Kumjian-Pask algebras of finitely-aligned higher-rank graphs
aa r X i v : . [ m a t h . R A ] D ec KUMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNEDHIGHER-RANK GRAPHS
LISA ORLOFF CLARK AND YOSAFAT E. P. PANGALELA
Abstract.
We extend the the definition of Kumjian-Pask algebras to include algebrasassociated to finitely aligned higher-rank graphs. We show that these Kumjian-Paskalgebras are universally defined and have a graded uniqueness theorem. We also provethe Cuntz-Kreiger uniqueness theorem; to do this, we use a groupoid approach. Asa consequence of the graded uniqueness theorem, we show that every Kumjian-Paskalgebra is isomorphic to the Steinberg algebra associated to its boundary path groupoid.We then use Steinberg algebra results to prove the Cuntz-Kreiger uniqueness theoremand also to characterize simplicity and basic simplicity. Introduction
In the 1990s, C ∗ -algebras of row-finite directed graphs were introduced in [7, 16, 17].Since their first appearance, these C ∗ -algebras have been intensively studied (for example,see [24]). Some of the earliest results about these algebras include the existence of a uni-versal family, the gauge-invariant uniqueness theorem, and the Cuntz-Krieger uniquenesstheorem.Higher-rank graph C ∗ -algebras were introduced by Kumjian and Pask in [15] as ageneralisation of the C ∗ -algebras of directed graphs. In [15], Kumjian and Pask limittheir focus to row-finite higher-rank graphs with no sources. Later, Raeburn, Sims andYeend extended the coverage by introducing C ∗ -algebras of locally convex, row-finitehigher-rank graphs in [21] and then finitely aligned higher-rank graphs in [22]. It is in thefinitely aligned setting where graphs that fail to be row-finite are considered. Once againRaeburn, Sims and Yeend establish the existence of a universal family, the gauge-invariantuniqueness theorem, and the Cuntz-Krieger uniqueness theorem.On the other hand, Leavitt path algebras were developed independently by Ara, Moreno,and Pardo in [4] and Abrams and Aranda Pino in [2]. A complex Leavitt path algebrais a purely algebraic structure constructed from a directed graph that sits densely insidethe graph C ∗ -algebra. Tomforde showed in [30] that one can generalise further and defineLeavitt path R -algebras where R is any commutative ring with identity. Tomforde provedthe existence of a universal family, the graded uniqueness theorem (which is the algebraicanalogue of the gauge-invariant uniqueness theorem), and the Cuntz-Krieger uniquenesstheorem for Leavitt path R -algebras. Tomforde’s proofs in [30] use techniques that aresimilar to those employed by Raeburn for Leavitt path C -algebras in [6] and in Tomforde’searlier paper [29] for Leavitt path K -algebras where K is an arbitrary field. Mathematics Subject Classification.
Key words and phrases.
Kumjian-Pask algebra, finitely aligned k -graph, Steinberg algebra.This research was done as part of the second author’s PhD thesis at the University of Otago underthe supervision of the first author and Iain Raeburn. Thank you to Iain for his guidance. Moving to higher-rank graphs, Kumjian-Pask R -algebras were introduced in [5] andinclude the class of Leavitt path algebras. Kumjian-Pask algebras are the algebraic ana-logue of the higher-rank graph C ∗ -algebras of [15]. As in [15], the authors of [5] considerrow-finite higher-rank graphs with no sources. Later, Clark, Flynn and an Huef devel-oped Kumjian-Pask algebras for locally convex, row-finite higher-rank graphs in [11]. Tocomplete the final algebraic piece, in this paper we introduce Kumjian-Pask algebras forfinitely aligned higher-rank graphs. We will establish the existence of a universal family,the graded-invariant uniqueness theorem, and the Cuntz-Krieger uniqueness theorem.Our motivation to consider this class of higher-rank graphs comes from our desireto establish an algebraic version of [19, Theorem 4.1]: there Pangalela shows that theToeplitz C ∗ algebra associated to a row-finite graph Λ can be realized as the graph C ∗ -algebra associated to a higher-rank graph constructed from Λ, called T Λ. In this setting T Λ has sources and is not locally convex.Let Λ be a finitely aligned k -graph and let R be a commutative ring with identity. Wedefine a Kumjian-Pask Λ-family (Definition 3.1) and show the existence of a universalKumjian-Pask algebra KP R (Λ) that is a Z k -graded R -algebra in Proposition 3.7. Wethen prove the graded-invariant uniqueness theorem in Theorem 4.1. Up to this point,our techniques mirror the C ∗ -algebraic techniques of [22]. However, the proof of theCuntz-Krieger uniqueness theorem of [22] is highly analytic so we must use an alternateapproach. We have chosen a groupoid approach.In Section 5, we introduce groupoids and Steinberg algebras . Then, given a finitelyaligned higher-rank graph Λ, we build the associated boundary-path groupoid G Λ as in[32]. We then use the graded-invariant uniqueness theorem (Theorem 4.1) to show thatthe Kumjian-Pask algebra KP R (Λ) is isomorphic to the Steinberg algebra A R ( G Λ ) inProposition 5.4. With this isomorphism in place, we aim to use results about Steinbergalgebras to establish results about Kumjian-Pask algebras.First we establish how certain properties of Λ translate to properties of G Λ ; we do thisin Section 6 and Section 7. Of interest in its own right, we show that a higher-rank graphΛ is aperiodic if and only if the boundary-path groupoid G Λ is effective in Proposition6.3. We also show in Proposition 7.1, that a higher-rank graph Λ is cofinal if and only if G Λ is minimal .Now in Section 8, we prove the Cuntz-Krieger uniqueness theorem. This theorem onlyapplies to Kumjian-Pask algebras associated to aperiodic graphs. The proof is a simpy anapplication of the Cuntz-Krieger uniqueness theorem for Steinberg algebras [9, Theorem3.2] which applies to effective groupoids. Note that our technique gives an alternate proofof the Cuntz-Krieger uniqueness theorem in the special cases of Leavitt path algebras in[30] and the row-finite Kumjian-Pask algebras of [5, 11].Finally, in Section 9, we give necessary and sufficient conditions for KP R (Λ) to bebasically simple in Theorem 9.3 and simple in Theorem 9.4. These two results are aconsequence of the characterisation of basic simplicity and simplicity of the Steinbergalgebra A R ( G Λ ) (see Theorem 4.1 and Corollary 4.6 of [9]).2. Background
Let N be the set of non-negative integers and let k be a positive integer. We write n ∈ N k as ( n , . . . , n k ) and for m, n ∈ N k , we write m ≤ n to denote m i ≤ n i for UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 3 ≤ i ≤ k . We also write m ∨ n for their coordinate-wise maximum and m ∧ n for theircoordinate-wise minimum. We denote the usual basis in N k by { e i } .A directed graph or 1 -graph E = ( E , E , r, s ) consists of countable sets of vertices E ,edges E and functions r, s : E → E , which denote range and source maps, respectively.We follow the conventions of [23] and write λµ to denote the composition of paths λ and µ with s ( λ ) = r ( µ ). Thus a path of length n ∈ N is a sequence λ = λ · · · λ n of edges λ i with s ( λ i ) = r ( λ i +1 ) for 1 ≤ i ≤ n −
1. We also have s ( λ ) := s ( λ n ) and r ( λ ) := r ( λ ). Remark . We use this convention of paths because we view the collection of paths asa category.2.1.
Higher-rank graphs.
For a positive integer k , we regard the additive semigroup N k as a category with one object. A higher-rank graph or k -graph Λ = (Λ , Λ , r, s ) is acountable small category Λ with a functor d : Λ → N k , called the degree map , satisfyingthe factorisation property : for every λ ∈ Λ and m, n ∈ N k with d ( λ ) = m + n , there areunique elements µ, ν ∈ Λ such that λ = µυ and d ( µ ) = m , d ( ν ) = n . We then write λ (0 , m ) for µ and λ ( m, m + n ) for ν .We write Λ to denote the set of objects in Λ and we identify each object v ∈ Λ with the identity morphism at the object, which, by the factorisation property, is theonly morphism with range and source v . We then regard elements of Λ as vertices . For n ∈ N k , we define Λ n := { λ ∈ Λ : d ( λ ) = n } and call the elements λ of Λ n paths of degree n . For each λ ∈ Λ we say λ has source s ( λ )and range r ( λ ). For v ∈ Λ , λ ∈ Λ and E ⊆ Λ, we define vE := { µ ∈ E : r ( µ ) = v } , λE := { λµ ∈ Λ : µ ∈ E, r ( µ ) = s ( λ ) } , Eλ := { µλ ∈ Λ : µ ∈ E, s ( µ ) = r ( λ ) } . Remark . In older references, for example [15, 21], v Λ is denoted by Λ ( v ). Example . Let k ∈ N and m ∈ ( N ∪ {∞} ) k . We defineΩ k,m := (cid:8) ( p, q ) ∈ N k × N k : p ≤ q ≤ m (cid:9) .This is a category with objects (cid:8) p ∈ N k : p ≤ m (cid:9) , range map r ( p, q ) = p , source map s ( p, q ) = q , and degree map d ( p, q ) = q − p . Then (Ω k,m , d ) is a k -graph.One way to visualise k -graphs is to use coloured graphs. By choosing k -different colours c , . . . , c k , we can view paths in Λ e i as edges of colour c i . For a k -graph Λ, we call itscorresponding coloured graph the skeleton of Λ. For further discussion about k -graphsand their skeletons, see [14].Let Λ be a k -graph. For λ, µ ∈ Λ, we say that τ is a minimal common extension of λ and µ if d ( τ ) = d ( λ ) ∨ d ( µ ) , τ (0 , d ( λ )) = λ and τ (0 , d ( µ )) = µ .Let MCE ( λ, µ ) denote the collection of all minimal common extensions of λ and µ . Thenwe write Λ min ( λ, µ ) := { ( ρ, τ ) ∈ Λ × Λ : λρ = µτ ∈ MCE ( λ, µ ) } . LISA ORLOFF CLARK AND YOSAFAT E. P. PANGALELA
Meanwhile, for E ⊆ Λ and λ ∈ Λ, we writeExt ( λ ; E ) := [ µ ∈ E (cid:8) ρ : ( ρ, τ ) ∈ Λ min ( λ, µ ) (cid:9) .A set E ⊆ v Λ is exhaustive if for every λ ∈ v Λ, there exists µ ∈ E such that Λ min ( λ, µ ) = ∅ . We define FE (Λ) := [ v ∈ Λ { E ⊆ v Λ \ { v } : E is finite and exhaustive } .For E ∈ FE (Λ), we write r ( E ) for the vertex v which satisfies E ⊆ v Λ.We say that Λ is finitely aligned if Λ min ( λ, µ ) is finite (possibly empty) for all λ, µ ∈ Λ.We see that every 1-graph is finitely aligned. As in [15, Definition 1.4], we say that a k -graph Λ is row-finite if v Λ n is finite for every v ∈ Λ and n ∈ N k . Note that for all λ, µ ∈ Λ, we have (cid:12)(cid:12) Λ min ( λ, µ ) (cid:12)(cid:12) = | MCE ( λ, µ ) | ≤ (cid:12)(cid:12) r ( λ ) Λ d ( λ ) ∨ d ( µ ) (cid:12)(cid:12) . Hence, every row-finite k -graph Λ is finitely aligned. On the other hand, a finitely aligned k -graph Λ is notnecessarily row-finite.For example, consider the 2-graph Λ which has skeleton • vef f ...where ef i = f i e for all positive integers i , the solid edge has degree (1 ,
0) and dashededges have degree (0 , | v Λ (0 , | = ∞ . On the otherhand, for λ, µ ∈ Λ, (cid:12)(cid:12) Λ min1 ( λ, µ ) (cid:12)(cid:12) is either 0 or 1, and then Λ is finitely aligned.Following [15, Definition 1.4], a k -graph Λ has no sources if v Λ n is nonempty for every v ∈ Λ and n ∈ N k . Meanwhile, recall from [21, Definition 3.9] that a k -graph Λ is locallyconvex if for all v Λ , 1 ≤ i, j ≤ k with i = j , λ ∈ v Λ e i and µ ∈ v Λ e j , the sets s ( λ ) Λ e j and s ( µ ) Λ e i are nonempty.Consider the 2-graph Λ with skeleton • v • v • v • v • v e f e f e UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 5 where e f = f e , solid edges have degree (1 ,
0) and dashed edges have degree (0 , v does not receive edges with degree (0 , v is a source of Λ . Furthermore,Λ fails to be locally-convex since e ∈ v Λ (1 , , f ∈ v Λ (0 , but s ( e ) Λ (0 , = ∅ . On theother hand, Λ is row-finite thus Λ is finitely aligned.Next consider the 2-graph Λ with skeleton • v • v • v • w • w ··· e f e f f e where ef i = f e i for all positive integers i , solid edges have degree (1 ,
0) and dashed edgeshave degree (0 , (cid:12)(cid:12) Λ min3 ( e, f ) (cid:12)(cid:12) = ∞ , then Λ is not finitely aligned. Hence, notevery k -graph is finitely aligned.To summarise, finitely aligned k -graphs generalise both row-finite k -graphs with nosources and locally convex row-finite k -graphs. However, this class of k -graphs does notcover all k -graphs. In this paper, we focus on finitely aligned k -graphs. For other examplesand further discussion, see [15, 19, 21, 22, 31].2.2. Paths and boundary paths.
Suppose that Λ is a finitely aligned k -graph. Recallfrom [21, Definition 3.1] that for n ∈ N k , we defineΛ ≤ n := { λ ∈ Λ : d ( λ ) ≤ n , and d ( λ ) i < n i implies s ( λ ) Λ e i = ∅} .Note that v Λ ≤ n = ∅ for all v ∈ Λ and n ∈ N k . This is because v is contained in v Λ ≤ n whenever v Λ ≤ n has no non-trivial paths of degree less than or equal to q . For furtherdiscussion, see [21, Remark 3.2].Following [13, Definition 5.10], we say that a degree-preserving functor x : Ω k,m → Λis a boundary path of Λ if for every n ∈ N k with n ≤ m and for E ∈ x ( n, n ) FE (Λ),there exists λ ∈ E such that x ( n, n + d ( λ )) = λ . We write ∂ Λ for the set of all boundarypaths. Note that for v ∈ Λ , v∂ Λ is nonempty [13, Lemma 5.15].
Remark . In the locally convex setting, the set Λ ≤∞ (as defined in [21, Definition 3.14])is referred to as the “boundary path space”. Indeed, if Λ is row-finite and locally convex,then Λ ≤∞ = ∂ Λ [31, Proposition 2.12]. However, more generally, Λ ≤∞ ⊆ ∂ Λ and the twocan be different (see [31, Example 2.11]).Let x ∈ ∂ Λ. If n ∈ N k and n ≤ d ( x ), we define σ n x by σ n x (0 , m ) = x ( n, n + m ) for all m ≤ d ( x ) − n , and by [13, Lemma 5.13.(1)], σ n x also belongs to ∂ Λ. We also write x ( n )for the vertex x ( n, n ). Then the range of boundary path x is the vertex r ( x ) := x (0).For λ ∈ Λ x (0), we also have λx ∈ ∂ Λ [13, Lemma 5.13.(2)].2.3.
Graded rings.
Suppose that G is an additive abelian group. A ring A is G -graded if there are additive subgroups { A g : g ∈ G } satisfying: A = M g ∈ G A g and for g, h ∈ G , A g A h ⊆ A g + h . LISA ORLOFF CLARK AND YOSAFAT E. P. PANGALELA If A and B are G -graded rings, a homomorphism π : A → B is G -graded if π ( A g ) ⊆ B g for g ∈ G .Let A be a G -graded ring. We say an ideal I of A is a G -graded ideal if { I ∩ A g : g ∈ G } is a grading of I . 3. Kumjian-Pask Λ -families Suppose that Λ is a finitely aligned k -graph and R is a commutative ring with identity1. For λ ∈ Λ, we call λ ∗ a ghost path ( λ ∗ is a formal symbol) and we define G (Λ) := { λ ∗ : λ ∈ Λ } .For v ∈ Λ , we define v ∗ := v . We also extend r and s to be defined on G (Λ) by r ( λ ∗ ) = s ( λ ) and s ( λ ∗ ) = r ( λ ) .We then define composition on G (Λ) by setting λ ∗ µ ∗ = ( µλ ) ∗ for λ, µ ∈ Λ; and write G (cid:0) Λ =0 (cid:1) the set of ghost paths that are not vertices. Note that the factorisation propertyof Λ induces a similar factorisation property on G (Λ). Definition 3.1. A Kumjian-Pask Λ -family { S λ , S µ ∗ : λ, u ∈ Λ } in an R -algebra A consistsof S : Λ ∪ G (cid:0) Λ =0 (cid:1) → A such that:(KP1) { S v : v ∈ Λ } is a collection of mutually orthogonal idempotents;(KP2) for λ, µ ∈ Λ with s ( λ ) = r ( µ ), we have S λ S µ = S λµ and S µ ∗ S λ ∗ = S ( λµ ) ∗ ;(KP3) S λ ∗ S µ = P ( ρ,τ ) ∈ Λ min ( λ,µ ) S ρ S τ ∗ for all λ, µ ∈ Λ; and(KP4) Q λ ∈ E (cid:0) S r ( E ) − S λ S λ ∗ (cid:1) = 0 for all E ∈ FE (Λ).
Remark . A number of aspects of these relations are worth commenting on:(i) In previous references about Leavitt path algebras and Kumjian-Pask algebras,people usually distinguish the vertex idempotents as “ P v ” (for example, see [1, 2,3, 4, 5, 11, 29, 30]). We do not follow this convention because we do not want tomake additional unnecessary cases in each proof.(ii) (KP2) in [5, 11] has more relations to check. However, using our notational con-vention, those relations can be simplified and are equivalent to our (KP2).(iii) The restriction to finitely aligned k -graphs is necessary for the sum in (KP3) tobe make sense (see [20]).(iv) In (KP3), we interpret the empty sum as 0, so S λ ∗ S µ = 0 whenever Λ min ( λ, µ ) = ∅ .We also have S λ ∗ S λ = S s ( λ ) .(v) (KP3-4) have been changed from those in [5, Definition 3.1] and [11, Definition3.1]. We do this because we need to adjust the relations to deal with situationwhere k -graph is not locally convex. For further discussion, see Appendix A of[22].The following lemma establishes some useful properties of a family satisfying (KP1-3). Proposition 3.3.
Let Λ be a finitely aligned k -graph, R be a commutative ring with ,and { S λ , S µ ∗ : λ, u ∈ Λ } be a family satisfying (KP1-3) in an R -algebra A . Then (a) S λ S λ ∗ S µ S µ ∗ = P λρ ∈ MCE( λ,µ ) S λρ S ( λρ ) ∗ for λ, µ ∈ Λ ; and { S λ S λ ∗ : λ ∈ Λ } is a com-muting family. (b) The subalgebra generated by { S λ , S µ ∗ : λ, u ∈ Λ } is span R { S λ S µ ∗ : λ, u ∈ Λ , s ( λ ) = s ( µ ) } . UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 7 (c)
For n ∈ N k and λ, µ ∈ Λ ≤ n , we have S λ ∗ S µ = δ λ,µ S s ( λ ) . (d) Suppose that rS v = 0 for all r ∈ R \ { } , v ∈ Λ and that λ, µ ∈ Λ with s ( λ ) = s ( µ ) . If r ∈ R \ { } and G ⊆ s ( λ ) Λ is finite non-exhaustive, then rS λ = 0 and rS λ (cid:0) Y ν ∈ G (cid:0) S s ( λ ) − S ν S ν ∗ (cid:1) (cid:1) S µ ∗ = 0 .Proof. To show (a), we take λ, µ ∈ Λ and then S λ S λ ∗ S µ S µ ∗ = S λ (cid:0) X ( ρ,τ ) ∈ Λ min ( λ,µ ) S ρ S τ ∗ (cid:1) S µ ∗ = X ( ρ,τ ) ∈ Λ min ( λ,µ ) S λρ S ( µτ ) ∗ = X ( ρ,τ ) ∈ Λ min ( λ,µ ) S λρ S ( λρ ) ∗ = X λρ ∈ MCE( λ,µ ) S λρ S ( λρ ) ∗ .Furthermore, S λ S λ ∗ S µ S µ ∗ = X λρ ∈ MCE( λ,µ ) S λρ S ( λρ ) ∗ = X µτ ∈ MCE( λ,µ ) S µτ S ( µτ ) ∗ = S µ S µ ∗ S λ S λ ∗ ,as required.Next we show (b). For λ, u ∈ Λ, we have S λ S µ ∗ = S λ S s ( λ ) S s ( µ ) S µ ∗ by (KP2). Thenby (KP1), S λ S µ ∗ = 0 implies s ( λ ) = s ( µ ). Therefore, the result follows from part (a),(KP2) and (KP3).To show (c), we take λ, µ ∈ Λ ≤ n . Suppose that S λ ∗ S µ = 0. By (KP3), there exists( ρ, τ ) ∈ Λ min ( λ, µ ) such that λρ = µτ and d ( λρ ) ≤ n . Since λ, µ ∈ Λ ≤ n , then ρ = s ( λ ) = τ and hence λ = µ .Finally, we show (d). Take r ∈ R \ { } and λ ∈ Λ. Suppose for contradiction that rS λ = 0. Then 0 = S λ ∗ ( rS λ ) = rS λ ∗ S λ = rS s ( λ ) ,which contradicts with rS v = 0 for all r ∈ R \ { } and v ∈ Λ . Hence, rS λ = 0.Now take r ∈ R \ { } , λ, µ ∈ Λ with s ( λ ) = s ( µ ) and finite non-exhaustive G ⊆ s ( λ ) Λ.Suppose for contradiction that rS λ (cid:0) Y ν ∈ G (cid:0) S s ( λ ) − S ν S ν ∗ (cid:1) (cid:1) S µ ∗ = 0.Since G is non-exhaustive, then there exists γ ∈ s ( λ ) Λ such that Ext ( γ ; G ) = ∅ . HenceΛ min ( ν, γ ) = ∅ for every ν ∈ G , and then by (KP3), S ν ∗ S γ = 0 for ν ∈ G . Therefore,0 = (cid:0) rS λ (cid:0) Y ν ∈ G (cid:0) S s ( λ ) − S ν S ν ∗ (cid:1) (cid:1) S µ ∗ (cid:1) S µγ = rS λ (cid:0) Y ν ∈ G (cid:0) S s ( λ ) − S ν S ν ∗ (cid:1) (cid:1) S γ = rS λ S γ = rS λγ ,which contradicts with rS λγ = 0. Hence, rS λ (cid:0) Q ν ∈ G (cid:0) S s ( λ ) − S ν S ν ∗ (cid:1) (cid:1) S µ ∗ = 0. (cid:3) Remark . For n ∈ N k , we have Λ n ⊆ Λ ≤ n . Hence, Proposition 3.3.(c) also implies thatfor n ∈ N k and λ, µ ∈ Λ n , we have S λ ∗ S µ = δ λ,µ S s ( λ ) . LISA ORLOFF CLARK AND YOSAFAT E. P. PANGALELA
Remark . Suppose that rS v = 0 for all r ∈ R \ { } , v ∈ Λ and that λ, µ ∈ Λwith s ( λ ) = s ( µ ). Then the contrapositive of Proposition 3.3.(d) says: if r ∈ R and G ⊆ s ( λ ) Λ is finite such that rS λ (cid:0) Q ν ∈ G (cid:0) S s ( λ ) − S ν S ν ∗ (cid:1) (cid:1) S µ ∗ = 0, then we have either r = 0 or G is exhaustive.Now we give an example of a Kumjian-Pask Λ-family in a particular algebra of endo-morphism. Proposition 3.6.
Let Λ be a finitely aligned k -graph and R be a commutative ring with .Let F R ( ∂ Λ) be the free module with basis the boundary path space. Then for every v ∈ Λ and λ, µ ∈ Λ \ Λ , there exist endomorphisms S v , S λ , S µ ∗ : F R ( ∂ Λ) → F R ( ∂ Λ) such thatfor x ∈ ∂ Λ , S v ( x ) = ( x if r ( x ) = v ; otherwise, S λ ( x ) = ( λx if s ( λ ) = r ( x ) ; otherwise, S µ ∗ ( x ) = ( σ d ( µ ) x if x (0 , d ( µ )) = µ ; otherwise.Furthermore, { S λ , S µ ∗ : λ, u ∈ Λ } is a Kumjian-Pask Λ -family in the R -algebra End ( F R ( ∂ Λ)) with rS v = 0 for all r ∈ R \ { } and v ∈ Λ .Proof. Take v ∈ Λ and λ, µ ∈ Λ \ Λ . First note that for x ∈ ∂ Λ and m ≤ d ( x ), we have σ m x ∈ ∂ Λ. Now define functions f v , f λ , and f µ ∗ : ∂ Λ → F R ( ∂ Λ) by f v ( x ) = ( x if r ( x ) = v ;0 otherwise, f λ ( x ) = ( λx if s ( λ ) = r ( x ) ;0 otherwise, f µ ∗ ( x ) = ( σ d ( µ ) x if x (0 , d ( µ )) = µ ;0 otherwise.The universal property of free modules gives nonzero endomorphisms S v , S λ , S µ ∗ : F R ( ∂ Λ) → F R ( ∂ Λ)extending f v , f λ , and f µ ∗ , as needed.Now we claim that { S λ , S µ ∗ : λ, u ∈ Λ } is a Kumjian-Pask Λ-family. To see (KP1), take v ∈ Λ and x ∈ ∂ Λ. Then we have S v ( x ) = x = S v ( x ) if r ( x ) = v , and S v ( x ) = 0 = S v ( x )otherwise. Hence S v = S v . Now take v, w ∈ Λ with v = w and x ∈ ∂ Λ. Since x ∈ w∂ Λimplies x / ∈ v∂ Λ, we have S v S w ( x ) = 0 for x ∈ ∂ Λ and S v S w = 0.Next we show (KP2). Take λ, µ ∈ Λ with s ( λ ) = r ( µ ). Then for x ∈ s ( µ ) ∂ Λ, wehave µx ∈ s ( λ ) ∂ Λ. Then S λ S µ ( x ) = λµx = S λµ ( x ) if x ∈ s ( µ ) ∂ Λ, and S λ S µ ( x ) = 0 = S λµ ( x ) otherwise. Hence S λ S µ = S λµ . Meanwhile, for x ∈ r ( λ ) ∂ Λ with x (0 , d ( λµ )) = λµ , we have d ( λµ ) ≤ d ( x ) and σ d ( λµ ) x ∈ s ( µ ) ∂ Λ. Furthermore, x (0 , d ( λµ )) = λµ , UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 9 implies x (0 , d ( λ )) = λ and then we have d ( λ ) ≤ d ( x ) and σ d ( λ ) x ∈ s ( λ ) ∂ Λ. Hence, S µ ∗ S λ ∗ ( x ) = S µ ∗ σ d ( λ ) x = σ d ( λ )+ d ( µ ) x = σ d ( λµ ) x = S ( λµ ) ∗ ( x )if x (0 , d ( λµ )) = λµ , and S µ ∗ S λ ∗ ( x ) = 0 = S ( λµ ) ∗ ( x ) otherwise. Therefore, S µ ∗ S λ ∗ = S ( λµ ) ∗ .Now we show (KP3). Take λ, µ ∈ Λ. If r ( λ ) = r ( µ ), then S λ ∗ S µ = 0 and Λ min ( λ, µ ) = ∅ , as required. Suppose r ( λ ) = r ( µ ). We have S λ ∗ S µ ( x ) = ( ( µx ) ( d ( λ ) , d ( µx )) if x ∈ s ( µ ) ∂ Λ and ( µx ) (0 , d ( λ )) = λ ;0 otherwise.Take x ∈ s ( µ ) ∂ Λ. Note that s ( µ ) = r ( τ ) for ( ρ, τ ) ∈ Λ min ( λ, µ ). First suppose( µx ) (0 , d ( λ )) = λ . Then for ( ρ, τ ) ∈ Λ min ( λ, µ ),( µx ) (0 , d ( λρ )) = λρ and ( µx ) (0 , d ( µτ )) = µτ .Hence x (0 , d ( τ )) = τ and S ρ S τ ∗ ( x ) = S ρ (0) = 0. Therefore X ( ρ,τ ) ∈ Λ min ( λ,µ ) S ρ S τ ∗ ( x ) = 0 . Next suppose ( µx ) (0 , d ( λ )) = λ . Since ( µx ) (0 , d ( λ )) = λ and ( µx ) (0 , d ( µ )) = µ ,there is τ ∈ s ( µ ) Λ such that ( ρ, τ ) ∈ Λ min ( λ, µ ) and ( µx ) (0 , d ( µτ )) = µτ . There-fore x (0 , d ( τ )) = τ . Note that this τ is unique by the factorisation property. Hencefor ( ρ ′ , τ ′ ) ∈ Λ min ( λ, µ ) such that ( ρ ′ , τ ′ ) = ( ρ, τ ), we have S ρ ′ S τ ′∗ ( x ) = 0. Also x (0 , d ( τ )) = τ , thus S ρ S τ ∗ ( x ) = S ρ ( x ( d ( τ ) , d ( x ))) = ρ [ x ( d ( τ ) , d ( x ))]= ρ [( µx ) ( d ( µτ ) , d ( µx ))]= ρ [( µx ) ( d ( λρ ) , d ( µx ))] (since µτ = λρ )= ( µx ) ( d ( λ ) , d ( µx ))and X ( ρ ′ ,τ ′ ) ∈ Λ min ( λ,µ ) S ρ S τ ∗ ( x ) = S ρ S τ ∗ ( x ) = ( µx ) ( d ( λ ) , d ( µx )) = S λ ∗ S µ ( x ) ,as required.Finally, we show (KP4). Take E ∈ FE (Λ). Take x ∈ r ( E ) ∂ Λ. Since E ∈ x (0) FE (Λ)and x is a boundary path, then there exists λ ∈ E such that x (0 , d ( λ )) = λ . This implies (cid:0) S r ( E ) − S λ S λ ∗ (cid:1) ( x ) = S r ( E ) ( x ) − S λ S λ ∗ ( x )= x − S λ ( x ( d ( λ ) , d ( x )))= x − x = 0.Hence (cid:0) Y λ ∈ E (cid:0) S r ( E ) − S λ S λ ∗ (cid:1) (cid:1) ( x ) = 0for x ∈ r ( E ) ∂ Λ, and Q λ ∈ E (cid:0) S r ( E ) − S λ S λ ∗ (cid:1) = 0.Thus { S λ , S µ ∗ : λ, u ∈ Λ } is a Kumjian-Pask Λ-family, as claimed. Now note that for v ∈ Λ , v∂ Λ is nonempty. This implies that for all r ∈ R \ { } and v ∈ Λ , rS v = 0. (cid:3) Using an alternate construction of a Kumjian-Pask Λ-family, we next show that thereis an R -algebra which is universal for Kumjian-Pask Λ-families. Theorem 3.7.
Let Λ be a finitely aligned k -graph and R be a commutative ring with . (a) There is a universal R -algebra KP R (Λ) generated by a Kumjian-Pask Λ -family { s λ , s µ ∗ : λ, u ∈ Λ } such that whenever { S λ , S µ ∗ : λ, u ∈ Λ } is a Kumjian-Pask Λ -family in an R -algebra A , then there exists a unique R -algebra homomorphism π S : KP R (Λ) → A such that π S ( s λ ) = S λ and π S ( s µ ∗ ) = S µ ∗ for λ, µ ∈ Λ . (b) We have rs v = 0 for all r ∈ R \ { } and v ∈ Λ . (c) The subsets KP R (Λ) n := span R { s λ s µ ∗ : λ, µ ∈ Λ , d ( λ ) − d ( µ ) = n } forms a Z k -grading of KP R (Λ) .Proof. We use an argument similar to [5, Theorem 3.4] and [11, Theorem 3.7]. To show(a), first we define X := Λ ∪ G (cid:0) Λ =0 (cid:1) and F R ( w ( X )) be the free algebra on the set w ( X )of words on X . Let I be the ideal of F R ( w ( X )) generated by elements of the followingsets:(i) { vw − δ v,w v : v, w ∈ Λ } ,(ii) { λ − µν, λ ∗ − ν ∗ µ ∗ : λ, µ, ν ∈ Λ and λ = µν } ,(iii) { λ ∗ µ − P ( ρ,τ ) ∈ Λ min ( λ,µ ) ρτ ∗ : λ, µ ∈ Λ } , and(iv) { Q λ ∈ E ( r ( E ) − λλ ∗ ) : E ∈ FE (Λ) } .Now define KP R (Λ) := F R ( w ( X )) /I and q : F R ( w ( X )) → F R ( w ( X )) /I be thequotient map. Define s λ := q ( λ ) for λ ∈ Λ, and s µ ∗ := q ( µ ∗ ) for µ ∗ ∈ G (cid:0) Λ =0 (cid:1) . Then { s λ , s µ ∗ : λ ∈ Λ , µ ∗ ∈ G (cid:0) Λ =0 (cid:1) } is a Kumjian-Pask Λ-family in KP R (Λ).Now let { S λ , S µ ∗ : λ, u ∈ Λ } be a Kumjian-Pask Λ-family in an R -algebra A . Define f : X → A by f ( λ ) := S λ for λ ∈ Λ, and f ( µ ∗ ) := S µ ∗ for µ ∗ ∈ G (cid:0) Λ =0 (cid:1) . The universalproperty of F R ( w ( X )) gives an unique R -algebra homomorphism φ : F R ( w ( X )) → A such that φ | X = f . Since { S λ , S µ ∗ : λ, u ∈ Λ } is a Kumjian-Pask Λ-family, then I ⊆ ker ( φ ). Thus there exists an R -algebra homomorphism π S : KP R (Λ) → A such that π S ◦ q = φ . The homomorphism π S is unique since the elements in X generate F R ( w ( X ))as an algebra. Furthermore, we have π S ( s λ ) = S λ for λ ∈ Λ and π S ( s µ ∗ ) = S µ ∗ for µ ∗ ∈ G (cid:0) Λ =0 (cid:1) , as required.To show (b), let { S λ , S µ ∗ : λ, u ∈ Λ } be the Kumjian-Pask Λ-family as in Proposi-tion 3.6. Then rS v = 0 for v ∈ Λ . Since π S ( rs v ) = rS v = 0 for all r ∈ R \ { } and v ∈ Λ , we have rs v = 0 for all r ∈ R \ { } and v ∈ Λ .Next we show (c). We first extend the degree map to w ( X ) by d ( w ) := P | w | i =1 d (( w i ))for w ∈ w ( X ). By [5, Proposition 2.7], F R ( w ( X )) is Z k -graded by the subgroups F R ( w ( X )) n := X w ∈ w ( X ) r w w : r w = 0 implies d ( w ) = n .Now we claim that the ideal I defined in (a) is a graded ideal. It suffices to showthat I is generated by elements in F R ( w ( X )) n for some n ∈ Z k . Since d ( v ) = 0 for v ∈ Λ , then the generators in (i) belong to F R ( w ( X )) . If λ = µν in Λ, then λ − µν belongs to F R ( w ( X )) d ( λ ) and λ ∗ − ν ∗ µ ∗ belongs to F R ( w ( X )) − d ( λ ) . For λ, µ ∈ Λ and
UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 11 ( ρ, τ ) ∈ Λ min ( λ, µ ), we have d ( ρ ) − d ( τ ) = ( d ( λ ) ∨ d ( µ ) − d ( λ )) − ( d ( λ ) ∨ d ( µ ) − d ( µ )) = − d ( λ ) + d ( µ )and then the generators in (iii) belong to F R ( w ( X )) − d ( λ )+ d ( µ ) . Finally, a word λλ ∗ hasdegree 0 and then the generators in (iv) belong to F R ( w ( X )) . Thus I is a graded ideal.Since I is graded, then KP R (Λ) = F R ( w ( X )) /I is graded by the subgroups( F R ( w ( X )) /I ) n := span R { q ( w ) : w ∈ w ( X ) , d ( w ) = n } .By Proposition 3.3.(b), we have KP R (Λ) = span R { s λ s µ ∗ : λ, u ∈ Λ , s ( λ ) = s ( µ ) } . Wehave to show thatKP R (Λ) n := span R { s λ s µ ∗ : λ, u ∈ Λ , d ( λ ) − d ( µ ) = n } = ( F R ( w ( X )) /I ) n .Take λ, u ∈ Λ with d ( λ ) − d ( µ ) = n . Then s λ s µ ∗ = q ( λ ) q ( µ ∗ ) = q ( λµ ∗ ) and d ( λµ ∗ ) = d ( λ ) − d ( µ ) = n . Hence s λ s µ ∗ ∈ ( F R ( w ( X )) /I ) n and KP R (Λ) n ⊆ ( F R ( w ( X )) /I ) n .To prove ( F R ( w ( X )) /I ) n ⊆ KP R (Λ) n , we first establish the following claim: Claim 3.8.
Let X := Λ ∪ G (cid:0) Λ =0 (cid:1) and q : F R ( w ( X )) → KP R (Λ) be the quotient map.Then for w ∈ w ( X ) , we have q ( w ) ∈ KP R (Λ) d ( w ) .Proof of Claim 3.8. We are modifying the proof of [5, Lemma 3.5] and [11, Lemma 3.8]using our version of (KP3). We prove the claim by induction on | w | . For | w | = 0, wehave w = v for some v ∈ Λ . Then q ( w ) = s v = s v s v ∗ and d ( v ) − d ( v ) = 0. So q ( w ) ∈ KP R (Λ) d ( w ) .For | w | = 1, we have two possibilities. First suppose w = λ for λ ∈ Λ. Then q ( w ) = s λ = s λ s s ( λ ) ∗ and d ( λ ) − d ( s ( λ )) = d ( λ ). So q ( w ) ∈ KP R (Λ) d ( w ) . Next suppose w = λ ∗ for λ ∈ Λ. Then q ( w ) = s λ ∗ = s s ( λ ) s λ ∗ and d ( s ( λ )) − d ( λ ) = − d ( λ ) = d ( λ ∗ ). So q ( w ) ∈ KP R (Λ) d ( w ) .For | w | = 2, we have four possibilities: w = λµ ∗ , w = λµ , w = µ ∗ λ ∗ , or w = λ ∗ µ . Forthe first three cases, we have q ( λµ ∗ ) = s λ s µ ∗ and d ( λ ) − d ( µ ) = d ( λµ ∗ ) , q ( λµ ) = s λµ s s ( µ ) ∗ and d ( λµ ) − d ( s ( µ )) = d ( λµ ) , q ( µ ∗ λ ∗ ) = s s ( µ ) s ( λµ ) ∗ and d ( s ( µ )) − d (( λµ ) ∗ ) = d ( µ ∗ λ ∗ ) ,as required. Suppose w = λ ∗ µ . By (KP3), we have q ( λ ∗ µ ) = s λ ∗ s µ = X ( ρ,τ ) ∈ Λ min ( λ,µ ) s ρ s τ ∗ .For ( ρ, τ ) ∈ Λ min ( λ, µ ), we have λρ = µτ and then d ( w ) = d ( µ ) − d ( λ ) = d ( ρ ) − d ( ρ ).So q ( w ) ∈ KP R (Λ) d ( w ) .Now suppose that n ≥ q ( y ) ∈ KP R (Λ) d ( y ) for every word y with | y | ≤ n . Let w be a word with | w | = n + 1 and q ( w ) = 0. If w contains a subword w i w i +1 = λµ , then λ and µ are composable in Λ since otherwise q ( λµ ) = 0. Now let w ′ be the word obtainedfrom w by replacing w i w i +1 with the single path λµ , and then q ( w ) = s w · · · s w i − s λ s µ s w i +2 s w n +1 = s w · · · s w i − s λµ s w i +2 s w n +1 = q ( w ′ ) .Since | w ′ | = n and d ( w ′ ) = d ( w ), the inductive hypothesis implies q ( w ) ∈ KP R (Λ) d ( w ) .A similar argument shows q ( w ) ∈ KP R (Λ) d ( w ) whenever w contains a subword w i w i +1 = µ ∗ λ ∗ . So suppose w contains no subword of the form λµ or µ ∗ λ ∗ . Since | w | ≥
3, either w w or w w has the form λ ∗ µ . By (KP3), we write q ( w ) as a sum of terms q ( y i ) with | y i | = n + 1 and d ( y i ) = d ( w ). Since | w | ≥
3, each nonzero summand q ( y i ) contains afactor of the form s γ s ρ or one of the form s τ ∗ s γ ∗ . Then the previous argument shows thatevery q ( y i ) ∈ KP R (Λ) d ( w ) and q ( w ) ∈ KP R (Λ) d ( w ) , as required. (cid:3) Claim 3.8Every element in ( F R ( w ( X )) /I ) n is in the form q ( w ) with w ∈ w ( X ) and d ( w ) = n , which, by Claim 3.8, belongs to KP R (Λ) n . Then ( F R ( w ( X )) /I ) n ⊆ KP R (Λ) n , asrequired. (cid:3) Definition 3.9.
Suppose that { S λ , S µ ∗ : λ, u ∈ Λ } is the Kumjian-Pask Λ-family in the R -algebra End ( F R ( ∂ Λ)) as in Proposition 3.6. We call the R -algebra homomorphism π S : KP R (Λ) → End ( F R ( ∂ Λ)) obtained from Theorem 3.7.(a), the boundary path repre-sentation of KP R (Λ).4. The graded-invariant uniqueness theorem
Throughout this section, Λ is a finitely aligned k -graph and R is a commutative ringwith identity 1. Theorem 4.1 (The graded-uniqueness theorem) . Let Λ be a finitely aligned k -graph, R bea commutative ring with , and A be a Z k -graded R -algebra. Suppose that π : KP R (Λ) → A is a Z k -graded R -algebra homomorphism such that π ( rs v ) = 0 for all r ∈ R \ { } and v ∈ Λ . Then π is injective. We start the proof of Theorem 4.1 by adapting some C ∗ -algebra results used to provethe gauge-invariant uniqueness theorem [22, Theorem 4.2] to Kumjian-Pask algebras.Although the argument is rather technical, the point is that most of the argument in C ∗ -algebra setting also works in our situation.First we recall from [22, Definition 2.5] that a Cuntz-Krieger Λ -family is a collection { T λ : λ ∈ Λ } of partial isometries (in other words, it satisfies T λ = T λ T ∗ λ T λ for λ ∈ Λ, see[23, Appendix A]) in a C ∗ -algebra B satisfying:(TCK1) { T v : v ∈ Λ } is a collection of mutually orthogonal projections;(TCK2) T λ T µ = T λµ whenever s ( λ ) = r ( µ );(TCK3) T ∗ λ T µ = P ( ρ,τ ) ∈ Λ min ( λ,µ ) T ρ T ∗ τ for all λ, µ ∈ Λ; and(CK) Q λ ∈ E (cid:0) T r ( E ) − T λ T ∗ λ (cid:1) = 0 for all E ∈ FE (Λ).For a finitely aligned k -graph Λ, there exists a universal C ∗ -algebra C ∗ (Λ) generated bythe universal Cuntz-Krieger Λ-family { t λ : λ ∈ Λ } . Now suppose that { S λ , S µ ∗ : λ, u ∈ Λ } is a Kumjian-Pask Λ-family in an R -algebra A and we define T λ := S λ for λ ∈ Λ and T ∗ µ := S µ ∗ for µ ∈ G (cid:0) Λ =0 (cid:1) . Then { T λ : λ ∈ Λ } is a collection satisfying T λ = T λ T ∗ λ T λ for λ ∈ Λ, (TCK1-3) and (CK). (Note that we do not say that { T λ : λ ∈ Λ } is a Cuntz-KriegerΛ-family, since we need a C ∗ -algebra containing T λ , T ∗ µ .) Similarly, a Cuntz-Krieger Λ-family in a C ∗ -algebra gives a Kumjian-Pask Λ-family. Thus one can translate proofsabout Cuntz-Krieger Λ-families to proofs about Kumjian-Pask Λ-families.The key ingredient to proof of Theorem 4.1 is proving that the uniqueness theorem holdson the core KP R (Λ) := span R { s λ s µ ∗ : d ( λ ) = d ( µ ) } (Theorem 4.4). First we establishsome preliminary results and notation. UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 13
Following [22, Lemma 3.2], for every finite set E ⊆ Λ, there exists a finite set F ⊆ Λwhich contains E and satisfies λ, µ, ρ, τ ∈ F , d ( λ ) = d ( µ ) , d ( ρ ) = d ( τ ) , s ( λ ) = s ( µ ) , and s ( ρ ) = s ( τ )(4.1) imply (cid:8) λα, τ β : ( α, β ) ∈ Λ min ( µ, ρ ) (cid:9) ⊆ F .We then write Π E := \ { F ⊆ Λ : E ⊆ F and F satisfies (4.1) } and Π E × d,s Π E for the set { ( λ, µ ) ∈ Π E × Π E : d ( λ ) = d ( µ ) , s ( λ ) = s ( µ ) } . Note thatΠ E is finite. Now recall from Notation 3.12 of [22] that for λ ∈ Π E , we write T ( λ ) := { ν ∈ s ( λ ) Λ : d ( ν ) = 0 , λν ∈ Π E } .Since λT ( λ ) ⊆ Π E and Π E is finite, then T ( λ ) is also finite.Now suppose that { S λ , S µ ∗ : λ, u ∈ Λ } is a Kumjian-Pask Λ-family in an R -algebra A .The argument of Lemma 3.2 of [22] shows that the set M S Π E := span R { S λ S µ ∗ : ( λ, µ ) ∈ Π E × d,s Π E } is closed under multiplication. For ( λ, µ ) ∈ Π E × d,s Π E , defineΘ ( S ) Π Eλ,µ := S λ (cid:0) Y ν ∈ T ( λ ) (cid:0) S s ( λ ) − S λν S ( λν ) ∗ (cid:1) (cid:1) S µ ∗ . Applying the argument of Proposition 3.9 and Proposition 3.11 of [22] gives the follow-ing.
Lemma 4.2.
Let { S λ , S µ ∗ : λ, u ∈ Λ } be a Kumjian-Pask Λ -family in an R -algebra A and E ⊆ Λ be finite. For ( λ, µ ) , ( ρ, τ ) ∈ Π E × d,s Π E , we have Θ ( S ) Π Eλ,µ
Θ ( S ) Π Eρ,τ = δ µ,ρ Θ ( S ) Π Eλ,τ , S λ S µ ∗ = X λν ∈ Π E Θ ( S ) Π Eλν,µν and M S Π E is spanned by the set { Θ ( S ) Π Eλ,µ : ( λ, µ ) ∈ Π E × d,s Π E } . Lemma 4.3.
Let Λ be a finitely aligned k -graph, R be a commutative ring with and E ⊆ Λ be finite. Suppose that π : KP R (Λ) → A is an R -algebra homomorphism such that π ( rs v ) = 0 for all r ∈ R \ { } and v ∈ Λ . Let ( λ, µ ) ∈ Π E × d,s Π E . Then the followingconditions are equivalent: (a) π (cid:0) Θ ( s ) Π Eλ,µ (cid:1) = 0 . (b) Θ ( s ) Π Eλ,µ = 0 . (c) T ( λ ) is exhaustive.Furthermore, for r ∈ R \ { } we have π (cid:0) r Θ ( s ) Π Eλ,µ (cid:1) = 0 if and only if r Θ ( s ) Π Eλ,µ = 0 and π is injective on M s Π E .Proof. By following the argument of Proposition 3.13 and Corollary 3.17 of [22], we havethe three equivalent conditions. Now take ( λ, µ ) ∈ Π E × d,s Π E and r ∈ R \ { } . If r Θ ( s ) Π Eλ,µ = 0, we trivially have π (cid:0) r Θ ( s ) Π Eλ,µ (cid:1) = 0. So suppose π (cid:0) r Θ ( s ) Π Eλ,µ (cid:1) = 0. Since π ( rs v ) = 0 for all r ∈ R \ { } and v ∈ Λ , then by Remark 3.5, π (cid:0) r Θ ( s ) Π Eλ,µ (cid:1) = 0 implies that T ( λ ) is exhaustive (since r = 0) and by (c) ⇒ (b), Θ ( s ) Π Eλ,µ = 0. So r Θ ( s ) Π Eλ,µ = 0, asrequired.Next we show that π is injective on M s Π E . Take a ∈ M s Π E such that π ( a ) = 0. Wehave to show a = 0. Since a ∈ M s Π E and M s Π E = span R { Θ ( s ) Π Eλ,µ : ( λ, µ ) ∈ Π E × d,s Π E } (Lemma 4.2), we write a = P ( λ,µ ) ∈ F r λ,µ Θ ( s ) Π Eλ,µ where F ⊆ Π E × d,s Π E is finite andfor all ( λ, µ ) ∈ F , we have r λ,µ ∈ R and Θ ( s ) Π Eλ,µ = 0. If T ( λ ) is exhaustive for some( λ, µ ) ∈ F , then by (c) ⇒ (b), Θ ( s ) Π Eλ,µ = 0, which contradicts Θ ( s ) Π Eλ,µ = 0. So T ( λ ) isnon-exhaustive for all ( λ, µ ) ∈ F . Since π ( a ) = 0, then for ( ρ, τ ) ∈ F , we have0 = π (cid:0) Θ ( s ) Π Eρ,ρ (cid:1) π ( a ) π (cid:0) Θ ( s ) Π Eτ,τ (cid:1) = π (cid:0) Θ ( s ) Π Eρ,ρ (cid:1) π (cid:0) X ( λ,µ ) ∈ F r λ,µ Θ ( s ) Π Eλ,µ (cid:1) π (cid:0) Θ ( s ) Π Eτ,τ (cid:1) = r ρ,τ π (cid:0) Θ ( s ) Π Eρ,τ (cid:1) = r ρ,τ Θ ( π ( s )) Π Eρ,τ (by Lemma 4.2).But now since π ( rs v ) = 0 for all r ∈ R \ { } and v ∈ Λ , then by Remark 3.5, r ρ,τ Θ ( π ( s )) Π Eρ,τ = 0 implies that r ρ,τ = 0 (since T ( ρ ) is non-exhaustive). Therefore, a = 0 and π is injective on M s Π E . (cid:3) A direct consequence of Lemma 4.3 is:
Theorem 4.4.
Let Λ be a finitely aligned k -graph and R be a commutative ring with .Suppose that π : KP R (Λ) → A is an R -algebra homomorphism such that π ( rs v ) = 0 forall r ∈ R \ { } and v ∈ Λ . Then π is injective on KP R (Λ) .Proof. Take a ∈ KP R (Λ) such that π ( a ) = 0. We have to show a = 0. Write a = P ( λ,µ ) ∈ F r λ,µ s λ s µ ∗ with d ( λ ) = d ( µ ) for ( λ, µ ) ∈ F . Define E := { λ, µ : ( λ, µ ) ∈ F } andthen a ∈ M s Π E . Since π is injective on M s Π E (Lemma 4.3), a = 0. (cid:3) Now we establish the last stepping stone result before proving Theorem 4.1.
Lemma 4.5.
Let I be a graded ideal of KP R (Λ) . Then I is generated as an ideal by theset I := I ∩ KP R (Λ) .Proof. We generalise the argument of [30, Lemma 5.1]. Take n ∈ Z k and write n = n − n such that n , n ∈ N k . We show that I n := I ∩ KP R (Λ) n is contained inKP R (Λ) n I KP R (Λ) n . Now take a ∈ I n and write a = P ( λ,µ ) ∈ F r λ,µ s λ s µ ∗ . Note that d ( λ ) − d ( µ ) = n for ( λ, µ ) ∈ F . Since n = n − n with n , n ∈ N k , then for every( λ, µ ) ∈ F , by the factorisation property, there exist λ , λ , µ , µ such that λ = λ λ , µ = µ µ , d ( λ ) = n , d ( µ ) = n , and d ( λ ) = d ( µ ) .Hence a = P ( λ λ ,µ µ ) ∈ F r λ λ ,µ µ s λ (cid:0) s λ s µ ∗ (cid:1) s µ ∗ . Take ( α α , β β ) ∈ F . Note that for ν, γ ∈ Λ with d ( ν ) = d ( γ ), by Remark 3.4, we have s ν ∗ s γ = 0 if ν = γ . Then s α ∗ as β = X ( λ λ ,µ µ ) ∈ F r λ λ ,µ µ (cid:0) s α ∗ s λ (cid:1) (cid:0) s λ s µ ∗ (cid:1) (cid:0) s µ ∗ s β (cid:1) = X ( α λ ,β µ ) ∈ F r α λ ,β µ s λ s µ ∗ UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 15 since d ( α ) = n = d ( λ ) and d ( β ) = n = d ( µ ) for ( λ λ , µ µ ) ∈ F . Since a ∈ I ,then s α ∗ as β ∈ I . Furthermore, since d ( λ ) = d ( µ ) for ( α λ , β µ ) ∈ F , then s α ∗ as β ∈ KP R (Λ) . Hence, for ( α α , β β ) ∈ F , X ( α λ ,β µ ) ∈ F r α λ ,β µ s λ s µ ∗ = s α ∗ as β ∈ I and X ( α λ ,β µ ) ∈ F r α λ ,β µ s α λ s ( β µ ) ∗ = s α (cid:0) s α ∗ as β (cid:1) s β ∗ ∈ KP R (Λ) n I KP R (Λ) n .Therefore, a = X ( λ λ ,µ µ ) ∈ F r λ λ ,µ µ s λ λ s ( µ µ ) ∗ = X { ( α ,β ):( α α ,β β ) ∈ F } X ( α λ ,β µ ) ∈ F r α λ ,β µ s α λ s ( β µ ) ∗ belongs to KP R (Λ) n I KP R (Λ) n , and I n ⊆ KP R (Λ) n I KP R (Λ) n .Now since I is a graded ideal I = L n ∈ Z k I n and I is generated as an ideal by I . (cid:3) Proof of Theorem 4.1.
Because π is graded, we have that ker π is a graded ideal. ByLemma 4.5, the ideal ker π is generated by the set ker π ∩ KP R (Λ) . Thus it sufficesto show π | KP R (Λ) : KP R (Λ) → A is injective. However, the injectivity follows fromTheorem 4.4. (cid:3) One immediate application of Theorem 4.1 is:
Proposition 4.6.
Let Λ be a finitely aligned k -graph. Let { s λ , s µ ∗ : λ, u ∈ Λ } be theuniversal Kumjian-Pask Λ -family for R = C and { t λ : λ ∈ Λ } be the universal Cuntz-Krieger Λ -family. Then there is an isomorphism π t : KP C (Λ) → span C (cid:8) t λ t ∗ µ : λ, µ ∈ Λ (cid:9) such that π t ( s λ ) = t λ and π t ( s µ ∗ ) = t ∗ µ for λ, u ∈ Λ . In particular, KP C (Λ) is isomorphicto a dense subalgebra of C ∗ (Λ) .Proof. Since { t λ : λ ∈ Λ } satisfies (TCK1-3) and (CK), then (cid:8) t λ , t ∗ µ : λ, µ ∈ Λ (cid:9) also sat-isfies (KP1-4) and is a Kumjian-Pask Λ-family in C ∗ (Λ). Thus the universal property ofKP C (Λ) gives a homomorphism π t from KP C (Λ) onto the dense subalgebra A := span C (cid:8) t λ t ∗ µ : λ, µ ∈ Λ (cid:9) of C ∗ (Λ).Next we show the injectivity of π t . By Theorem 4.1, it suffices to show that π t is a Z k -graded algebra homomorphism. We claim that A is graded by A n := span C (cid:8) t λ t ∗ µ : λ, µ ∈ Λ , d ( λ ) − d ( µ ) = n (cid:9) .Note that for λ, µ, ρ, τ ∈ Λ with d ( λ ) − d ( µ ) = n and d ( ρ ) − d ( τ ) = m , we have t λ t ∗ µ t ρ t ∗ τ = t λ (cid:0) X ( µ ′ ,ρ ′ ) ∈ Λ min ( µ,ρ ) t µ ′ t ∗ ρ ′ (cid:1) t ∗ τ (by (TCK3))= X ( µ ′ ,ρ ′ ) ∈ Λ min ( µ,ρ ) t λµ ′ t ∗ τρ ′ and for ( µ ′ , ρ ′ ) ∈ Λ min ( µ, ρ ), d ( λµ ′ ) − d ( τ ρ ′ ) = d ( λ ) + d ( µ ′ ) − d ( τ ) − d ( ρ ′ )= d ( λ ) + ( d ( µ ) ∨ d ( ρ ) − d ( µ )) − d ( τ ) − ( d ( µ ) ∨ d ( ρ ) − d ( ρ ))= ( d ( λ ) − d ( µ )) − ( d ( τ ) − d ( ρ ))= n + m .Hence A n A m ⊆ A n + m . Since each spanning element t λ t ∗ µ belongs to A d ( λ ) − d ( µ ) , everyelement a of A can be written as a finite sum P a n with a n ∈ A n . For a n ∈ A n such thata finite sum P a n = 0, then we have each a n = 0 by following the argument of [5, Lemma7.4]. Thus (cid:8) A n : n ∈ Z k (cid:9) is a grading of A , as claimed. Then π t is a Z k -graded and byTheorem 4.1, π t is injective. (cid:3) Steinberg algebras
Steinberg algebras were introduced by Steinberg in [28] and are algebraic analogues ofgroupoid C ∗ -algebras. In [12], Clark and Sims show that for every 1-graph E , its Leavittpath algebra is isomorphic to a Steinberg algebra. In this section, we show that for everyfinitely aligned k -graph Λ, its Kumjian-Pask algebra is isomorphic to a Steinberg algebra(Proposition 5.4). We start out with an introduction to groupoids and Steinberg algebrasin general.A groupoid G is a small category in which every morphism has an inverse. For agroupoid G , we write r ( a ) and s ( a ) to denote the range and source of a ∈ G . Because r ( a ) = s ( a − ) for a ∈ G , then r and s have the common image. We call this commonimage the unit space of G and denote it G (0) . A pair ( a, b ) ∈ G × G is said composable if s ( a ) = r ( b ). We then use notation G (2) to denote the collection of composable pairs in G . For A, B ⊆ G , we write AB := (cid:8) ab : a ∈ A, b ∈ B, ( a, b ) ∈ G (2) (cid:9) .We say G is a topological groupoid if G is endowed with a topology such that compositionand inversion on G are continuous. We also call an open set U ⊆ G an open bisection if s and r restricted to U are homeomorhisms into G (0) . Finally, we call G ample if G has abasis of compact open bisections. Remark . Note that if G is ample, then G is locally compact and ´etale. In fact, G is Hausdorff ample if and only if G is locally compact, Hausdorff and ´etale with totallydisconnected unit space.Now suppose that G is a Hausdorff ample groupoid and R is a commutative ring with1. As in [9, Section 2.2], the Steinberg algebra associated to G is A R ( G ) := { f : G → R : f is locally constant and has compact support } where addition and scalar multiplication are defined pointwise, and convolution is givenby ( f ⋆ g ) ( a ) := X r ( a )= r ( b ) f ( b ) g (cid:0) b − a (cid:1) . In [28], Steinberg writes R G to denote A R ( G ). UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 17
Furthermore, for compact open bisections U and V , we have the characteristic function1 U ∈ A R ( G ) and 1 U ⋆ V = 1 UV [28, Proposition 4.3]. Note that for f ∈ A R ( G ), supp ( f ) is clopen ([9, Remark 2.1]). Example . To each finitely aligned k -graph Λ, we define the associated boundary-pathgroupoid G Λ from [32, Definition 4.8] as follows. WriteΛ ∗ s Λ := { ( λ, µ ) ∈ Λ × Λ : s ( λ ) = s ( µ ) } .The objects of G Λ are Obj ( G Λ ) := ∂ Λ.The morphisms areMor ( G Λ ) := { ( λz, d ( λ ) − d ( µ ) , µz ) ∈ ∂ Λ × Z k × ∂ Λ :( λ, µ ) ∈ Λ ∗ s Λ , z ∈ s ( λ ) ∂ Λ } = { ( x, m, y ) ∈ ∂ Λ × Z k × ∂ Λ : there exists p, q ∈ N k such that p ≤ d ( x ) , q ≤ d ( y ) , p − q = m and σ p x = σ q y } .The range and source maps are given by r ( x, m, y ) := x and s ( x, m, y ) := y , and compo-sition is defined such that(( x , m , y ) , ( y , m , y )) ( x , m + m , y ) .Fianlly inversion is given by ( x, m, y ) ( y, − m, x ).Next, we show how to realise G Λ as a topological groupoid. For ( λ, µ ) ∈ Λ ∗ s Λ andfinite non-exhaustive subset G ⊆ s ( λ ) Λ, we write Z Λ ( λ ) := λ∂ Λ, Z Λ ( λ \ G ) := Z Λ ( λ ) \ (cid:16) [ ν ∈ G Z Λ ( λν ) (cid:17) , Z Λ ( λ ∗ s µ ) := { ( x, d ( λ ) − d ( µ ) , y ) ∈ G Λ : x ∈ Z Λ ( λ ) , y ∈ Z Λ ( µ )and σ d ( λ ) x = σ d ( µ ) y } ,and Z Λ ( λ ∗ s µ \ G ) := Z Λ ( λ ∗ s µ ) \ (cid:16) [ ν ∈ G Z Λ ( λν ∗ s µν ) (cid:17) . The sets Z Λ ( λ ∗ s µ \ G ) form a basis of compact open bisections for a second-countable,Hausdorff topology on G Λ under which it is an ample groupoid. Further, the sets Z Λ ( λ \ G )form a basis of compact open sets for G (0)Λ . Remark . A number of notes of this example:(i) We think of G (0)Λ = ∂ Λ as a subset of G Λ under the correspondence x ( x, , x ).(ii) In [32], Yeend defines Z Λ ( λ \ G ) and Z Λ ( λ ∗ s µ \ G ) where G is finite. However,if G is exhaustive, then Z Λ ( λ \ G ) and Z Λ ( λ ∗ s µ \ G ) are empty sets. Thus ourdefinitions make sure that both Z Λ ( λ \ G ) and Z Λ ( λ ∗ s µ \ G ) are non-empty.Next we generalise [10, Proposition 4.3] as follows: Proposition 5.4.
Let Λ be a finitely aligned k -graph and G Λ be its boundary-path groupoidas defined in Example 5.2. Let R be a commutative ring with . Then there is an isomor-phism π T : KP R (Λ) → A R ( G Λ ) such that π T ( s λ ) = 1 Z Λ ( λ ∗ s s ( λ )) and π T ( s µ ∗ ) = 1 Z Λ ( s ( µ ) ∗ s µ ) for λ, u ∈ Λ . The only part of the proof of Proposition 5.4 that requires much additional work isshowing the surjectivity of π T . For this, we establish the following two lemmas. Theselemmas show that the characteristic function associated to a compact open set in G Λ canbe written as a sum of elements in the form 1 Z Λ ( λ ∗ s µ \ G ) . Lemma 5.5.
Let ( λ, µ ) , ( λ ′ , µ ′ ) ∈ Λ ∗ s Λ , G ⊆ s ( λ ) Λ , and G ′ ⊆ s ( λ ′ ) Λ . Define F :=Λ min ( λ, λ ′ ) ∩ Λ min ( µ, µ ′ ) . Then (*) Z Λ ( λ ∗ s µ \ G ) ∩ Z Λ ( λ ′ ∗ s µ ′ \ G ′ ) = G ( γ,γ ′ ) ∈ F Z Λ ( λγ ∗ s µ ′ γ ′ \ [Ext ( γ ; G ) ∪ Ext ( γ ′ ; G ′ )]) .Proof. We generalise the argument of [12, Example 3.2] for 1-graphs. First we show thatthe collection { Z Λ ( λγ ∗ s µ ′ γ ′ \ [Ext ( γ ; G ) ∪ Ext ( γ ′ ; G ′ )]) : ( γ, γ ′ ) ∈ F } is disjoint. It suffices to show that the collection { Z Λ ( λγ ∗ s µ ′ γ ′ ) : ( γ, γ ′ ) ∈ F } is disjoint. Suppose for contradiction that there exist ( γ, γ ′ ) , ( γ ′′ , γ ′′′ ) ∈ F such that( γ, γ ′ ) = ( γ ′′ , γ ′′′ ) and V := Z Λ ( λγ ∗ s µ ′ γ ′ ) ∩ Z Λ ( λγ ′′ ∗ s µ ′ γ ′′′ ) = ∅ . Note that if γ = γ ′′ ,then λ ′ γ ′ = λγ (since ( γ, γ ′ ) ∈ Λ min ( λ, λ ′ ) )= λγ ′′ (since γ = γ ′′ )= λ ′ γ ′′′ (since ( γ ′′ , γ ′′′ ) ∈ Λ min ( λ, λ ′ ) )and γ ′ = γ ′′′ by the factorisation property, which contradicts ( γ, γ ′ ) = ( γ ′′ , γ ′′′ ). The sameargument shows that γ ′ = γ ′′′ implies γ = γ ′′ . Hence γ = γ ′′ and γ ′ = γ ′′′ . Meanwhile,since ( γ, γ ′ ) , ( γ ′′ , γ ′′′ ) ∈ F , then d ( γ ) = d ( γ ′′ ) and d ( γ ′ ) = d ( γ ′′′ ). Take ( x, m, y ) ∈ V .Then x ∈ Z Λ ( λγ ) and x ∈ Z Λ ( λγ ′′ ). Since d ( γ ) = d ( γ ′′ ), then d ( λγ ) = d ( λγ ′′ ) and γ = x ( d ( λ ) , d ( λγ )) = x ( d ( λ ) , d ( λγ ′′ )) = γ ′′ , which contradicts γ = γ ′′ . Hence thecollection { Z Λ ( λγ ∗ s µ ′ γ ′ ) : ( γ, γ ′ ) ∈ F } is disjoint, and so is { Z Λ ( λγ ∗ s µ ′ γ ′ \ [Ext ( γ ; G ) ∪ Ext ( γ ′ ; G ′ )]) : ( γ, γ ′ ) ∈ F } .Next we show the right inclusion of (*). Write U := Z Λ ( λ ∗ s µ \ G ) ∩ Z Λ ( λ ′ ∗ s µ ′ \ G ′ )and take ( x, m, y ) ∈ U . We show ( x, m, y ) ∈ Z Λ ( λγ ∗ s µ ′ γ ′ \ [Ext ( γ ; G ) ∪ Ext ( γ ′ ; G ′ )])for some ( γ, γ ′ ) ∈ F . Because x ∈ Z Λ ( λ ) and x ∈ Z Λ ( λ ′ ), then d ( x ) ≥ d ( λ ) ∨ d ( λ ′ ) andthere exists ( γ, γ ′ ) ∈ Λ min ( λ, λ ′ ) such that(5.1) x ∈ Z Λ ( λγ ) . Using a similar argument, there exists ( γ ′′ , γ ′′′ ) ∈ Λ min ( µ, µ ′ ) such that(5.2) y ∈ Z Λ ( µγ ′′ ) . UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 19
We claim that γ = γ ′′ and γ ′ = γ ′′′ . To see this, note that m = d ( λ ) − d ( µ ) = d ( λ ′ ) − d ( µ ′ ) and d ( γ ) = d ( λ ) ∨ d ( λ ′ ) − d ( λ ) = ( d ( µ ) + m ) ∨ ( d ( µ ′ ) + m ) − ( d ( µ ) + m )= ( d ( µ ) ∨ d ( µ ′ )) + m − ( d ( µ ) + m ) = d ( µ ) ∨ d ( µ ′ ) − d ( µ ) = d ( γ ′′ ) .Since ( x, m, y ) ∈ Z Λ ( λ ∗ s µ \ G ), then σ d ( λ ) x = σ d ( µ ) y and γ = (cid:0) σ d ( λ ) x (cid:1) (0 , d ( γ )) = (cid:0) σ d ( µ ) y (cid:1) (0 , d ( γ ′ )) = γ ′′ .Using a similar argument, we also get γ ′ = γ ′′′ proving the claim.Next we show that ( x, m, y ) ∈ Z Λ ( λγ ∗ s µ ′ γ ′ ). By (5.1) and (5.2), we have x ∈ Z Λ ( λγ )and y ∈ Z Λ ( µγ ′′ ). Since γ = γ ′′ , γ ′ = γ ′′′ , ( γ ′′ , γ ′′′ ) ∈ Λ min ( µ, µ ′ ), then µγ ′′ = µγ = µ ′ γ ′ and y ∈ Z Λ ( µ ′ γ ′ ) . On the other hand, since ( x, m, y ) ∈ Z Λ ( λ ∗ s µ \ G ), then σ d ( λ ) x = σ d ( µ ) y and σ d ( λγ ) x = σ d ( µγ ) y = σ d ( µ ′ γ ′ ) y since µγ = µ ′ γ ′ . Since m = d ( λ ) − d ( µ ) = d ( λγ ) − d ( µ ′ γ ′ ), then ( x, m, y ) ∈ Z Λ ( λγ ∗ s µ ′ γ ′ ),as required.Finally we show that ( x, m, y ) / ∈ Z Λ ( λγν ∗ s µ ′ γ ′ ν ) for all ν ∈ Ext ( γ ; G ) ∪ Ext ( γ ′ ; G ′ ).Suppose for a contradiction that there exists ν ∈ Ext ( γ ; G ) ∪ Ext ( γ ′ ; G ′ ) such that( x, m, y ) ∈ Z Λ ( λγν ∗ s µ ′ γ ′ ν ). Without loss of generality, suppose ν ∈ Ext ( γ ; G ). Thenthere exists ν ′ ∈ G such that γν ∈ Z Λ ( ν ′ ). Since x ∈ Z Λ ( λγν ), y ∈ Z Λ ( µ ′ γ ′ ν ) = Z Λ ( µγν ), and γν ∈ Z Λ ( ν ′ ), then x ∈ Z Λ ( λν ′ ) and y ∈ Z Λ ( µν ′ ) where ν ′ ∈ G . Thiscontradicts ( x, m, y ) ∈ Z Λ ( λ ∗ s µ \ G ). Hence( x, m, y ) ∈ Z Λ ( λγ ∗ s µ ′ γ ′ \ [Ext ( γ ; G ) ∪ Ext ( γ ′ ; G ′ )])and U ⊆ G ( γ,γ ′ ) ∈ F Z Λ ( λγ ∗ s µ ′ γ ′ \ [Ext ( γ ; G ) ∪ Ext ( γ ′ ; G ′ )]) .Next we show the left inclusion of (*). Take ( γ, γ ′ ) ∈ F and(5.3) ( x, m, y ) ∈ Z Λ ( λγ ∗ s µ ′ γ ′ \ [Ext ( γ ; G ) ∪ Ext ( γ ′ ; G ′ )]) .We show ( x, m, y ) belongs to both Z Λ ( λ ∗ s µ \ G ) and Z Λ ( λ ′ ∗ s µ ′ \ G ′ ). Without loss ofgenerality, it suffices to show ( x, m, y ) ∈ Z Λ ( λ ∗ s µ \ G ). First we show that ( x, m, y ) ∈ Z Λ ( λ ∗ s µ ). Note that we have µγ = µ ′ γ ′ and m = d ( λγ ) − d ( µ ′ γ ′ ) = d ( λ ) − d ( µ ). Onthe other hand, ( x, m, y ) ∈ Z Λ ( λγ ∗ s µ ′ γ ′ ) also implies x ∈ Z Λ ( λγ ) and y ∈ Z Λ ( µ ′ γ ′ ) = Z Λ ( µγ ). Furthermore, σ ( λ ) x = [ x ( d ( λ ) , d ( λγ ))] (cid:2) σ ( λγ ) x (cid:3) = γ (cid:2) σ ( λγ ) x (cid:3) (since x ( d ( λ ) , d ( λγ )) = γ )= γ [ σ ( µ ′ γ ′ ) y ] (since σ ( λγ ) x = σ ( µ ′ γ ′ ) y )= [ y ( d ( µ ) , d ( µγ ))] [ σ ( µ ′ γ ′ ) y ] (since y ( d ( µ ) , d ( µγ )) = γ )= [ y ( d ( µ ) , d ( µγ ))] [ σ ( µγ ) y ] (since µγ = µ ′ γ ′ )= σ ( µ ) y and then ( x, m, y ) ∈ Z Λ ( λ ∗ s µ ), as required.To complete the proof, we have to show ( x, m, y ) / ∈ Z Λ ( λν ∗ s µν ) for all ν ∈ G . Supposefor contradiction that there exists ν ∈ G such that ( x, m, y ) ∈ Z Λ ( λν ∗ s µν ). In particular, x ∈ Z Λ ( λν ). Since x ∈ Z Λ ( λγ ) and x ∈ Z Λ ( λν ), then there exists ν ′ ∈ Ext ( γ ; { ν } ) suchthat x ∈ Z Λ ( λγν ′ ). Hence σ ( λγν ′ ) x = σ ( µγν ′ ) y (since σ ( λ ) x = σ ( µ ) y )= σ ( µ ′ γ ′ ν ′ ) y (since µγ = µ ′ γ ′ ), (cid:0) σ ( µ ) y (cid:1) (0 , d ( γν ′ )) = (cid:0) σ ( λ ) x (cid:1) (0 , d ( γν ′ )) (since σ ( λ ) x = σ ( µ ) y )(5.4) = x ( d ( λ ) , d ( λγν ′ ))= γν ′ (since x ∈ Z Λ ( λγν ′ ) ),and y (0 , d ( µ ′ γ ′ ν ′ )) = y (0 , d ( µγν ′ )) (since µγ = µ ′ γ ′ )= µγν ′ (by (5.4))= µ ′ γ ′ ν ′ (since µγ = µ ′ γ ′ ).Furthermore, d ( λγν ′ ) − d ( µ ′ γ ′ ν ′ ) = d ( λγ ) − d ( µ ′ γ ′ )= d ( λγ ) − d ( µγ ) (since µγ = µ ′ γ ′ )= d ( λ ) − d ( µ ) = m .Hence ( x, m, y ) ∈ Z Λ ( λγν ′ ∗ s µ ′ γ ′ ν ′ ) for some ν ′ ∈ Ext ( γ ; { ν } ) ⊆ Ext ( γ ; G ), whichcontradicts (5.3). The conclusion follows. (cid:3) Lemma 5.6.
Let { Z Λ ( λ i ∗ s µ i \ G i ) } ni =1 be a finite collection of compact open bisectionsets and U := n [ i =1 Z Λ ( λ i ∗ s µ i \ G i ) .Then U ∈ span R (cid:8) Z Λ ( λ ∗ s µ \ G ) : ( λ, µ ) ∈ Λ ∗ s Λ , G ⊆ s ( λ ) Λ (cid:9) .Proof. It is trivial for n = 1. Now let n = 2 and F := Λ min ( λ , λ ) ∩ Λ min ( µ , µ ) . If F = ∅ , then 1 U = 1 Z Λ ( λ ∗ s µ \ G ) + 1 Z Λ ( λ ′ ∗ s µ ′ \ G ′ ) .Otherwise, by Proposition 5.5, we have1 U = 1 Z Λ ( λ ∗ s µ \ G ) + 1 Z Λ ( λ ′ ∗ s µ ′ \ G ′ ) − X ( γ,γ ′ ) ∈ F Z γ,γ ′ where Z γ,γ ′ := Z Λ ( λγ ∗ s µ ′ γ ′ \ Ext ( γ ; G ) ∪ Ext ( γ ′ ; G ′ )), as required. For n ≥
3, by usingthe inclusion-exclusion principle and de Morgan’s law, 1 U can be written as a sum ofelements in the form 1 Z Λ ( λ ∗ s µ \ G ) . (cid:3) Proof of Proposition 5.4.
Define T λ := 1 Z Λ ( λ ∗ s s ( λ )) . Then by [13, Theorem 6.13] (or [32,Example 7.1]), { T λ , T µ ∗ : λ, u ∈ Λ } is a Kumjian-Pask Λ-family in A R ( G Λ ). Hence, thereexists a homomorphism π T : KP R (Λ) → A R ( G Λ ) such that π T ( s λ ) = T λ and π T ( s µ ∗ ) = T µ ∗ for λ, µ ∈ Λ by Theorem 3.7(a).
UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 21
To see that π T is injective, first we show that π T is graded. Take λ, µ ∈ Λ. Then s λ s µ ∗ ∈ KP R (Λ) d ( λ ) − d ( µ ) and π T ( s λ s µ ∗ ) = 1 Z Λ ( λ ∗ s µ ) = 1 { ( x,d ( λ ) − d ( µ ) ,y ):( λ,µ ) ∈ Λ ∗ s Λ ,z ∈ s ( λ ) ∂ Λ } ∈ A R ( G Λ ) d ( λ ) − d ( µ ) .Since for every n ∈ Z k , KP R (Λ) n is spanned by elements in the form s λ s µ ∗ (Theorem3.7.(c)), then for n ∈ Z k , π T (KP R (Λ) n ) ⊆ A R ( G Λ ) n and π T is graded. Since π T ( rs v ) = r Z Λ ( v ∗ s v ) = 0 for all r ∈ R \ { } and v ∈ Λ , and π T is graded, then by Theorem 4.1, π T is injective, as required.Finally we show the surjectivity of π T . Take f ∈ A R ( G Λ ). By [9, Lemma 2.2], f canbe written as P U ∈ F a U U where a U ∈ R , each U is in the form S ni =1 Z Λ ( λ i ∗ s µ i \ G i ) forsome n ∈ N , and F is finite set of mutually disjoint elements. Hence, to show f ∈ im ( π T ),it suffices to show 1 U ∈ im ( π T )where U := S ni =1 Z Λ ( λ i ∗ s µ i \ G i ) for some n ∈ N and collection { Z Λ ( λ i ∗ s µ i \ G i ) } ni =1 .By Lemma 5.6, 1 U can be written as the sum of elements in the form 1 Z Λ ( λ ∗ s µ \ G ) . On theother hand, for ( λ, µ ) ∈ Λ ∗ s Λ and finite G ⊆ s ( λ ) Λ, we have T λ (cid:0) Y ν ∈ G (cid:0) T s ( λ ) − T ν T ν ∗ (cid:1) (cid:1) T µ ∗ = 1 Z Λ ( λ ∗ s s ( λ )) (cid:0) Y ν ∈ G (cid:0) Z Λ ( s ( λ ) ∗ s s ( λ )) − Z Λ ( ν ∗ s ν ) (cid:1) (cid:1) Z Λ ( s ( µ ) ∗ s µ ) (5.5) = 1 Z Λ ( λ ∗ s s ( λ )) (cid:0) Y ν ∈ G (cid:0) Z Λ ( s ( λ ) ∗ s s ( λ ) \{ ν } ) (cid:1) (cid:1) Z Λ ( s ( µ ) ∗ s µ ) = 1 Z Λ ( λ ∗ s s ( λ )) (cid:0) Q ν ∈ G Z Λ ( s ( λ ) ∗ s s ( λ ) \{ ν } ) (cid:1) Z Λ ( s ( µ ) ∗ s µ ) = 1 Z Λ ( λ ∗ s s ( λ )) (cid:0) Z Λ ( s ( λ ) ∗ s s ( λ ) \ G ) (cid:1) Z Λ ( s ( µ ) ∗ s µ ) = 1 Z Λ ( λ ∗ s µ \ G ) since s ( λ ) = s ( µ ). Hence, 1 Z Λ ( λ ∗ s µ \ G ) belongs to im ( π T ) and then so does 1 U , as required.Therefore, π T is surjective and then is an isomorphism. (cid:3) Remark . Finitely aligned k -graphs include 1-graphs and row-finite k -graphs with nosources. Further, in these cases, the boundary path groupoid G Λ of Example 5.2 coincideswith G E of [12] and G Λ of [10]. Thus, we have generalised Example 3.2 of [12] andProposition 4.3 of [10]. For locally convex row-finite k -graphs, our construction gives aSteinberg algebra model of the Kumjian-Pask algebras of [11].6. Aperiodic higher-rank graphs and effective groupoids
In this section and Section 7, we investigate the relationship between a k -graph Λ andits boundary-path groupoid G Λ as constructed in Example 5.2. We expect the Cuntz-Krieger uniqueness theorem (Theorem 8.1) to apply only to aperiodic finitely aligned k -graphs (definition below). On the other hand, effective groupoids (definition below) areneeded in the hypothesis of the Cuntz-Krieger uniqueness theorem for Steinberg algebras(Theorem 8.2). In this section, our main result is Proposition 6.3 which says that a finitelyaligned k -graph Λ is aperiodic if and only if the boundary-path groupoid G Λ is effective.We say a boundary path x is aperiodic if for all λ, µ ∈ Λ r ( x ), λ = µ implies λx = µx .We say a finitely aligned k -graph Λ is aperiodic if for each v ∈ Λ , there exists an aperiodicboundary path x with r ( x ) = v . Remark . There are several equivalent ways to define the aperiodicity condition forfinitely aligned k -graphs (see [13, 18, 22, 26]). However, those definitions are equivalent by[18, Proposition 3.6] and [26, Proposition 2.11]. The definition we use is called Condition(B ′ ) in [13, Remark 7.3] and [26, Definition 2.1.(ii)]. Remark . For 1-graphs, the aperiodicity condition is known as Condition (L), which,using our conventions, says that every cycle has an entry (see [1, 3, 7, 16, 23, 29, 30]).Next let G be a toplogical groupoid. Define Iso ( G ) the isotropy groupoid of G byIso ( G ) := { a ∈ G : s ( a ) = r ( a ) } .We then say G is effective if the interior of Iso ( G ) is G (0) . See [8, Lemma 3.1] for someequivalent characterisations. Proposition 6.3.
Let Λ be a finitely aligned k -graph. Then Λ is aperiodic if and only ifthe boundary-path groupoid G Λ is effective.Proof. ( ⇒ ) First suppose that Λ is aperiodic. We trivially have G (0)Λ belongs to the interiorof Iso ( G Λ ). Now we show the reverse inclusion. Take a an interior point of Iso ( G Λ ). Thenthere exits Z Λ ( λ ∗ s µ \ G ) such that Z Λ ( λ ∗ s µ \ G ) ⊆ Iso ( G Λ ) and a ∈ Z Λ ( λ ∗ s µ \ G ).We show λ = µ .Note that since a ∈ Z Λ ( λ ∗ s µ \ G ), then Z Λ ( λ ∗ s µ \ G ) is not empty and by Remark5.3.(ii), G is not exhaustive. Hence, there exists ν ∈ s ( λ ) Λ such that Λ min ( ν, γ ) = ∅ for γ ∈ G . Because Λ is aperiodic, there exists a aperiodic boundary path x ∈ s ( ν ) ∂ Λ . We claim that the boundary path νx is also aperiodic. Suppose for contradiction thatthere exists λ ′ , µ ′ ∈ Λ r ( νx ) such that λ ′ = µ ′ and(6.1) λ ′ ( νx ) = µ ′ ( νx ) . Since λ ′ , µ ′ , ν ∈ Λ, by unique the factorisation property we have λ ′ = µ ′ implies λ ′ ν = µ ′ ν .Now because x is aperiodic, λ ′ ν = µ ′ ν implies λ ′ ν ( x ) = µ ′ ν ( x ), which contradicts (6.1).Hence, νx is aperiodic, as claimed.Since λνx ∈ Z Λ ( λ ) \ Z Λ ( λγ ) and µνx ∈ Z Λ ( µ ) \ Z Λ ( µγ ) for γ ∈ G , we have( λνx, d ( λ ) − d ( µ ) , µνx ) ∈ Z Λ ( λ ∗ s µ \ G ) .Thus Z Λ ( λ ∗ s µ \ G ) ⊆ Iso ( G Λ ), and hence λνx = µνx . Since νx is aperiodic, we have λ ( νx ) = µ ( νx ) which implies λ = µ . Therefore, G Λ is effective.( ⇐ ) Now suppose that Λ is not aperiodic. Then there exists v ∈ Λ such that for allboundary path x ∈ v∂ Λ, x is not aperiodic. Claim 6.4.
For x ∈ v∂ Λ , we have x G Λ x = { x } .Proof of Claim 6.4. Take x ∈ v∂ Λ. Since x is not aperiodic, then there exist λ, µ ∈ Λ r ( x )such that λ = µ and λx = µx . If d ( λ ) = d ( µ ), then λ = ( λx ) (0 , d ( λ )) = ( µx ) (0 , d ( µ )) = µ ,which contradicts with λ = µ .So suppose d ( λ ) = d ( µ ). Note that for 1 ≤ i ≤ k such that d ( λ ) i = d ( µ ) i , we have d ( x ) i = ∞ (since λx = µx ). Hence(( d ( λ ) ∨ d ( µ )) − d ( λ )) ∨ (( d ( λ ) ∨ d ( µ )) − d ( µ )) ≤ d ( x ) . UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 23
Write p := ( d ( λ ) ∨ d ( µ )) − d ( λ ) and q := ( d ( λ ) ∨ d ( µ )) − d ( µ ). Then σ p x = σ p (cid:0) σ d ( λ ) ( λx ) (cid:1) = σ d ( λ ) ∨ d ( µ ) ( λx )= σ d ( λ ) ∨ d ( µ ) ( µx ) (since λx = µx )= σ q (cid:0) σ d ( µ ) ( µx ) (cid:1) = σ q x and p = q (since d ( λ ) = d ( µ )). This implies ( x, p − q, x ) ∈ G Λ \ G (0)Λ and x G Λ x = { x } . (cid:3) Claim 6.4Since x G Λ x = { x } for all x ∈ v∂ Λ, then Z Λ ( v ) ∩ { z ∈ G (0)Λ : z G Λ z = { z }} = ∅ and { z ∈ G (0)Λ : z G Λ z = { z }} is not dense in G (0)Λ . Since G Λ is locally compact, second-countable, Hausdorff and ´etale, then by [24, Proposition 3.6.(b)], G Λ is not effective, asrequired. (cid:3) Remark . In fact, for a finitely aligned k -graph Λ, the following five conditions areequivalent:(a) G Λ is effective.(b) G Λ is topologically principal in that the set of units with trivial isotropy is densein G (0) .(c) G Λ satisfies Condition (1) of Theorem 5.1 of [25].(d) Λ has no local periodicity as defined in [26].(e) Λ is aperiodic.In [24, Proposition 3.6], Renault shows that for a locally compact, second-countable,Hausdorff, ´etale G , G is effective if and only if it is topologically principle. Since theboundary-path groupoid G Λ is locally compact, second-countable, Hausdorff and ´etale,then (a) ⇔ (b). Meanwhile, in [32, Theorem 5.2], Yeend proves (b) ⇔ (c). [Note that Yeenduses notion “ essentially free ” instead of “topologically principal”.] Lemma 5.6 of [25]gives (c) ⇔ (d). Finally, (d) ⇔ (e) follows from [26, Proposition 2.11].7. Cofinal higher-rank graphs and minimal groupoids
In this section, we show that a finitely aligned k -graph Λ is cofinal if and only if theboundary-path groupoid G Λ is minimal (Proposition 7.1). Later, we use this relationshipto study the simplicity of Kumjian-Pask algebras in Section 9.Recall from [27, Definition 8.4] that we say a k -graph Λ is cofinal if for all v ∈ Λ and x ∈ ∂ Λ, there exists n ≤ d ( x ) such that v Λ x ( n ) = ∅ .In a groupoid G , a subset U ⊆ G (0) is called invariant if s ( a ) ∈ U implies r ( a ) ∈ U for all a ∈ G . Note that U is invariant if and only if G (0) \ U is invariant. We thensay a topological groupoid G is minimal if G (0) has no nontrivial open invariant subsets.Equivalently, G is minimal if for each x ∈ G (0) , the orbit [ x ] := s ( x G ) is dense in G (0) . Proposition 7.1.
Let Λ be a finitely aligned k -graph. Then Λ is cofinal if and only if theboundary-path groupoid G Λ is minimal.Proof. ( ⇒ ) Suppose that Λ is cofinal. Take x ∈ G (0)Λ . We have to show that [ x ] is dense in G (0)Λ . Take a nonempty open set Z Λ ( λ \ G ) and we claim that Z Λ ( λ \ G ) ∩ [ x ] = ∅ . Since Z Λ ( λ \ G ) is nonempty, we have that G is not exhaustive (see Remark 5.3.(i)). Then there exists ν ∈ s ( λ ) Λ such that Λ min ( ν, γ ) = ∅ for γ ∈ G . Now consider the vertex s ( λν ) and the boundary path x . Since Λ is cofinal, then there exists n ≤ d ( x ) such that s ( λν ) Λ x ( n ) = ∅ . Take µ ∈ s ( λν ) Λ x ( n ). Because x is a boundary path, so is σ n x .Hence, y := λνµ [ σ n x ]is also a boundary path. It is clear that y ∈ Z Λ ( λ ) and since Λ min ( ν, γ ) = ∅ for γ ∈ G ,we have y / ∈ Z Λ ( λγ ) for γ ∈ G . Hence, y ∈ Z Λ ( λ \ G ).On the other hand, since y = λνµ [ σ n x ], then ( x, n − d ( λνµ ) , y ) ∈ G Λ and y ∈ [ x ].Therefore, Z Λ ( λ \ G ) ∩ [ x ] = ∅ . Thus, [ x ] is dense in G (0)Λ and G Λ is minimal.( ⇐ ) Suppose that Λ is not cofinal. Then there exist v ∈ Λ and x ∈ ∂ Λ such that forall n ≤ d ( x ) , we have v Λ x ( n ) = ∅ . We claim Z Λ ( v ) ∩ [ x ] = ∅ . Suppose for contradictionthat Z Λ ( v ) ∩ [ x ] = ∅ . Take y ∈ Z Λ ( v ) ∩ [ x ]. Because y ∈ [ x ], then there exists p, q ∈ N k such that ( x, p − q, y ) ∈ G Λ . This implies σ p x = σ q y . Since y ∈ Z Λ ( v ), then r ( y ) = v .Hence, σ p x = σ q y and r ( y ) = v imply that y (0 , q ) belongs to v Λ x ( p ), which contradictswith v Λ x ( n ) = ∅ for all n ≤ d ( x ). Therefore, Z Λ ( v ) ∩ [ x ] = ∅ , as claimed, and [ x ] is notdense in G (0)Λ . Thus, G Λ is not minimal. (cid:3) The Cuntz-Krieger uniqueness theorem
Throughout this section, Λ is a finitely aligned k -graph and R is a commutative ringwith identity 1. Theorem 8.1 (The Cuntz-Krieger uniqueness theorem) . Let Λ be an aperiodic finitelyaligned k -graph, R be a commutative ring with . Suppose that π : KP R (Λ) → A is an R -algebra homomorphism such that π ( rs v ) = 0 for all r ∈ R \ { } and v ∈ Λ . Then π isinjective. We show Theorem 8.1 by using the Cuntz-Krieger uniqueness theorem for Steinbergalgebras [9, Theorem 3.2]. First we verify an alternate formulation of the Cuntz-Kriegeruniqueness theorem for Steinberg algebras that will be useful.
Theorem 8.2.
Let G be an effective, Hausdorff, ample groupoid, and R be a commutativering with . Let B be a basis of compact open bisection for the topology on G . Let φ : A R ( G ) → A be an R -algebra homomorphism. Suppose that ker ( φ ) = 0 . Then thereexist r ∈ R \ { } and B ∈ B such that B ⊆ G (0) and φ ( r B ) = 0 .Proof. Since ker ( φ ) = 0, then by [9, Theorem 3.2], there exist r ∈ R \ { } and a nonemptycompact open subset K ⊆ G (0) such that φ ( r K ) = 0. Since K is open, then there is B ∈ B such that B ⊆ K . Hence, B ⊆ G (0) and0 = φ ( r K ) φ (1 B ) = φ ( r KB ) = φ ( r K ∩ B ) = φ ( r B ) . (cid:3) Proof of Theorem 8.1.
First note that G Λ is a Hausdorff and ample groupoid that is effec-tive by Proposition 6.3. Thus it satisfies the hypothesis of Theorem 8.2. Now recall theisomorphism π T : KP R (Λ) → A R ( G Λ ) as in Proposition 5.4. Then π T ( s λ ) = 1 Z Λ ( λ ∗ s s ( λ )) and π T ( s µ ∗ ) = 1 Z Λ ( s ( µ ) ∗ s µ ) for λ, u ∈ Λ. Define φ := π ◦ π − T . To show the injectivity of π , it suffices to show that φ is injective. Suppose for contradiction that φ is not injective.By Theorem 8.2, there exist r ∈ R \ { } and Z Λ ( λ \ G ) such that φ (cid:0) r Z Λ ( λ \ G ) (cid:1) = 0. Since UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 25 Z Λ ( λ \ G ) can be identified as 1 Z Λ ( λ ∗ s λ \ G ) (Remark 5.3.(i)), then by following the argumentof (5.5), we get φ (cid:0) r Z Λ ( λ \ G ) (cid:1) = π (cid:0) rs λ (cid:0) Y ν ∈ G (cid:0) s s ( λ ) − s ν s ν ∗ (cid:1) (cid:1) s λ ∗ (cid:1) and then(8.1) π (cid:0) rs λ (cid:0) Y ν ∈ G (cid:0) s s ( λ ) − s ν s ν ∗ (cid:1) (cid:1) s λ ∗ (cid:1) = 0.On the other hand, since π ( rs v ) = 0 for all r ∈ R \ { } and v ∈ Λ , and G is finitenon-exhaustive, then by Proposition 3.3.(d), π (cid:0) rs λ (cid:0) Y ν ∈ G (cid:0) s s ( λ ) − s ν s ν ∗ (cid:1) (cid:1) s λ ∗ (cid:1) = 0,which contradicts (8.1). The conclusion follows. (cid:3) One application of Theorem 8.1 is:
Corollary 8.3.
Let Λ be finitely aligned k -graph and R be a commutative ring with .Then Λ is aperiodic if and only if the boundary-path representation π S : KP R (Λ) → End ( F R ( ∂ Λ)) is injective.
To show Corollary 8.3, we establish some results and notation.Following [26, Definition 2.3], for a finitely aligned k -graph Λ, we say Λ has no localperiodicity if for every v ∈ Λ and every n = m ∈ N k , there exists x ∈ v∂ Λ such thateither d ( x ) (cid:3) n ∨ m or σ n x = σ m x . If no local aperiodicity fails at v ∈ Λ , then thereare n = m ∈ N k such that σ n x = σ m x for all x ∈ v∂ Λ. In this case, we say Λ has localperiodicity n, m at v ∈ Λ . Lemma 8.4 ([26, Lemma 2.9]) . Let Λ be a finitely aligned k -graph which has local pe-riodicity n, m at v ∈ Λ . Then d ( x ) ≥ n ∨ m and σ n x = σ m x for every x ∈ v∂ Λ . Fix x ∈ v∂ Λ and set µ = x (0 , m ) , α = x ( m, m ∨ n ) , and ν = ( µα ) (0 , n ) . Then µαy = ναy for every y ∈ s ( α ) ∂ Λ .Proof of Corollary 8.3. ( ⇒ ) Suppose that Λ is aperiodic. By Proposition 3.6, we have π S ( rs v ) = 0 for all r ∈ R \ { } and v ∈ Λ . Since Λ is aperiodic, then by Theorem 8.1, π S is injective.( ⇐ ) Suppose that Λ is not aperiodic. We are following the argument of [5, Lemma 5.9].Since Λ is not aperiodic, by [26, Proposition 2.11], there exist v ∈ Λ and n = m ∈ N k such that Λ has local periodicity n, m at v ∈ Λ . Let µ, ν, α be as in Lemma 8.4 anddefine a := s µα s ( µα ) ∗ − s να s ( µα ) ∗ . We claim that a ∈ ker ( π S ) \ { } .First we show that a = 0. Suppose for contradiction that a = 0. Then s µα s ( µα ) ∗ = s να s ( µα ) ∗ . Note that d (cid:0) s µα s ( µα ) ∗ (cid:1) = d ( µα ) − d ( µα ) = 0 and d (cid:0) s να s ( µα ) ∗ (cid:1) = d ( να ) − d ( µα ) = d ( ν ) + d ( α ) − d ( µ ) − d ( α ) = n − m = 0.Hence s µα s ( µα ) ∗ = s να s ( µα ) ∗ = 0. Thus, 0 = s ( µα ) ∗ (cid:0) s µα s ( µα ) ∗ (cid:1) s µα = s s ( µα ) = s s ( µα ) , whichcontradicts Theorem 3.7.(b). Hence a = 0. Now we show that a ∈ ker ( π S ). Take y ∈ ∂ Λ, and it suffices to show π S ( a ) ( y ) = 0.Recall that π S ( s λ ) = S λ and π S ( s µ ∗ ) = S µ ∗ where S λ ( y ) = ( λy if s ( λ ) = r ( y ) ;0 otherwise, and S µ ∗ ( y ) = ( σ d ( µ ) y if y (0 , d ( µ )) = µ ;0 otherwise.First suppose that y (0 , d ( µα )) = µα . Then S ( µα ) ∗ ( y ) = 0 and π S ( a ) ( y ) = S µα S ( µα ) ∗ ( y ) − S να S ( µα ) ∗ ( y ) = 0. Next suppose that y (0 , d ( µα )) = µα . Then π S ( a ) ( y ) = ( S µα − S να ) (cid:0) σ d ( µα ) y (cid:1) .Since y ∈ ∂ Λ, then σ d ( µα ) y ∈ s ( α ) ∂ Λ and by Lemma 8.4, µα (cid:0) σ d ( µα ) y (cid:1) = να (cid:0) σ d ( µα ) y (cid:1) and hence π S ( a ) ( y ) = 0. Thus, a ∈ ker ( π S ) \ { } , as claimed, and π S is not injective. (cid:3) Basic simplicity and simplicity
As in [30], we say an ideal I in KP R (Λ) is basic if whenever r ∈ R \ { } and v ∈ Λ ,we have rs v ∈ I implies s v ∈ I . We also say that KP R (Λ) is basically simple if its onlybasic ideals are { } and KP R (Λ).In this section, we investigate necessary and sufficient conditions for KP R (Λ) to bebasically simple (Theorem 9.3) and to be simple (Theorem 9.4). We show that bothresults can be viewed as a consequences of basic simplicity and simplicity characterisationsof Steinberg algebras. Therefore, we state necessary and sufficient conditions for theSteinberg algebra A R ( G ) to be basically simple and to be simple in the following twotheorems. Theorem 9.1 ([9, Theorem 4.1]) . Let G be an Hausdorff, ample groupoid and R be acommutative ring with . Then A R ( G ) is basically simple if and only if G is effective andminimal. Theorem 9.2 ([9, Corollary 4.6]) . Let G be an Hausdorff, ample groupoid and R be acommutative ring with . Then A R ( G ) is simple if and only if R is a field and G iseffective and minimal. Now we are ready to prove our results in this section.
Theorem 9.3.
Let Λ be a finitely aligned k -graph and let R be a commutative ring with . Then KP R (Λ) is basically simple if and only if Λ is aperiodic and cofinal.Proof. ( ⇒ ) First suppose that KP R (Λ) is basically simple. By Proposition 5.4, A R ( G Λ )is also basically simple and then by Theorem 9.1, G Λ is effective and minimal. On theother hand, G Λ is effective implies that Λ is aperiodic (Proposition 6.3), and G Λ is minimalimplies that Λ is cofinal (Proposition 7.1). The conclusion follows.( ⇐ ) Next suppose that Λ is aperiodic and cofinal. By Proposition 6.3 and Proposition7.1, G Λ is effective and minimal and then by Theorem 9.1, A R ( G Λ ) is basically simple.Since A R ( G Λ ) is isomorphic to KP R (Λ) (Proposition 5.4), then KP R (Λ) is also basicallysimple, as required. (cid:3) Theorem 9.4.
Let Λ be a finitely aligned k -graph and let R be a commutative ring with . Then KP R (Λ) is simple if and only if R is a field and Λ is aperiodic and cofinal. UMJIAN-PASK ALGEBRAS OF FINITELY-ALIGNED HIGHER-RANK GRAPHS 27
Proof. ( ⇒ ) First suppose that KP R (Λ) is simple. Then KP R (Λ) is also basically simpleand Theorem 9.3 implies that Λ is aperiodic and cofinal. On the other hand, since KP R (Λ)is simple, then by Proposition 5.4, A R ( G Λ ) is also simple and by Theorem 9.2, R is a field,as required.( ⇐ ) Next suppose that R is a field and Λ is aperiodic and cofinal. By Proposition6.3 and Proposition 7.1, G Λ is effective and minimal. Hence, by Theorem 9.2, A R ( G Λ ) issimple and by Proposition 5.4, so is KP R (Λ). (cid:3) References [1] G. Abrams,
Leavitt path algebras: the first decade , Bull. Math. Sci. (2015), 59–120.[2] G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph , J. Algebra (2005), 319–334.[3] G. Abrams and G. Aranda Pino,
The Leavitt path algebras of arbitrary graphs,
Houston J. Math. (2008), 423–442.[4] P. Ara, M.A. Moreno and E. Pardo, Nonstable K -theory for graph algebras , Algebr. Represent.Theory (2007), 157–178.[5] G. Aranda Pino, J. Clark, A. an Huef and I. Raeburn, Kumjian-Pask algebras of higher-rank graphs ,Trans. Amer. Math. Soc. (2013), 3613–3641.[6] G. Aranda Pino, M. Siles Molina and F. Perera Dom`enech, editors, Graph Algebras: bridging thegap between analysis and algebra. Notes from the “Workshop on Graph Algebras”, Universidad deM´alaga, 2006.[7] T. Bates, D. Pask, I. Raeburn and W. Szyma´nski,
The C ∗ -algebra of row-finite graphs , New York J.Math. (2000), 307–324.[8] J. Brown, L.O. Clark, C. Farthing and A. Sims, Simplicity of algebras associated to ´etale groupoids ,Semigroup Forum (2014), 433–452.[9] L.O. Clark and C. Edie-Michell, Uniqueness theorems for Steinberg algebras , Algebr. Represent.Theory (2015), 907–916.[10] L.O. Clark, C. Farthing, A. Sims and M. Tomforde, A groupoid generalisation of Leavitt path algebras ,Semigroup Forum (2014), 501–517.[11] L.O. Clark, C. Flynn and A. an Huef, Kumjian-Pask algebras of locally convex higher-rank graphs ,J. Algebra (2014), 445–474.[12] L.O. Clark and A. Sims,
Equivalent groupoids have Morita equivalent Steinberg algebras , J. PureAppl. Algebra (2015), 2062–2075.[13] C. Farthing, P.S. Muhly and T. Yeend,
Higher-rank graph C ∗ -algebras: an inverse semigroup andgroupoid approach , Semigroup Forum (2005), 159–187.[14] R. Hazlewood, I. Raeburn, A. Sims and S.B.G. Webster, Remarks on some fundamental results abouthigher-rank graphs and their C ∗ -algebras , Proc. Edinb. Math. Soc. (2013), 575–597.[15] A. Kumjian and D. Pask, Higher rank graph C ∗ -algebras , New York J. Math. (2000), 1–20.[16] A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs , Pasific J. Math. (1998), 161–174.[17] A. Kumjian, D. Pask, I. Raeburn and J. Renault,
Graphs, groupoids, and Cuntz-Krieger algebras , J.Funct. Anal. (1997), 505–541.[18] P. Lewin and A. Sims,
Aperiodicity and cofinality for finitely aligned higher-rank graphs , Math. Proc.Cambridge Philos. Soc. (2010), 333–350.[19] Y.E.P. Pangalela,
Realising the Toeplitz algebra of a higher-rank graph as a Cuntz-Krieger algebra ,arXiv:1507.07610 [math.OA], 2015.[20] I. Raeburn and A. Sims,
Product systems of graphs and the Toeplitz algebras of higher-rank graphs ,J. Operator Theory (2005), 399–429.[21] I. Raeburn, A. Sims and T. Yeend, Higher-rank graphs and their C ∗ -algebras , Proc. Edinb. Math.Soc. (2003), 99–115.[22] I. Raeburn, A. Sims and T. Yeend, The C ∗ -algebras of finitely aligned higher-rank graphs , J. Funct.Anal. (2004), 206–240.[23] I. Raeburn, Graph Algebras, CBMS Regional Conference Series in Math., vol. 103, American Math-ematical Society, 2005. [24] J. Renault, Cartan subalgebras in C ∗ -algebras , Ir. Math. Soc. Bull. (2008), 29–63.[25] J. Renault, A. Sims, D.P. Williams and T. Yeend, Uniqueness theorems for topological higher-rankgraph C ∗ -algebras , arXiv:0906.0829v3 [math.OA], 2012.[26] J. Shotwell, Simplicity of finitely aligned k -graph C ∗ -algebras , J. Operator Theory (2012), 335–347.[27] A. Sims, Gauge-invariant ideals in the C ∗ -algebras of finitely aligned higher-rank graphs , Canad. J.Math. (2006), 1268–1290.[28] B. Steinberg, A Groupoid approach to discrete inverse semigroup algebras , Adv. Math. (2010),689–727.[29] M. Tomforde,
Uniqueness theorems and ideal structure for Leavitt path algebras , J. Algebra (2007), 270–299.[30] M. Tomforde,
Leavitt path algebras with coefficients in a commutative ring , J. Pure Appl. Algebra (2011), 471–484.[31] S.B.G. Webster,
The path space of a higher-rank graph , Studia Math., (2011), 155–185.[32] T. Yeend,
Groupoid models for the C ∗ -algebras of topological higher-rank graphs , J. Operator Theory (2007), 95–120. Lisa Orloff Clark and Yosafat E. P. Pangalela, Department of Mathematics andStatistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand
E-mail address ::