof. Let A := pC ∗ (Λ II ) p and A := C ∗ (Λ II ). Then A is unital and A is stably unital[31, Remarks 1.6.(v)]. Moreover, both are stably finite by Corollary 6.7 so their K -groups are equipped with the canonical partial ordering [3, § τ induces a state K ( τ ) on K ( A ), whose range is Z [ ]. Since A is a full corner in A theinclusion mapping i : A ֒ → A induces an order-isomorphism K ( i ) : K ( A ) → K ( A ).The partial isometry ( s e i,j + s e i,j )( s f i +1 ,j + s f i +1 ,j ) ∗ implements a Murray von-Neumannequivalence in A between s v ( i,j ) and s v ( i +1 ,j ) for all i, j ∈ Z . It follows, by induction, that s v ( i,j ) ∼ M . vN s v ( k,j ) for all i, j, k ∈ Z , and hence their K -classes are equal. Moreover, foreach i, j, k ∈ Z such that k ≥ j we have[ s v ( i,j ) ] = X λ ∈ v ( i,j ) Λ ( k − j ) e [ s λ s ∗ λ ] = | v ( i,j ) Λ ( k − j ) e II | [ s v ( i,k ) ] = 2 k − j [ s v ( i,k ) ] , since, for each λ ∈ v ( i,j ) Λ ( k − j ) e II , s ( λ ) = v ( l,k ) for some l ∈ Z , and s λ s ∗ λ is Murrayvon-Neumann equivalent to s ( λ ) in A .Let i, j ∈ Z . If j <
0, then [ s v ( i,j ) ] = 2 − j [ s v ( i, ] = 2 − j [ s v (0 , ]. If j ≥ s v ( i,j ) ] = [ s v (0 ,j ) ] = [ s λ s ∗ λ ] for some λ ∈ v (0 , Λ je II . By Lemma 6.11 K ( A ) is generated by { [ s v ] : v ∈ Λ } . Therefore it is also generated by { [ s λ s ∗ λ ] : λ ∈ s v (0 , Λ II } . Thus K ( A )is generated by the pre-image under K ( i ), namely { [ s λ s ∗ λ ] : λ ∈ s v (0 , Λ II } .Let λ, µ ∈ v (0 , Λ II such that s ( λ ) = v ( i,j ) , s ( µ ) = v ( k,l ) for some i, j, k, l ∈ Z with j ≤ k .Then the following equations are satisfied in K ( A ): [ s λ s ∗ λ ] = [ s v ( i,j ) ] = 2 k − j [ s v ( i,k ) ] =2 k − j [ s µ s ∗ µ ]. It follows that [ s λ s ∗ λ ] = 2 k − j [ s µ s ∗ µ ] in K ( A ) also.The above implies that we can write each x ∈ K ( A ) as x = P nj =0 x j [ s λ j s ∗ λ j ] where λ j ∈ v (0 , Λ II v (0 ,j ) for all j = 1 , , . . . , n . Furthermore, x = ( P nj =0 n − j x j )[ s λ n s ∗ λ n ] sothat each element in K ( A ) can be written as an integer multiple of [ s λ s ∗ λ ] for some λ ∈ v (0 , Λ II .We will show that K ( τ ) is injective. Suppose that K ( τ )( x ) = 0 for some x ∈ K ( A ). Now x = m [ s λ s ∗ λ ] for some m ∈ Z and λ ∈ v (0 , Λ II . So we have 0 = mK ( τ )([ s λ s ∗ λ ]) = mτ ( s λ s ∗ λ ). Thus m = 0 since τ is faithful, so that x = 0, asrequired. Since we already showed that projections with the same trace are Murray-vonNeumann equivalent, it follows that K -equivalence is the same as Murray-von Neumannequivalence.We have established that K ( τ ) is a positive isomorphism, so to show that it is anisomorphism of ordered groups, it suffices to prove that Z [ ] + ⊆ K ( τ )( K ( A ) + ). Let y ∈ Z [ ] + then y = m − n for some m, n ∈ N . But m − n = K ( τ )( m [ s λ s ∗ λ ]) for some λ ∈ v (0 , Λ II , thus y = K ( τ )( x ) for some x ∈ K ( A ) + , as required.As A is strongly Morita equivalent to A , it remains to prove that K ( A ) is isomorphicto the trivial group. This follows immediately from [21, Proposition 3.16] since Λ II andΛ I share the same skeleton. (cid:3) F k -GRAPH C ∗ -ALGEBRAS 29 Remark . It follows from the above the unique normalised traces on C ∗ ( H Λ II ) andon C ∗ ( H Λ I ) determine an order-isomorphism K ( C ∗ (Λ II )) ∼ = K ( C ∗ (Λ I )) which carriesthe class of each vertex projection in C ∗ (Λ II ) to the class of the corresponding vertexprojection in C ∗ (Λ I ).All of the above is strong evidence suggesting that C ∗ (Λ II ) is isomorphic to C ∗ (Λ I ).However, since each vertex v i,j receives both a periodic infinite path x i,j and an aperiodicinfinite path y i,j , every open set in the unit space G (0)Λ of the k -graph groupoid of [31]contains both a point with trivial isotropy and a point with nontrivial isotropy. So G Λ is topologically free but not principal: the set of units with trivial isotropy is dense in G (0) , but not the whole of G (0) . Let D := span { s λ s ∗ λ : λ ∈ Λ II } . It follows from [43,Proposition 5.11] that the pair ( C ∗ (Λ II ) , D ) is a Cartan pair but not a C ∗ -diagonal, andin particular that D does not have unique extension of pure states to C ∗ (Λ II ). Since thecanonical maximal abelian subalgebra in an AF algebra is always a diagonal in the senseof Kumjian and in particular always has the extension property, it follows that if C ∗ (Λ II )is AF, then its canonical diagonal subalgebra (as an AF algebra) is not conjugate to theCartan subalgebra D . It must, however, be in some sense locally conjugate: the rangeof the trace on C ∗ (Λ II ) is exhausted on D , and since trace equivalence coincides withMurray-von Neumann equivalence for projections in AF algebras, it would follow thateach projection in the AF diagonal was Murray-von Neumann equivalent to a projectionin D .We have been unable to determine whether C ∗ (Λ II ) is indeed an AF algebra, and leavethis interesting question open. Remark . Our results in this paper constitute only a start on the problem of whena k -graph algebra is AF. There are numerous examples upon which our results shedlittle light, and it seems likely that substantially different techniques are required tounderstand them. One such is the unique 2-graph X with skeleton as illustrated inFigure 3 discussed on [20, pages 52 and 53]. We pass up, for now, analysis of this...... ...... ...... ...... ...... ...... . . . . . .. . . . . .. . . . . .. . . . . . Figure 3.
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