Irreducibility and monicity for representations of k-graph C^*-algebras
aa r X i v : . [ m a t h . OA ] F e b Irreducibility and monicity for representations of k -graph C ∗ -algebras Carla Farsi, Elizabeth Gillaspy, and Daniel Gon¸calvesFebruary 8, 2021
Abstract
The representations of a k -graph C ∗ -algebra C ∗ (Λ) which arise from Λ-semibranchingfunction systems are closely linked to the dynamics of the k -graph Λ. In this paper,we undertake a systematic analysis of the question of irreducibility for these represen-tations. We provide a variety of necessary and sufficient conditions for irreducibility,as well as a number of examples indicating the optimality of our results. We also ex-plore the relationship between irreducible Λ-semibranching representations and purelyatomic representations of C ∗ (Λ). Throughout the paper, we work in the setting of row-finite source-free k -graphs; this paper constitutes the first analysis of Λ-semibranchingrepresentations at this level of generality. Keywords and phrases: C ∗ -algebras, monic representations, higher-rank graphs, k -graphs,Λ-semibranching function systems. Contents
On atomic irreducible representations of C ∗ (Λ) Understanding the irreducible representations of a C ∗ -algebra enables an analysis of its spec-trum and primitive ideal space, as well as its representation theory. In addition, longstandingopen questions such as Naimark’s problem [41] use irreducible representations to (conjec-turally) determine how similar two C ∗ -algebras are. However, for non-type I C ∗ -algebras A , it is well known [25, 26] that the natural Borel structure on the space b A of irreduciblerepresentations of A is not countably separated. As many higher-rank graph C ∗ -algebras arenot type I, this lack of a “reasonable” parametrization of their irreducible representationsobligates us to restrict our attention to specific subfamilies of representations when exploringthe question of irreducibility. In this paper, we study the irreducibility of the representationsof higher-rank graph C ∗ -algebras C ∗ (Λ) which arise from Λ-semibranching function systems.For graph C ∗ -algebras and related constructions, the representations arising from branch-ing systems [39, 28, 29, 27, 22] provide a key insight into the structure of the C ∗ -algebra.Intuitively, a branching system is a family of subsets of a measure space ( X, µ ) which reflectsthe structure of the graph C ∗ -algebra C ∗ (Λ), so that one obtains a natural representation of C ∗ (Λ) on L ( X, µ ). Indeed, this representation encodes the natural dynamics of the prefixingand coding maps on the space Λ ∞ of infinite paths in the graph Λ: the fact that C (Λ ∞ ) isa subalgebra of C ∗ (Λ) implies that the structure of C (Λ ∞ ), and in particular the dynamicsof changing an infinite path x x x . . . to x x x x . . . or to x x . . . , must also be reflectedin the branching system on ( X, µ ). Consequently, the study of representations arising frombranching systems also enhances our understanding of the symbolic dynamics associated toa graph or higher-rank-graph.In some settings, in fact, every representation arises from a branching system: the thirdauthor together with D. Royer identify in [28, 29] a class of graphs for which every rep-resentation of the graph C ∗ -algebra is unitarily equivalent to a representation arising froma branching system. For higher-rank graphs Λ, branching systems were introduced as Λ-semibranching function systems by the first and second authors together with S. Kang andJ. Packer in [22], where the associated representations were used to construct wavelets andto analyze the KMS states of the higher-rank graph C ∗ -algebra C ∗ (Λ). Subsequent workby Farsi, Gillaspy, Kang, and Packer together with P. Jorgensen [18] showed that a largeclass of representations of C ∗ (Λ) – the so-called monic representations – all arise from Λ-semibranching function systems, and these authors provide in [19] a more detailed analysis ofthe structure of Λ-semibranching function systems in the case when the associated measurespace is atomic. The question of when a Λ-semibranching representation is faithful has beenexplored in [22, 27, 20]. However, even in the previously mentioned studies, there has notbeen a specific emphasis on irreducible representations, which instead constitute the focusof this research. Although the primitive ideal space of higher-rank graph C ∗ -algebras is wellunderstood [8, 36], the broad applicability of Λ-semibranching function systems has inspiredus to explore the question of when the associated representations of C ∗ (Λ) are irreducible.Higher-rank graphs (or k -graphs) are a k -dimensional generalization of directed graphs21-graphs). Introduced by Kumjian and Pask in [37], k -graphs provide a framework forcombinatorial constructions of C ∗ -algebras and shift spaces, beyond the setting of graphand Cuntz–Krieger C ∗ -algebras, and graph and Markov shift spaces. In fact, in the decadessince their introduction, k -graph C ∗ -algebras have led to advances in symbolic dynamics[47, 9] and noncommutative geometry [33], as well as insights into the dimension theory of C ∗ -algebras [45]. Higher-rank graph C ∗ -algebras share many of the important properties ofCuntz and Cuntz–Krieger C ∗ -algebras (of which they are generalizations [11, 12, 16, 38]),including Cuntz–Krieger uniqueness theorems and realizations as groupoid C ∗ -algebras.As we mentioned above the main objective of this paper is to provide a variety of nec-essary and sufficient conditions for a Λ-semibranching function system to give rise to anirreducible representation of C ∗ (Λ). In our work we will establish a strong link beteween therepresentation theory of k -graph C ∗ -algebras and the symbolic dynamics associated to a k -graph, by detecting irreducibility of a representation arising from Λ-semibranching functionssystems in terms of conditions on the coding maps. We focus on the setting of row-finitesource-free k -graphs, whose Λ-semibranching function systems have not yet been analyzedin the literature. There are nontrivial structural differences between the finite k -graph caseand the row-finite one: for example, the infinite path space in the row-finite case is typicallynon-compact. To obtain our main results, we consequently need to extend a number ofresults which had previously been established in the literature only for finite k -graphs.We now describe the content of this paper. In addition to reviewing the relevant back-ground material (including definitions of higher-rank graphs and their Λ-semibranching func-tion systems) in Section 2, we also establish two new results in that section. Namely, Proposi-tion 2.15 shows that, given an invariant subset E ⊆ X and a Λ-semibranching representationon ( X, µ ), we also obtain a Λ-semibranching representation on L ( E, µ ) under mild hypothe-ses. Theorem 2.19 describes a method for constructing measures on the infinite path spaceof Λ ∞ , ensuring that we have a supply of examples of Λ-semibranching function systems.Then, Section 3 establishes that for row-finite higher-rank graphs, the characterization ofthose Λ-semibranching representations which are unitarily equivalent to representations onthe infinite path space Λ ∞ is analogous to the known characterization for finite higher-rankgraphs. In more detail, Theorem 3.6 and Theorem 3.8 give respectively a representation-theoretic and a measure-theoretic characterization of when a Λ-semibranching representationis equivalent to one arising from Λ ∞ . Even when restricted to the setting of finite higher-rankgraphs, Theorem 3.8 is stronger than the existing results in the literature. We also explorethe relationship between monic representations and the periodicity of Λ in Section 3, andobtain in Corollary 3.12 a new necessary condition for the monicity of branching represen-tations for directed graphs containing cycles without entrance.Section 4 is the main contribution of this paper and where we establish connections be-tween irreducibility of a representation arising from Λ-semibranching function system and thedynamics associated to a k -graph. Here we provide a variety of necessary (Propositions 4.6and 4.13, and Theorem 4.8) and sufficient (Theorems 4.15 and 4.20, and Proposition 4.26)conditions for a Λ-semibranching function system to give rise to an irreducible representa-tion of C ∗ (Λ). Many of these necessary and sufficient conditions deal with the ergodicityof the dynamics of the Λ-semibranching function system. We also wish to highlight Theo-rem 4.8, which shows that only cofinal k -graphs admit irreducible Λ-semibranching repre-sentations. Although we provide examples which indicate that our necessary conditions are3ot in general sufficient, when the measure space X of the Λ-semibranching function systemis Λ ∞ , the situation is different: Proposition 4.6 and Theorem 4.15 combine to imply thata Λ-semibranching representation on (Λ ∞ , µ ) is irreducible precisely when the coding mapsare jointly ergodic with respect to µ . Motivated in part by Theorem 4.8, we also providesufficient conditions in Theorems 4.15 and 4.20 for irreducibility of a Λ-semibranching rep-resentation arising from a proper subset X of Λ ∞ . In particular, Example 4.22 shows that,using Theorem 4.20, one can use a Λ-semibranching function system to obtain an irreduciblerepresentation of C ∗ (Λ) even if Λ is not necessarily cofinal.We conclude this paper in Section 5 by studying the irreducibility of atomic Λ-semibranchingrepresentations. Indeed, Theorem 5.3 shows that any irreducible Λ-semibranching represen-tation on an atomic measure space is purely atomic in the sense of [19], and Theorem 5.5shows in particular that irreducible representations on atomic measure spaces are monic.Finally, in Section 5.1, we present an application of Λ-semibranching representations in thecontext of the Naimark problem for graph algebras. Acknowledgments
C.F. was partially supported by the Simons Foundation Collaboration Grant for Mathe-matics
In this section we recall the definition of higher-rank graphs and their C ∗ -algebras from [37],together with results on Λ-semibranching function systems and their associated representa-tions that extend those established for finite higher-rank graphs in [22] and [18]. Let N = { , , , . . . } denote the monoid of natural numbers under addition, and let k ∈ N with k ≥
1. We write e , . . . , e k for the standard basis vectors of N k , where e i is the vectorof N k with 1 in the i -th position and 0 everywhere else. Definition 2.1. [37, Definition 1.1] A higher-rank graph or k -graph is a countable smallcategory Λ with a degree functor d : Λ → N k satisfying the factorization property : for anymorphism λ ∈ Λ and any m, n ∈ N k such that d ( λ ) = m + n ∈ N k , there exist uniquemorphisms µ, ν ∈ Λ such that λ = µν and d ( µ ) = m , d ( ν ) = n .When discussing k -graphs, we use the arrows-only picture of category theory; thus, ob-jects in Λ are identified with identity morphisms, and the notation λ ∈ Λ means λ is amorphism in Λ. Recall that a small category is one in which the collection of arrows is a set. xample . ,1. The higher-rank graphs with k = 1 (the 1-graphs) correspond to the categories whoseobjects are the vertices of a directed graph E , and whose morphisms are the finitepaths in E . In this case, d ( λ ) ∈ N is the number of edges in λ .2. The k -graph Ω k has Obj(Ω k ) = N k and Mor(Ω k ) = { ( m, n ) ∈ N k × N k : m ≤ n } . Wehave d ( m, n ) = n − m .We often regard k -graphs as a k -dimensional generalization of directed graphs, so we callmorphisms λ ∈ Λ paths in Λ, and the objects (identity morphisms) are often called vertices .For n ∈ N k , we write Λ n := { λ ∈ Λ : d ( λ ) = n } (1)With this notation, note that Λ is the set of objects (vertices) of Λ, and we will callelements of Λ e i (for any i ) edges . We write r, s : Λ → Λ for the range and source maps inΛ respectively. For vertices v, w ∈ Λ , we define v Λ w := { λ ∈ Λ : r ( λ ) = v, s ( λ ) = w } and v Λ n := { λ ∈ Λ : r ( λ ) = v, d ( λ ) = n } . Our focus in this paper is on row-finite k -graphs with no sources. We say that Λ has nosources or is source-free if v Λ n = ∅ for all v ∈ Λ and n ∈ N k . It is well known that thisis equivalent to the condition that v Λ e i = ∅ for all v ∈ Λ and all basis vectors e i of N k . A k -graph Λ is row-finite if (cid:16) v Λ n (cid:17) < ∞ , ∀ v ∈ Λ , ∀ n ∈ N k . (2)For m, n ∈ N k , we write m ∨ n for the coordinatewise maximum of m and n . Given λ, η ∈ Λ, we writeΛ min ( λ, η ) := { ( α, β ) ∈ Λ × Λ : λα = ηβ, d ( λα ) = d ( λ ) ∨ d ( η ) } . (3)If k = 1, then Λ min ( λ, η ) will have at most one element; this need not be true if k > min ( λ, ν ) isMCE( λ, ν ) = { ξ ∈ Λ : ∃ ( α, β ) ∈ Λ min ( λ, ν ) such that ξ = λα = νβ } . Observe that if r ( λ ) = r ( ν ) then MCE( λ, ν ) = ∅ = Λ min ( λ, ν ); if r ( λ ) = r ( ν ) then r ( ξ ) = r ( λ )for all ξ ∈ MCE( λ, ν ). Definition 2.3 ([37] Definitions 2.1) . Let Λ be a k -graph. An infinite path in Λ is a k -graphmorphism (degree-preserving functor) x : Ω k → Λ, and we write Λ ∞ for the set of infinitepaths in Λ. Since Ω k has a terminal object (namely 0 ∈ N k ) but no initial object, we thinkof our infinite paths as having a range r ( x ) := x (0) but no source. For each m ∈ N k , wehave a shift map σ m : Λ ∞ → Λ ∞ given by σ m ( x )( p, q ) = x ( p + m, q + m ) (4)for x ∈ Λ ∞ and ( p, q ) ∈ Ω k . 5t is well-known that the collection of cylinder sets Z ( λ ) = { x ∈ Λ ∞ : x (0 , d ( λ )) = λ } , for λ ∈ Λ, form a compact open basis for a locally compact Hausdorff topology on Λ ∞ if Λ isrow-finite: see Section 2 of [37]. The cylinder sets also generate the standard Borel structure B o (Λ ∞ ) on Λ ∞ . In particular, if we enumerate the vertices, say { v i } i ∈ N of Λ , recall that the σ -algebra A of the disjoint union Λ ∞ = F n ∈ N v n Λ ∞ is defined to be A = n A ⊆ Λ ∞ such that A ∩ v j Λ ∞ Borel, ∀ j ∈ N o . We also have a partially defined “prefixing map” σ λ : Z ( s ( λ )) → Z ( λ ) for each λ ∈ Λ: σ λ ( x ) = λx = ( p, q ) λ ( p, q ) , q ≤ d ( λ ) x ( p − d ( λ ) , q − d ( λ )) , p ≥ d ( λ ) λ ( p, d ( λ )) x (0 , q − d ( λ )) , p < d ( λ ) < q Definition 2.4.
The orbit of an infinite path x isOrbit( x ) = { y ∈ Λ ∞ : ∃ m, n ∈ N k s.t. σ n ( x ) = σ m ( y ) } = { λσ n ( x ) : n ∈ N k , λ ∈ Λ } . (5)We say that a k -graph Λ is cofinal if, and only if, for all x ∈ Λ ∞ and v ∈ Λ , there exists y ∈ Orbit( x ) with r ( y ) = v .Now we introduce the C ∗ -algebra associated to a row-finite, source-free k -graph Λ. Definition 2.5. ([37, Definition 1.5]) Let Λ be a row-finite k -graph with no sources. A Cuntz–Krieger Λ -family is a collection { t λ : λ ∈ Λ } of partial isometries in a C ∗ -algebrasatisfying(CK1) { t v : v ∈ Λ } is a family of mutually orthogonal projections,(CK2) t λ t η = t λη if s ( λ ) = r ( η ),(CK3) t ∗ λ t λ = t s ( λ ) for all λ ∈ Λ,(CK4) for all v ∈ Λ and n ∈ N k , we have t v = P λ ∈ v Λ n t λ t ∗ λ . The Cuntz–Krieger C ∗ -algebra C ∗ (Λ) associated to Λ is the universal C ∗ -algebra generatedby a Cuntz–Krieger Λ-family, in the sense that for any Cuntz–Krieger Λ-family { t λ : λ ∈ Λ } ,there is an onto ∗ -homomorphism C ∗ (Λ) → C ∗ ( { t λ : λ ∈ Λ } ). We will usually write s λ forthe generator of C ∗ (Λ) corresponding to λ ∈ Λ.Since the sum of two projections is a projection iff the summands are mutually orthogonal,(CK4) implies that t λ t ∗ λ ⊥ t η t ∗ η if λ = η . Also, conditions (CK2) - (CK4) implies that for all λ, η ∈ Λ, we have t ∗ λ t η = X ( α,β ) ∈ Λ min ( λ,η ) t α t ∗ β . (6)It follows that C ∗ (Λ) = span { s α s ∗ β : α, β ∈ Λ , s ( α ) = s ( β ) } . .2 Λ -semibranching function systems, Λ -projective systems, andrepresentations In [22], separable representations of C ∗ (Λ) (when Λ is finite) were constructed by using Λ-semibranching function systems on measure spaces. Intuitively, a Λ-semibranching functionsystem is a way of encoding the Cuntz-Krieger relations at the measure-space level: theprefixing map τ λ corresponds to the partial isometry s λ ∈ C ∗ (Λ). The construction of aΛ-semibranching function system from [22, Section 3.1] extends verbatim to the row-finitecase; we provide the details below. Definition 2.6. [39, Definition 2.1] Let (
X, µ ) be a measure space, and let I be a finite orcountable set of indices. Suppose that, for each i ∈ I , we have a measurable subset D i ⊆ X ,with 0 < µ ( D i ) < ∞ for all i , and a measurable map σ i : D i → X . The family { σ i } i ∈ I is a semibranching function system if the following hold.1. Writing R i = σ i ( D i ), we have µ ( X \ [ i R i ) = 0 , µ ( R i ∩ R j ) = 0 for i = j, and µ ( R i ) < ∞ for all i .2. The Radon–Nikodym derivative Φ i := d ( µ ◦ σ i ) dµ is strictly positive µ -a.e. on D i .A measurable map σ : X → X is called a coding map for the family { σ i } i ∈ I if σ ◦ σ i = id D i for all i .Since µ ( R i ) = µ ◦ σ i ( D i ) = R D i d ( µ ◦ σ i ) dµ dµ , and µ ( D i ) >
0, the hypothesis that the Radon–Nikodym derivative is strictly positive implies that 0 < µ ( R i ) always. Definition 2.7. [22, Definition 3.2] Let Λ be a row-finite source-free k -graph and let ( X, µ )be a measure space. A Λ -semibranching function system on (
X, µ ) is a collection { D λ } λ ∈ Λ of measurable subsets of X , together with a family of prefixing maps { τ λ : D λ → X } λ ∈ Λ ,and a family of coding maps { τ m : X → X } m ∈ N k , such that(a) For each m ∈ N k , the family { τ λ : d ( λ ) = m } is a semibranching function system, withcoding map τ m .(b) If v ∈ Λ , then τ v = id .(c) Let R λ = τ λ ( D λ ). For each λ ∈ Λ , ν ∈ s ( λ )Λ, we have R ν ⊆ D λ (up to a set of measure0), and τ λ τ ν = τ λν a.e.(Note that this implies that up to a set of measure 0, D λν = D ν whenever s ( λ ) = r ( ν )).(d) The coding maps satisfy τ m ◦ τ n = τ m + n for any m, n ∈ N k . (Note that this impliesthat the coding maps pairwise commute.)7s established in [22] in the case of finite k -graphs, any Λ-semibranching function systemgives rise to a representation of C ∗ (Λ) via ‘prefixing’ and ‘chopping off’ operators that satisfythe Cuntz–Krieger relations. For the convenience of the reader, we recall the formula forthese Λ-semibranching representations of C ∗ (Λ). The following theorem is an extension of[22, Theorem 3.5] to the row-finite case (cf. also [27, Theorem 3.5]); the proof for finite k -graphs given in [22] extends verbatim to the row-finite setting. Theorem 2.8. [22, Theorem 3.5], [27, Theorem 3.5] Let Λ be a row-finite k -graph with nosources and suppose that we have a Λ -semibranching function system on a measure space ( X, µ ) with prefixing maps { τ λ : λ ∈ Λ } and coding maps { τ m : m ∈ N k } . For each λ ∈ Λ ,define an operator S λ on L ( X, µ ) by S λ ξ ( x ) = χ R λ ( x )(Φ λ ( τ d ( λ ) ( x ))) − / ξ ( τ d ( λ ) ( x )) . Then the operators { S λ : λ ∈ Λ } form a Cuntz–Krieger Λ -family and hence generate arepresentation π of C ∗ (Λ) on L ( X, µ ) . We now recall the definition of a Λ-projective system from [18]. Roughly speaking, a Λ-projective system on (
X, µ ) consists of a Λ-semibranching function system plus some extrainformation (encoded in the functions f λ below). Definition 2.9.
Let Λ be a row-finite k -graph with no sources. A Λ -projective system on ameasure space ( X, µ ) is a Λ-semibranching function system on (
X, µ ), with prefixing maps { τ λ : D λ → R λ } λ ∈ Λ and coding maps { τ n : n ∈ N k } together with a family of functions { f λ } λ ∈ Λ ⊆ L ( X, µ ) satisfying the following conditions:(a) For any λ ∈ Λ, we have 0 = d ( µ ◦ ( τ λ ) − ) dµ = | f λ | ;(b) For any λ, ν ∈ Λ, we have f λ · ( f ν ◦ τ d ( λ ) ) = f λν . Recall from [18, Remark 3.3] that the functions f λ vanish outside R λ , because the sameis true for the Radon–Nikodym derivative d ( µ ◦ ( τ λ ) − ) dµ .Condition (b) of Definition 2.9 is necessary in order to associate a Cuntz–Krieger Λ-family to a Λ-projective system. To be precise, we have the following Proposition, whichwas established for finite k -graphs in [18, Proposition 3.4]. Proposition 2.10.
Let Λ be a row-finite, source-free k -graph. Suppose that a measure space ( X, µ ) admits a Λ -semibranching function system with prefixing maps { τ λ : λ ∈ Λ } andcoding maps { τ n : n ∈ N k } . Suppose that { f λ } λ ∈ Λ is a collection of functions satisfyingCondition (a) of Definition 2.9. Then the maps { τ λ } , { τ n } and { f λ } λ form a Λ -projectivesystem on ( X, µ ) if and only if the operators T λ ∈ B ( L ( X, µ )) given by T λ ( f ) = f λ · ( f ◦ τ d ( λ ) ) (7) form a Cuntz–Krieger Λ -family with each T λ nonzero (and hence give a representation of C ∗ (Λ) ).Proof. The proof given in [18, Proposition 3.4] for finite higher-rank graphs holds verbatimfor row-finite k -graphs. 8e call the representation given in Equation (7) a Λ -projective representation . Example . For any Λ-semibranching function system on (
X, µ ), there is a natural choiceof an associated Λ-projective system; namely, for λ ∈ Λ n we define f λ ( x ) := Φ λ ( τ n ( x )) − / χ R λ ( x ) . (8)Condition (a) is satisfied because of the hypothesis that the Radon–Nikodym derivatives bestrictly positive µ -a.e. on their domain of definition. Since the operators S λ ∈ B ( L ( X, µ ))of Theorem 2.8 are given by S λ ( f ) = f λ · ( f ◦ τ n ) , and Theorem 2.8 establishes that { S λ } λ ∈ Λ is a Cuntz–Krieger Λ-family, Proposition 2.10shows that Equation (8) indeed describes a Λ-projective system. Remark . If { T λ } λ ∈ Λ is a Λ-projective representation, then one computes that T ∗ λ f = χ D λ · ( f ◦ τ λ ) f λ ◦ τ λ . It now follows, using the fact that τ λ ◦ τ d ( λ ) | R λ = id , that T λ T ∗ λ = M χ Rλ . (9)Moreover, Example 2.11 tells us that Equation (9) also holds for any Λ-semibranching rep-resentation.The following lemma will be used in Proposition 2.15 below, as well as later in Lemma4.18. Lemma 2.13.
Let { τ n , τ λ } λ,n be a Λ -semibranching function system on ( X, µ ) . If µ ( B ) = 0 then µ ( τ n ( B )) = 0 for any n ∈ N k .Proof. Observe that τ n ( B ) = a.e. F d ( λ )= n ( τ λ ) − ( B ). Moreover, if µ ( B ) = 0, Z ( τ λ ) − ( B ) d ( µ ◦ τ λ ) dµ dµ = ( µ ◦ τ λ )( τ − λ ( B )) = µ ( B ) = 0 . However, the definition of a Λ-semibranching function system requires Φ λ = d ( µ ◦ τ λ ) dµ > D s ( λ ) . In other words, µ ( τ − λ ( B )) = 0 for all λ ∈ Λ n , so µ ( τ n ( B )) = 0.Proposition 2.15 below shows that restricting a Λ-projective system to a subspace ( A, µ )of (
X, µ ) will still give a λ -projective system, as long as the subspace is invariant. Definition 2.14.
Let (
X, µ ) be a measure space, and T : X → X a function. We say that B ⊂ X is invariant with respect to T if µ ( T − ( B )∆ B ) = 0.Given a measurable subset A ∈ Σ of a measure space ( X, Σ , µ ), we write µ A := µ ( · ∩ A )for the measure given by restriction to A . We take the σ -algebra of µ A -measurable sets tobe { B ∩ A : B ∈ Σ } . 9 roposition 2.15. Suppose there is a Λ -semibranching function system { τ λ , τ n } on ( X, µ ) .If A ⊆ X is invariant with respect to τ n for all n , and µ ( A ∩ D v ) is nonzero for all v , thenthe restriction of a Λ -projective system on ( X, µ ) to ( X, µ A ) is again a Λ -projective system.Proof. We first check that if { τ λ } λ ∈ Λ is a Λ-semibranching function system on ( X, µ ), then { τ λ } λ also gives a Λ-semibranching function system on ( X, µ A ). By hypothesis we have µ A ( D v ) > v ∈ Λ , and for any n ∈ N k , µ A X \ [ d ( λ )= n R λ = µ A ∩ X \ [ d ( λ )= n R λ ≤ µ X \ [ d ( λ )= n R λ = 0 . We now argue that, for any λ ∈ Λ, the set Y λ := { y ∈ D s ( λ ) : d ( µ A ◦ τ λ ) dµ A ( y ) = 0 } has µ A -measure zero. As A is invariant, i.e., µ (( τ d ( λ ) ) − ( A )∆ A ) = 0, the sets { x ∈ A : τ d ( λ ) ( x ) A } = a.e. { x ∈ A : for all z ∈ A, d ( η ) = d ( λ ) , x = τ η ( z ) } and { x A : τ d ( λ ) ( x ) ∈ A } = a.e. { x A : x = τ η ( z ) for some z ∈ A, d ( η ) = d ( λ ) } have measure zero.Thus, their intersection with any measurable subset τ λ ( D ) of R λ also has measure zero,so the fact that τ λ is injective a.e. implies that (writing x = τ λ ( z ) for z ∈ D )0 = µ ( { τ λ ( z ) ∈ A : z ∈ D \ A } ) = µ ( { τ λ ( z ) A : z ∈ A ∩ D } ) . In other words, if D ⊆ D s ( λ ) ∩ A , then µ ( τ λ ( D ∩ A ) ∩ A ) = µ ( τ λ ( D ∩ A )) = µ ( τ λ ( D )) = µ ( τ λ ( D ) ∩ A ) . We conclude that for any measurable set D ⊆ D s ( λ ) ∩ A ,0 < Z D d ( µ ◦ τ λ ) dµ dµ = µ ( τ λ ( D )) = µ ( τ λ ( D ) ∩ A ) = µ A ( τ λ ( D )) = Z D d ( µ A ◦ τ λ ) dµ A dµ A . As µ | A = µ A , the uniqueness of the Radon-Nikodym derivatives implies that d ( µ ◦ τ λ ) dµ = d ( µ A ◦ τ λ ) dµ A a.e. on D s ( λ ) ∩ A. (10)Recall from Condition (2) of Definition 2.6 that X λ := (cid:26) x : d ( µ ◦ τ λ ) dµ ( x ) = 0 (cid:27) has µ -measure 0. In other words, Y λ = X λ ∩ A up to sets of measure zero. Since µ ( X λ ) = 0it follows that µ A ( Y λ ) = 0 as claimed. because τ d ( λ ) ◦ τ λ = a.e. id | D s ( λ )
10e have thus established that the maps { τ λ } λ ∈ Λ on ( X, µ A ) satisfy Condition (a) ofDefinition 2.7, and Condition (b) holds by construction. The fact that µ A ( Y ) ≤ µ ( Y ) forall Y ⊆ X gives us Condition (c), and Condition (d) holds on ( X, µ A ) because we have notchanged the definition of any of the maps. It follows that { τ λ } λ ∈ Λ induces a Λ-semibranchingfunction system on ( X, µ A ).To see that a Λ-projective system on ( X, µ ) restricts to one on (
X, µ A ), suppose thatwe have functions { f λ } λ ∈ Λ satisfying Definition 2.9 with respect to µ . That is, each f λ issupported on R λ and | f λ | = d ( µ ◦ ( τ λ ) − ) dµ , µ -a.e. on R λ .Let D ⊆ R λ be µ A -measurable; then there exists a µ -measurable set B ⊆ R λ such that D = A ∩ B . As D ⊆ A , Z D d ( µ ◦ ( τ λ ) − ) dµ dµ = Z D d ( µ ◦ ( τ λ ) − ) dµ dµ A . Since D = A ∩ B ⊆ R λ and A is invariant, τ − λ ( D ) = { x : τ λ ( x ) ∈ D = B ∩ A } satisfies τ − λ ( D ) \ A = { x : τ λ ( x ) ∈ D, x A } ⊆ { x : τ λ ( x ) ∈ A, x A }⊆ τ d ( λ ) ( { y ∈ R λ ∩ A : τ d ( λ ) ( y ) A } ) ⊆ τ d ( λ ) ( A ∆( τ d ( λ ) ) − ( A ))has measure zero by Lemma 2.13. Furthermore, as D ⊆ B, ( τ λ ) − ( D ) \ ( τ λ ) − ( B ) = ∅ . Weconclude that τ − λ ( D ) ⊆ a.e. A ∩ ( τ λ ) − ( B ). Similarly,( A \ τ − λ ( A )) ∩ τ − λ ( B ) = { x ∈ A : τ λ ( x ) ∈ B \ A } = τ d ( λ ) ( { y : y A, τ d ( λ ) ( y ) ∈ A } ∩ B ) ⊆ τ d ( λ ) (( A ∆( τ d ( λ ) ) − ( A )) ∩ B )has measure 0, and so A ∩ τ − λ ( B ) \ (cid:0) τ − λ ( D ) = τ − λ ( A ) ∩ τ − λ ( B ) (cid:1) has measure zero. Consequently, ( τ λ ) − ( D ) = a.e. A ∩ ( τ λ ) − ( B ) . Consequently, Z D d ( µ ◦ ( τ λ ) − ) dµ dµ = µ ( A ∩ ( τ λ ) − ( B )) = µ A (( τ λ ) − ( B )) = Z D d ( µ A ◦ ( τ λ ) − ) dµ A dµ A . As D was an arbitrary µ A -measurable set and Radon-Nikodym derivatives are unique, itfollows that d ( µ ◦ ( τ λ ) − ) dµ (cid:12)(cid:12)(cid:12)(cid:12) A = d ( µ A ◦ ( τ λ ) − ) dµ A . Consequently, if the functions { f λ } λ ∈ Λ give a Λ-projective system on ( X, µ ), so that | f λ | = ( µ ◦ ( τ λ ) − ) dµ , then their restrictions f λ | A satisfy | f λ | A | = d ( µ A ◦ ( τ λ ) − ) dµ A . The fact that the restric-tions f λ | A satisfy Condition (b) of Definition 2.9 is immediate from the assumption thatCondition (b) holds for the functions f λ . 11 .3 Measures on the infinite path space: the Carath´eodory/Kolmogorovextension theorem In this section we will present some results that will guarantee the existence of (projection-valued) measures on the infinite path space Λ ∞ of a row-finite source-free k -graph. Indeed,it will turn out that by using the Carath´eodory/Kolmogorov extension theorems and theirprojection-valued analogues, it will be sufficient to define our measures on cylinder sets.Recall that a measure µ on a measure space ( X, B ) is σ -finite if there exists a sequenceof subsets S n ∈ B with X = S n S n and µ ( S n ) < ∞ , ∀ n. Also recall that a family S ofsubsets of a set X is called a semiring of sets if it contains the empty set, A ∩ B ∈ S forall A, B ∈ S and, for every pair of sets
A, B ∈ S with A ⊆ B , the set B \ A is the unionof finitely many disjoint sets in S . If X ∈ S , then S is called a semialgebra. Semiringsand semialgebras canonically generate associated rings and algebras of sets by taking finiteunions. In particular a semiring (resp. semialgebra) is a ring (resp. algebra) if and only if isclosed under finite unions. Theorem 2.16 (Carath´eodory/Kolmogorov) . [2, Theorem 1.3.10] If F is an algebra ofsubsets of X , and µ is a countably additive function on F such that X is σ -finite with respectto ( F , µ ) , then µ extends uniquely to a measure on the σ -algebra generated by F . We also note the following projection-valued measure extension of the above result:
Lemma 2.17. [3, Theorem 7] If F is a bounded projection-valued measure defined on a ringof sets R , there exists one and only one (necessarily bounded) projection-valued measure E on σ ( R ) , the σ -ring generated by R , such that E is an extension of F . If R is an algebra,then σ ( R ) equals the σ -algebra generated by R . We will apply the above results to construct (projection-valued) measures on Λ ∞ . Inparticular, in our applications we will often take X = v Λ ∞ for a fixed vertex v ∈ Λ ,and the algebra S to be the collection of sets formed by taking finite intersections andunions of the cylinder sets Z ( λ ) with r ( λ ) = v . To check that S is an algebra, notice that X = v Λ ∞ = Z ( v ) is in S and use the following lemma, whose proof is a straightforwardapplication of the definitions given above. Lemma 2.18.
Let Λ be a row-finite k-graph and v be a vertex in Λ . If α, β ∈ v Λ then:1. Z ( α ) ∩ Z ( β ) = F ( λ,ν ) ∈ Λ min ( α,β ) Z ( αλ ) is a finite disjoint union of cylinder sets;2. Z ( α ) \ Z ( β ) = Z ( α ) \ F ξ ∈ MCE ( α,β ) Z ( ξ ) = F { Z ( αλ ) : d ( αλ ) = d ( α ) ∨ d ( β ) but αλ M CE ( α, β ) } is also a finite disjoint union of cylinder sets;3. Z ( α ) ∪ Z ( β ) is therefore also a finite disjoint union of cylinder sets. Before discussing projection-valued measures on Λ ∞ , we pause to reassure the readerthat there do indeed exist real-valued measures on the infinite path space of row-finite k -graphs. One approach to constructing such measures is to find a vector ξ ∈ R Λ > which is an Ash [2] uses the word “field” instead of “algebra.” A i of Λ. Given such an eigenvector, write β i for theeigenvalue of A i with respect to ξ , and for n = ( n , . . . , n k ) ∈ Z k , write β n := β n · · · β n k k .Then, if we define µ ( Z ( λ )) := β − d ( λ ) ξ s ( λ ) , one can compute that µ is countably additive on the algebra S of finite unions of cylindersets, and hence, by Theorem 2.16, induces a measure on the σ -algebra generated by thecylinder sets. This is the content of the next theorem, which arose from discussions withSooran Kang. Theorem 2.19.
Suppose that Λ is a row-finite k -graph with no sources. If there exists avector ξ ∈ R Λ > which is an eigenvector for each adjacency matrix A i of Λ , then the formula µ ( Z ( λ )) := β − d ( λ ) ξ s ( λ ) (11) defines a measure on the Borel σ -algebra of Λ ∞ .Proof. We will first show that µ is well defined and finitely additive on cylinder sets; thatis, if Z ( λ ) = ⊔ pi =1 Z ( η i ) then P pi =1 β − d ( η i ) ξ s ( η i ) = β − d ( λ ) ξ s ( λ ) .Suppose Z ( λ ) = ⊔ pi =1 Z ( η i ). Since Z ( λ ) = ⊔ pi =1 Z ( η i ), MCE( λ, η i ) = ∅ for all 1 ≤ i ≤ p .(In fact, for any n ≥ d ( λ ) ∨ d ( η i ) and for any β ji ∈ s ( η i )Λ n − d ( η i ) , the fact that Z ( λ ) = ⊔ pi =1 Z ( η i ) implies that η i β ji is an extension of λ . Consequently, there must exist a corre-sponding α ji ∈ s ( λ )Λ n − d ( λ ) so that λα ji = η i β ji . In other words, for any n ≥ d ( λ ) ∨ d ( η i ),there is a bijection between Λ min ( λ, η i ) and s ( η i )Λ n − d ( η i ) .)Since Λ is row-finite, for each i , the set Λ min ( λ, η i ) is finite; write its elements as { ( α ji , β ji ) } j ∈ J ,where J is a finite index set. Let n = ∨ i,j d ( λα ji ) = ∨ i,j d ( η i β ji ) . Then we have Z ( λ ) = p G i =1 G ( α ji ,β ji ) ∈ Λ min ( λ,η i ) G ξ ij ∈ s ( α ji )Λ n − d ( λαji ) Z ( λα ji ξ ij ) . Observe that, for each i, j , d ( α ij ξ ij ) = n − d ( λα ij ) + d ( α ij ) = n − d ( λ ). Moreover, for any m ∈ N k , Z ( λ ) = ⊔ γ ∈ s ( λ )Λ m Z ( λγ ). Taking m = n − d ( λ ) tells us that G γ ∈ s ( λ )Λ m Z ( λγ ) = Z ( λ ) = p G i =1 G ( α ji ,β ji ) ∈ Λ min ( λ,η i ) G ξ ij ∈ s ( α ) i j )Λ n − d ( λαji ) Z ( λα ji ξ ij ) . (12)Since both sides of the above equality are disjoint unions of cylinder sets of the same degree,the list of cylinder sets on the left must be precisely equal to the list of cylinder sets on theright. That is, each cylinder set Z ( λγ ) must equal Z ( λα ji ξ ij ) for precisely one path α ji ξ ij .From the fact that ξ is an eigenvector for each adjacency matrix A i with eigenvalue β i ,we easily compute that for any m ∈ N k , µ ( Z ( λ )) = X γ ∈ s ( λ )Λ m µ ( Z ( λγ )) . (13)13t now follows from Equations (13) and (12) that µ ( Z ( λ )) = p X i =1 X ( α ji ,β ji ) ∈ Λ min ( λ,η i ) X ξ ij ∈ s ( α ji )Λ n − d ( λαji ) µ ( Z ( λα ji ξ ij )) . (14)Since λα ji = η i β ji , we get µ ( Z ( λ )) = p X i =1 X ( α ji ,β ji ) ∈ Λ min ( λ,η i ) X ξ ij ∈ s ( α ) i j )Λ n − d ( ηiβji ) µ ( Z ( η i β ji ξ ij )) . (15)On the other hand, for fixed 1 ≤ i ≤ p , Z ( η i ) = G ( α ji ,β ji ) ∈ Λ min ( λ,η i ) G ξ ij ∈ s ( α ji )Λ n − d ( ηiβji ) Z ( η i β ji ξ ij ) , (16)where n = ∨ i,j d ( λα ji ) = ∨ i,j d ( η i β ji ). Again, d ( β ji ξ ij ) = n − d ( η i β ji ) + d ( β ji ) = n − d ( η i ) is thesame for all ξ ij . In other words, we can apply Equation (13) to η i instead of λ , using thedecomposition of Z ( η i ) from Equation (16) and setting m = n − d ( η i ). It follows that µ ( Z ( η i )) = X ( α ji ,β ji ) ∈ Λ min ( λ,η i ) X ξ ij ∈ s ( α ji )Λ n − d ( ηiβji ) µ ( Z ( η i β ji ξ ij )) . (17)Now combining (14), (15) and (17), we obtain p X i =1 µ ( Z ( η i )) = p X i =1 X ( α ji ,β ji ) ∈ Λ min ( λ,η i ) X ξ ij ∈ s ( α ji )Λ n − d ( ηiβji ) µ ( Z ( η i β ji ξ ij ))= p X i =1 X ( α ji ,β ji ) ∈ Λ min ( λ,η i ) X ξ ij ∈ s ( α ji )Λ n − d ( λαji ) µ ( Z ( λα ji ξ ij ))= µ ( Z ( λ )) . In other words, µ is indeed well-defined and finitely additive on cylinder sets.We now show that µ is countably additive, and hence a measure on the algebra S offinite disjoint unions of cylinder sets. By construction, if S ∈ S and S = ⊔ i ∈ N Z ( λ i ), we aredefining µ ( S ) = P i µ ( Z ( λ i )). To see that µ is a measure, we merely need to check that µ is additive on countable disjoint unions. Thus, suppose F i ∈ N Z ( λ i ) = F j ∈ N Z ( η j ). For eachfixed i ∈ N we have Z ( λ i ) = G j ∈ N Z ( λ i ) ∩ Z ( η j ) , and Z ( λ i ) ∩ Z ( η j ) = G ζ ∈ MCE( λ i ,η j ) Z ( ζ ) . The fact that Λ is row-finite ensures that MCE( λ i , η j ) is finite for all i, j , and also that eachcylinder set Z ( λ ) is compact and open. It follows that, since F j ∈ N Z ( η j ) is a cover for Z ( λ i ),there are only finitely many indices j such that Z ( λ i ) ∩ Z ( η j ) = ∅ . µ , we have X i ∈ N µ ( Z ( λ i )) = X i ∈ N X j ∈ N µ ( Z ( λ i ) ∩ Z ( η j ))= X i,j ∈ N X ζ ∈ MCE( λ i ,η j ) µ ( Z ( ζ ))= X j ∈ N µ ( Z ( η j )) , as desired. Note that we are able to interchange the order of the summation over i and j since,for fixed i (or equivalently for fixed j ), only finitely many of the intersections Z ( λ i ) ∩ Z ( η j )are non-empty.Now, we use Carath´eodory’s Theorem (Theorem 2.16) to extend µ uniquely to give ameasure (also denoted µ ) on the Borel σ -algebra of Λ ∞ , as desired.The above analysis begs the question of when the adjacency matrices of Λ admit acommon positive eigenvector ξ . When Λ is finite and strongly connected, [35, Corollary4.2] guarantees that a unique such eigenvector (of ℓ norm 1) exists. For row-finite (notnecessarily finite) k -graphs, if k = 1, Thomsen identified in [49] when an infinite directedgraph Λ will admit such an eigenvector ξ . While we anticipate that much of Thomsen’sanalysis could be extended to the setting of higher-rank graphs, for the moment we simplypresent one example where this can be done. Example . Define matrices A := , A := Let I denote the 4 × S := A I . . .I A I . . . I A I . . . ... . . . . . . , T := A I . . .I A I . . . I A I . . . ... . . . . . . Let Λ be the infinite 2-graph with vertex matrices
S, T . Notice that Λ consists of countablymany copies { Γ n } n ∈ N of the Ledrappier 2-graph Γ (cf. Example 5.4 of [23]); consecutive copiesare linked by four “forward” edges and four “backward” edges of each color. Moreover, thevector ξ = (1 , , , , , . . . ) given by ξ ℓ + m = ℓ + 1 (if 0 ≤ m ≤
3) is an eigenvector forboth S and T , with eigenvalue 4.Thus, using the Carath´eodory/Kolmogorov Extension Theorem (Theorem 2.16 above)we obtain a measure µ on Λ ∞ which extends the measure defined on cylinder sets by µ ( Z ( λ )) = (4 , − d ( λ ) ξ s ( λ ) , ∀ λ ∈ Λ . Monicity
The main results of this section are Theorem 3.6 and Theorem 3.8, which describe whena Λ-projective representation is equivalent to one arising from the infinite path space of Λ.Theorem 3.6 gives a representation-theoretic description, while Theorem 3.8 explains theequivalence at the level of the measure space (
X, µ ) underlying the Λ-projective representa-tion. In both cases, versions of these results were known for finite k -graphs, but Theorem 3.8is stronger than the previously established results, even in the finite case. We conclude thissection by providing a necessary condition for a Λ-projective representation to be monic inProposition 3.9, and we discuss the relationship of this condition to the existence of cycleswithout entry and their higher-rank generalizations.We first introduce, given a representation of C ∗ (Λ) , an associated projection-valued mea-sure on Λ ∞ , which will prove to be an invaluable tool in this section.Enumerate the vertices of the k-graph, say { v i } i ∈ N . Recall that the σ -algebra A of thedisjoint union Λ ∞ = F n ∈ N v n Λ ∞ is defined to be A = n A ⊆ Λ ∞ such that A ∩ v j Λ ∞ Borel, ∀ j ∈ N o . Lemma 2.17 is central to the proof of the following Proposition.
Proposition 3.1.
Let Λ be a row-finite k -graph with no sources. Given a representation π : C ∗ (Λ) → H , { π ( s λ ) = t λ } λ ∈ Λ , of a k -graph C ∗ -algebra C ∗ (Λ) on a separable Hilbert space H , there is a unique regular projection-valued measure P on the Borel σ -algebra B o (Λ ∞ ) ofthe infinite path space Λ ∞ which is defined on cylinder sets by P ( Z ( λ )) = t λ t ∗ λ for all λ ∈ Λ . (18) Moreover, the restriction π of the representation { t λ } λ ∈ Λ to the subalgebra C (Λ ∞ ) is givenby π ( f ) = Z Λ ∞ f ( x ) dP ( x ) , ∀ f ∈ C (Λ ∞ ) . (19) Proof.
First we will deal with the compact spectrum case by restricting to v Λ ∞ , where v ∈ Λ .Denote by S the algebra of finite unions of cylinder sets { Z ( λ ) : λ ∈ v Λ } . Given A ∈ S ,write it as a finite union of disjoint cylinder sets (this can be done by Lemma 2.18), say A = F ni =1 Z ( λ i ), and define P ( A ) = P ni =1 t λ i t ∗ λ i . Without loss of generality, since Λ isrow-finite and source-free, we may assume that each λ i has the same degree. Thus, (CK4)guarantees that the projections making up P ( A ) are mutually orthogonal, so that P ( A ) is aprojection for any such A .Next we prove that P is well defined (and hence a projection-valued measure on S ).Suppose that A can be written in two ways as a finite disjoint union of cylinder sets, A =16 Z ( λ i ) = F Z ( η j ). Then for each fixed i ∈ N we have Z ( λ i ) = G j Z ( λ i ) ∩ Z ( η j ) , and that Z ( λ i ) ∩ Z ( η j ) = G ζ ∈ MCE( λ i ,η j ) Z ( ζ ) . The fact that Λ is row-finite ensures that MCE( λ i , η j ) is finite for all i, j .Set m i = W j d ( λ i ) ∨ d ( η j ) ∈ N k to be the coordinate-wise maximum degree of all elementsin { MCE( λ i , η j ) } j , and write n i,j = m i − ( d ( λ i ) ∨ d ( η j )) ∈ N k . Note that since F i Z ( λ i ) = F j Z ( η j ), we in particular have that Z ( λ i ) ⊆ F j Z ( η j ). Thus every α ∈ s ( λ i )Λ m i − d ( λ i ) mustsatisfy Z ( λ i α ) ⊆ F j Z ( η j ). By the definition of m i , we then have that λ i α = ζ β for aunique ζ ∈ MCE( λ i , η j ) , β ∈ s ( ζ )Λ n i,j . Moreover, if ζ ∈ MCE( λ i , η j ) , β ∈ s ( ζ )Λ n i,j then thefact that ζ extends λ i and our choice of the degrees of ζ , β imply that ζ β = λ i α for some α ∈ s ( λ i )Λ m i − d ( λ i ) .Since t λ is a partial isometry for all λ ∈ Λ, it follows by (CK3) that t λ = t λ t ∗ λ t λ = t λ π ( s s ( λ ) ) = t λ X β ∈ s ( λ )Λ n π ( s β s ∗ β ) = X β ∈ s ( λ )Λ n t λβ t ∗ β for any n ∈ N k . It now follows by using (CK3), (CK4), and (CK2) that P ( G i Z ( λ i )) = X i t λ i t ∗ λ i = X i t λ i X α ∈ s ( λ i )Λ mi − d ( λi ) t α t ∗ α t ∗ λ i = X i X α ∈ s ( λ i )Λ mi − d ( λi ) t λ i α t ∗ λ i α = X i X j X ζ ∈ MCE( λ i ,η j ) X β ∈ s ( ζ )Λ ni,j t ζβ t ∗ ζβ = X i X j X ζ ∈ MCE( λ i ,η j ) t ζ t ∗ ζ . Since all of the sums in question are finite, we can rearrange the order of summation; then,by a symmetric argument to the one given above (replacing m i with n j = W i d ( η j ) ∨ d ( λ i )and α ∈ s ( λ i )Λ m i − d ( λ i ) with γ ∈ s ( η j )Λ n j − d ( η j ) ) we see that P ( G i Z ( λ i )) = X j X i X ζ ∈ MCE( λ i ,η j ) t ζ t ∗ ζ = X j t η j t ∗ η j = P ( G j Z ( η j )) . By Lemma 2.17 P extends to a unique measure on the Borel σ -algebra of v Λ ∞ . Nowenumerate the vertices of Λ, say { v i } i ∈ N . Recall that the σ -algebra A of the disjoint unionΛ ∞ = F n ∈ N v n Λ ∞ is defined to be A = n A ⊆ Λ ∞ such that A ∩ v j Λ ∞ Borel, ∀ j ∈ N o . Define the wanted projection-valued measure P on Λ ∞ by P ( A ) = X j P ( A ∩ v j Λ ∞ ) . (20)17ote that P ( A ) = P j P ( A ∩ v j Λ ∞ ) converges in the weak or strong operator topologies.Moreover, if we take a compact set C ⊆ Λ ∞ Borel, then C is included in a finite union ofthe sets v j Λ ∞ , and so the projection-valued measure defined in Equation (20), when evaluatedat C , coincides with the projection-valued measure coming from taking disjoint unions offinitely many v j Λ ∞ in this case. The uniqueness of the measure follows from the uniquenessof the extension to each B o ( v n Λ ∞ ) , and from the fact that every operator on L (Λ ∞ ) must beof the form given in Equation (20), since the spaces L ( v n Λ ∞ ) are orthogonal. The regularityof the measure P follows from the fact that Λ ∞ is σ -compact and metrizable, so all Borelsets are Baire sets. Consequently, [3, Theorem 18] implies that P is regular.Next we show the integral description of the representation. Following [34, Section 37],define for f bounded Z Λ ∞ f ( x ) dP ( x ) := A f , where A f ∈ B ( H ) is the unique operator such that for all ξ, ζ ∈ H , h A f ξ, ζ i = R Λ ∞ f ( x ) dP ξ,ζ ( x ),for the complex measure P ξ,ζ defined by P ξ,ζ ( Z ( λ )) = h t λ t ∗ λ ξ, ζ i . (21)Note that our definition of A f ensures that for all γ ∈ Λ ,A χ Z ( γ ) = π ( χ Z ( γ ) ) . Suppose f n → f ∈ C (Λ ∞ ) and f n = P k n i =1 α ni χ Z ( λ i ) , where Z ( λ i ) ∩ Z ( λ j ) = ∅ for all i = j .Then for all ǫ >
0, there is N ∈ N such that if n ≥ N we have sup {| f ( x ) | : x S k n i =1 Z ( λ i ) } <ǫ } and sup {| f ( x ) − α ni | : x ∈ Z ( λ i ) } < ǫ , and so |h ( A f − A f n ) ξ, ζ i| = (cid:12)(cid:12)(cid:12)(cid:12)Z Λ ∞ ( f − f n ) dP ξ,ζ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ |h ξ, ζ i| . Since our construction of f n ensures that A f n = π ( f n ), the above inequality becomes |h ( A f − π ( f n )) ξ, ζ i| ≤ ǫ |h ξ, ζ i| for all ξ, ζ ∈ H and all n ≥ N . It follows that the sequence ( π ( f n )) n ∈ N converges to A f innorm: if n ≥ N , k A f − π ( f n ) k = k A f − A f n k = sup k ξ k =1 k ( A f − A f n ) ξ k = sup k ξ k =1 h ( A f − A f n ) ξ, ( A f − A f n ) ξ i≤ sup k ξ k =1 sup {|h ( A f − A f n ) ξ, ζ i| : k ζ k = k ( A f − A f n ) ξ k}≤ sup k ξ k =1 sup { ǫ |h ξ, ζ i| : k ζ k = k ( A f − A f n ) ξ k} ≤ sup k ξ k =1 ǫ k ξ k k ( A f − A f n ) ξ k = ǫ k A f − A f n k , and consequently k A f − A f n k ≤ ǫ for large enough n .18n the other hand, the continuity of π implies that π ( f n ) → π ( f ). It follows that A f = Z Λ ∞ f ( x ) dP ( x ) = π ( f ) , as claimed. Remark . In [18, Proposition 3.8] the existence of the projection valued measure P wasproved for finite k -graphs using the Kolmogorov extension theorem. Proposition 3.1 abovegives an alternative approach to the proof of the existence of this projection valued measure.We now record some properties of the projection-valued measure associated to a repre-sentation of C ∗ (Λ). The proofs are very similar to the proofs recorded in [18] in the case offinite k -graphs. Proposition 3.3. ([18, Proposition 3.9 and Definition 4.1]) Let Λ be a row-finite, source-free k -graph, and fix a representation { t λ : λ ∈ Λ } of C ∗ (Λ) .(a) For λ, η ∈ Λ with s ( λ ) = r ( η ) , we have t λ P ( Z ( η )) t ∗ λ = P ( σ λ ( Z ( η ))) = P ( Z ( λη )) ;(b) For any fixed n ∈ N k , we have X λ ∈ s ( η )Λ n t λ P ( σ − λ ( Z ( η ))) t ∗ λ = P ( Z ( η )); (c) For any λ, η ∈ Λ with r ( λ ) = r ( η ) , we have t λ P ( σ − λ ( Z ( η ))) = P ( Z ( η )) t λ ;(d) When λ ∈ Λ n , we have t λ P ( Z ( η )) = P (( σ n ) − ( Z ( η ))) t λ . We now observe that the definition of monic representation from [18], originally given in thecontext of finite k -graphs, also makes sense for row-finite graphs. Definition 3.4. (cf. [18, Definition 4.1]) Let Λ be a row-finite k -graph with no sources. Arepresentation { t λ : λ ∈ Λ } of Λ on a Hilbert space H is called monic if t λ = 0 for all λ ∈ Λ,and there exists a vector ξ ∈ H such thatspan { t λ t ∗ λ ξ : λ ∈ Λ } = H . We say that such a vector ξ is a monic vector for the representation.Notice that a monic vector for { t λ } λ is a cyclic vector for the restriction π of the repre-sentation generated by { t λ } λ ∈ Λ to C (Λ ∞ ). We therefore obtain a Borel measure µ π on Λ ∞ given by µ π ( Z ( λ )) = h ξ, P ( Z ( λ )) ξ i = h ξ, t λ t ∗ λ ξ i . (22)An important feature of a monic vector is that its support must have full measure (cf.[18, Example 4.7]). 19 roposition 3.5. Let ( X, µ ) be a σ -finite measure space. If a Λ -projective representation of C ∗ (Λ) on L ( X, µ ) is monic then the support of any of its monic vectors ξ can differ from X by at most a set of measure 0.Proof. Recall that if { t λ : λ ∈ Λ } is a Λ-projective representation, then t λ t ∗ λ = M χ Rλ .Moreover, the support of any function in span { χ R λ ξ : λ ∈ Λ } is contained in the support of ξ . So, if X = supp( ξ ) ⊔ S for a set S with positive measure, the function χ A , where A ⊆ S isany measurable set of finite positive measure in the complement of the support of ξ , is notin span { χ R λ ξ : λ ∈ Λ } . Consequently, { t λ } λ cannot be a monic Λ-projective representationif S = X \ supp( ξ ) has positive measure.Note that the hypothesis that ( X, µ ) be σ -finite is necessary to guarantee the existenceof a set A as in the above proof. If we assume µ ( R v ) < ∞ for all v then σ -finiteness isautomatic.The following theorem is the main result of this section, which generalizes the finite casegiven in [18, Theorem 4.2]. The condition µ ( Z ( v n )) < ∞ for all n in the second part of thetheorem below cannot be removed, see Remark 3.7. Theorem 3.6.
Let Λ be a row-finite k -graph with no sources. If π : C ∗ (Λ) → H , with π ( s λ ) = t λ , is a monic representation of C ∗ (Λ) on a Hilbert space H , then { t λ } λ ∈ Λ is unitar-ily equivalent to a representation { S λ } λ ∈ Λ associated to a Λ -projective system on (Λ ∞ , µ π ) associated to the standard coding and prefixing maps σ n , σ λ of Definition 2.3.Conversely, if we have a representation π of C ∗ (Λ) on L (Λ ∞ , µ ) with µ ( Z ( v )) < ∞ for all v ∈ Λ , which arises from a Λ -projective system associated to the standard codingand prefixing maps σ n , σ λ , then the representation is monic, and L (Λ ∞ , µ ) is isometric with L (Λ ∞ , µ π ) .Proof. The proof is very similar to the proof of [18, Theorem 4.2], and because of that wewill only give a quick sketch of it, which highlights the differences between the finite and row-finite cases. The construction of a Λ-projective system on Λ ∞ from a monic representation π proceeds exactly as in [18, Theorem 4.2]. First, one defines the measure µ π on cylindersets Z ( λ ) ⊆ Λ ∞ by µ π ( Z ( λ )) := h ξ, π ( χ Z ( λ ) ) ξ i , where ξ is the cyclic vector for π . We use Carathe´eodory’s theorem to extend µ π to ameasure on Λ ∞ . Then the proof of [18, Theorem 4.2] can be followed to construct a Λ-projective system { f λ } λ ∈ Λ associated to the usual coding and prefixing maps { σ n , σ λ } n,λ onΛ ∞ , such that the affiliated Λ-projective representation is immediately seen to be unitarilyequivalent to π .For the converse, suppose that we have a representation π of C ∗ (Λ) on L (Λ ∞ , µ ) whicharises from a Λ-projective system. Then, in particular, π ( s λ s ∗ λ ) = M χ Z ( λ ) . Enumerate the vertices in Λ as { v n } n ∈ N and define ξ = X v n ∈ Λ χ Z ( v n ) n p µ ( Z ( v n )) ∈ L (Λ ∞ , µ ) . µ ( Z ( v )) > v by Condition (b) of Definition 2.7.) Then ξ is cyclic for C (Λ ∞ ). To see this, it suffices to show that χ Z ( λ ) ∈ π ( C (Λ ∞ ))( ξ ) for all λ ∈ Λ. Given λ ∈ Λ with r ( λ ) = v n , we compute: π ( n p µ ( Z ( v n )) χ Z ( λ ) ) ξ = n p µ ( Z ( v n )) π ( s λ s ∗ λ ) ξ = χ Z ( λ ) , since Z ( λ ) ∩ Z ( v ) = ∅ only when v = r ( λ ). The fact that L (Λ ∞ , µ ) is isometric with L (Λ ∞ , µ π ) now follows from the first part of the proof. Remark . ,1. In the proof of the converse direction above, if µ (Λ ∞ ) < ∞ then we could alternativelytake ξ = χ Λ ∞ to be our cyclic vector.2. As shown in the proof above, the measure µ π on Λ ∞ associated to a monic represen-tation π : C ∗ (Λ) → H with monic vector ξ must satisfy µ π ( Z ( λ )) = h ξ, π ( s λ s ∗ λ ) ξ i H < ∞ , ∀ λ ∈ Λ . Additionally, H is isometric to L (Λ ∞ , µ π ) (see the proof of [18, Theorem 4.2]). Ifin particular H = L (Λ ∞ , µ ), then L (Λ ∞ , µ ) and L (Λ ∞ , µ π ) are isometric, and so µ ( Z ( λ )) = µ π ( Z ( λ )) < ∞ , ∀ λ ∈ Λ ⇐⇒ µ ( Z ( v )) = µ π ( Z ( v )) < ∞ . This shows that µ ( Z ( v )) = µ π ( Z ( v )) < ∞ is a necessary condition for the monicity of a representation π : C ∗ (Λ) → L (Λ ∞ , µ ).Next we extend to row-finite graphs [18, Theorem 4.5], which shows that a Λ-semibranchingsystem on ( X, µ ) induces a monic representation of C ∗ (Λ), with monic vector the charac-teristic function of the whole space, if and only if the associated range sets generate the σ -algebra F of X . We also improve the aforementioned theorem by dropping the conditionon the monic vector.The following Theorem is the only result in this paper which depends explicitly on the σ -algebra F , so for this result only, we denote the associated Hilbert space by L ( X, F , µ ). Theorem 3.8.
Let Λ be a row-finite, source-free k -graph and let { t λ } λ ∈ Λ be a Λ -semibranchingrepresentation of C ∗ (Λ) on L ( X, F , µ ) with µ ( X ) < ∞ , where F denotes the σ -algebra on X . Let R be the collection of sets which are modifications of range sets R λ by sets of measurezero; that is, each element Y ∈ R has the form Y = R λ ∪ S or Y = R λ \ S for some set S of measure zero. Let σ ( R ) be the σ -algebra generated by R . The representation { t λ } λ ∈ Λ is monic if and only if σ ( R ) = F . In particular, for a monic Λ -semibranchingrepresentation { t λ } λ ∈ Λ , the set S := n n X i =1 a i t λ i t ∗ λ i X k k χ D vk | n ∈ N , λ i ∈ Λ , a i ∈ C o = n n X i =1 a i χ R λi | n ∈ N , λ i ∈ Λ , a i ∈ C o is dense in L ( X, F , µ ) . roof. Observe first that, if λ ∈ v k Λ, then for any Λ-semibranching representation { t λ } λ ∈ Λ ,Remark 2.12 establishes that t λ t ∗ λ X k k χ D vk = M χ Rλ X k k χ D vk = 1 k χ R λ . In other words, the two descriptions of S given in the statement of the theorem are equivalent.For the forward implication, suppose that the Λ-semibranching representation { t λ } λ ∈ Λ ismonic and that ξ is a monic vector for the representation. By Proposition 3.5, the supportof ξ is equal to X a.e.. Moreover, Remark 2.12 establishes that M := span { t λ t ∗ λ ( ξ ) : λ ∈ Λ } = span { χ R λ ξ : λ ∈ Λ } , which is dense in L ( X, F , µ ). Furthermore, the support of any function in M belongs to σ ( R ) since the support of ξ is X a.e.. Therefore for any f ∈ L ( X, F , µ ) there is a sequence( f j ) j , with f j ∈ M , such that lim j →∞ Z X | f j − f | dµ = 0 . In particular, ( f j ) → f in measure. The rest of the proof follows now exactly as in the proofof Theorem 4.5 in [18].For the converse, suppose σ ( R ) = F ; we will show that χ X is a monic vector for therepresentation by showing that S is dense in L ( X, F , µ ). Fix f ∈ L ( X, σ ( R ) , µ ) and ǫ > φ = P ni =1 a i χ X i , with X i ∈ σ ( R ), such that R X | φ − f | dµ < ǫ , andset A := max | a i | . By [13, Lemma A.2.1(ii)], for each i , there is an element B i of the algebra e R of sets generated by R such that µ ( B i ∆ X i ) < ǫn ( n − A . That is, each B i is a finite unionof elements of R , so for each i there exists a finite collection { R λ ij } j of range sets such that µ (cid:0) B i ∆( ∪ j R λ ij ) (cid:1) = 0 . In other words, if we define ψ = P ni =1 a i χ B i , then ψ ∈ S . Moreover, Z X | ψ − φ | dµ = Z X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i a i ( χ B i \ X i − χ X i \ B i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ≤ Z X i | a i | χ X i ∆ B i ! dµ = X i Z X i ∆ B i | a i | dµ + 2 X i = j | a i | | a j | µ (( X i ∆ B i ) ∩ ( X j ∆ B j )) < ǫ. The Cauchy-Schwarz inequality now implies that Z X | ψ − f | dµ ≤ Z X ( | ψ − φ | + | φ − f | ) dµ ≤ Z X | ψ − φ | dµ + Z X | φ − f | dµ + 2 Z X | φ − ψ | | φ − f | dµ< ǫ + ǫ + 2 √ ǫ. It follows that S is dense in L ( X, F , µ ) as claimed.We conclude this section with a necessary condition for monicity of a representation of a k -graph C ∗ -algebra. 22 roposition 3.9. Let Λ be a row-finite source-free k -graph and fix a Λ -semibranching func-tion system on a σ -finite measure space ( X, µ ) . Suppose that a vertex v of Λ satisfies ∀ τ ∈ v Λ : R τ = R vτ = D v . (23) If there is a measurable subset X v of D v with < µ ( X v ) < µ ( D v ) , then no Λ -projectiverepresentation arising from this Λ -semibranching function system will be monic.Proof. Let { t λ } λ ∈ Λ be a Λ-projective representation of C ∗ (Λ) on L ( X, µ ), and write π forthe induced representation of C (Λ ∞ ). Recall from Remark 2.12 that t λ t ∗ λ ( f ) = π ( χ Z ( λ ) )( f ) = χ R λ f for any finite path λ ∈ Λ and any f ∈ L ( X, µ ).Suppose that v and X v ⊆ D v are as in the statement of the theorem. We will show thatno vector ξ ∈ L ( X, µ ) can be monic. Recall from Proposition 3.5 that if ξ is monic for π then X \ supp( ξ ) has measure zero. Choose, therefore, ξ ∈ L ( X, µ ) with a.e. full support.Let M = span { χ R λ ξ : λ ∈ Λ } = span { t λ t ∗ λ ( ξ ) } = π ( C (Λ ∞ ))( ξ ) . Equation (23) implies that, if r ( λ ) = v , then χ R λ ξ | D v = ξ | D v . So if f ∈ M then the supportof f | D v is either D v or has measure zero. Therefore χ X v / ∈ M and hence ξ is not monic. Remark . By Definition 2.7, if we have a Λ-semibranching function system on (
X, µ ),then for every fixed m ∈ N k , D v = [ λ ∈ v Λ m R λ , and each R λ has positive measure. It follows that Equation (23) is satisfied iffFor every m ∈ N k there is precisely one path of degree m having range v . (24) Definition 3.11.
In a directed graph (1-graph) E , a cycle is a finite path e e · · · e n with s ( e n ) = r ( e ). We say that a cycle has an entrance if there is an i ≤ n and an edge e with r ( e ) = s ( e i ) but e = e i +1 . Corollary 3.12.
Let E be a row-finite source-free directed graph which contains a cyclewithout entrance, and let ( X, µ ) be σ -finite. If a Λ -projective representation of C ∗ ( E ) on L ( X, µ ) is monic, then for every vertex v lying on a cycle without entrance, D v has nomeasurable subsets X v with < µ ( X v ) < µ ( D v ) .Proof. By definition, if v lies on a cycle without entrance, then v satisfies Equation (24).The result now follows from Proposition 3.9. Remark . In [18, Example 4.7], the authors present an example of a Λ-semibranchingrepresentation associated to a 1-graph, on L ([0 , χ [0 , is not monic. Indeed, in that graph, there is a vertex v = v whichsupports a loop with no entrance, such that D v = (1 / , emark . Although Evans and Sims generalized the concept of “cycle without entrance”to higher-rank graphs in [17] (see [17, Remark 3.6]), we do not have an analogue of Corollary3.12 for higher-rank cycles without entrance: the 2-graph of [17, Example 6.1] containsa generalized cycle without entrance but does not satisfy Equation (24). The cycline pairsintroduced in [6, Definition 4.3] can also be viewed as a generalization of the notion of “cycleswithout entrance” to higher-rank graphs – by [6, Remark 4.9], a generalized cycle withoutentrance is a cycline pair if every extension of that generalized cycle without entrance is alsoa generalized cycle without entrance. However, even this stronger condition is not enoughto guarantee Equation (24). Section 5.11 of [40] (cf. also [24, Section 5.1]) exhibits a 2-graphwith one vertex for which Equation (24) fails, but which has nontrivial cycline pairs such as( e , f ). In this section, we describe a variety of necessary and sufficient conditions for a Λ-projectiverepresentation to be irreducible. We begin by defining the notions of ergodicity that we willuse to analyze the irreducibility of Λ-projective representations, and describing them in bothmeasure- and operator-theoretic terms. Proposition 4.5 then identifies the commutant of aΛ-projective representation on (
X, µ ) in terms of invariant functions on X .Section 4.1 highlights three necessary conditions for a Λ-projective representation to beirreducible, but shows by example that these conditions are not sufficient. However, inSection 4.2 we identify a variety of conditions under which a Λ-projective representation onthe infinite path space Λ ∞ must be irreducible. The final subsection, Section 4.3, includes avariety of other results related to the irreducibility of Λ-semibranching representations. Definition 4.1.
Let (
X, µ ) be a measure space. A function T : X → X is ergodic withrespect to µ if whenever µ ( A ∆ T − ( A )) = 0 (i.e., A is invariant as in Definition 2.14) we haveeither µ ( A ) = 0 or µ ( X \ A ) = 0. A family of maps { T i } i ∈ I is jointly ergodic with respect to µ if whenever µ ( A ∆ T − i ( A )) = 0 for all i , we have either µ ( A ) = 0 or µ ( X \ A ) = 0. Remark . If µ ( X ) < ∞ , this definition agrees with the definition of ergodicity used inearlier papers such as [18]. Definition 4.3.
Let (
X, µ ) be a measure space, and T : X → X a function. A measurablefunction f is invariant with respect to T if f ◦ T = f a.e.. We say that f is jointly invariant with respect to a family of maps { T i } i ∈ I if f ◦ T i = f a.e.. for all i.The following lemma is undoubtedly well-known to experts; we include it here for com-pleteness and ease of reference. Lemma 4.4.
Let ( X, µ ) be a measure space, and fix T : X → X and T i : X → X , i ∈ I .Then,(a) T is ergodic if, and only if, every invariant measurable function f : X → C is constanta.e.. b) The family { T i } is jointly ergodic if, and only if, every jointly invariant measurablefunction f : X → C is constant a.e..Proof. We prove (a) and leave the proof of (b) to the reader, as it is essentially the same.Suppose first that every invariant measurable function on X is constant. Choose A ⊆ X such that µ ( A ∆ T − ( A )) = 0. Then χ A ◦ T = χ A a.e. and so χ A is constant a.e.. Thereforeeither χ A = 0 a.e. (and so µ ( A ) = 0) or χ A = 1 a.e., in which case µ ( A c ) = 0.Suppose now that T is ergodic and let f be a measurable invariant function. For each r ∈ R , set I r = { x ∈ X : Im f ( x ) > r } , R r = { x ∈ X : Re f ( x ) > r } , E r ∈ { I r , R r } . Notice that each E r , for r ∈ R , is measurable and invariant with respect to T , and that if s > r then I s ⊆ I r and R s ⊆ R r . Since T is ergodic, we obtain that, for all r , µ ( E r ) = 0or µ ( X \ E r ) = 0. Before we proceed, we quickly observe that a real-valued function f isconstant if, and only if, there exists C such that for all r ≥ C , µ ( R r ) = 0, and for all r < C , µ ( X \ R r ) = 0.Now, let K I = sup { r : µ ( I r ) = 0 } and K R = sup { r : µ ( R r ) = 0 } . First, we will showthat K I < ∞ ; a similar argument will also establish that K R < ∞ . If x ∈ T n ∈ N I n then f ( x ) = + ∞ . Consequently, µ ( T n ∈ N I n ) = 0 (in fact T n ∈ N I n = ∅ ). As f is invariant and T is ergodic, for every r , either µ ( I r ) = 0 or µ ( X \ I r ) = 0. Notice that if µ ( I r ) = 0 for some r then µ ( I s ) = 0 for all s > r and in this case K I is finite. Suppose that µ ( X \ I r ) = 0for all r . Then µ (cid:0) X \ T n ∈ N I n (cid:1) = µ (cid:0)S n ∈ N X \ I n (cid:1) = 0, which implies that µ ( X ) = 0, acontradiction.Moreover, µ ( I K I ) = 0, since I K I = S n ∈ N I K I + n and µ ( I K I + n ) = 0 for all n . We nowuse the characterization of a constant function given above. For r ≥ K I we have that µ ( I r ) ≤ µ ( I K I ) = 0 and, for r < K I , there exists r such that r < r < K I and µ ( X \ I r ) = 0(from the definition of sup and the fact that the invariant set I r satisfies µ ( I r ) = 0). Since I r ⊆ I r , this implies that µ ( X \ I r ) = 0. In other words, whenever r < K I , there is a set offull measure on which Im f ( x ) > r . It follows that Im( f ) = K I a.e.. A similar argument willshow that Re( f ) = K R a.e., completing the proof that f = K R + iK I is constant a.e..One of the equivalent ways to describe irreducibility of a representation is via its com-mutant. Therefore, we give a description of the commutant of a representation associatedto a Λ-projective system below. Proposition 4.5.
Let Λ be a row-finite k -graph with no sources. Suppose that we have a Λ -projective system on a locally compact, σ -compact, Hausdorff space ( X, µ ) , where µ is aRadon measure, and let { T λ : λ ∈ Λ } be the associated representation of C ∗ (Λ) on L ( X, µ ) .Then, the commutant of the operators { T λ : λ ∈ Λ } consists of multiplication operators, andcontains the set of multiplication operators by functions h with h ◦ τ n = h µ -a.e., for all n ∈ N k .Proof. Let T ∈ B ( L ( X, µ )) be an operator in the commutant of the Λ-projective represen-tation. Then T commutes with the representation π of the abelian subalgebra C (Λ ∞ ) of C ∗ (Λ), where π ( χ Z ( λ ) ) = T λ T ∗ λ = M χ Rλ .
25y [1, Theorem 6.3.4 and Proposition 6.3.6], the fact that µ is a Radon measure on the locallycompact, σ -compact, Hausdorff space X , enables us to invoke the proof of [42, Proposition4.7.6] to conclude that T = M h , for some h ∈ L ∞ ( X ). It is easy to check that any multipli-cation operator T = M h , where h ◦ τ n = h a.e. for all n ∈ N k , commutes with each operator T λ . In this subsection we describe three necessary conditions for the irreducibility of a Λ-projective representation of C ∗ (Λ) arising from Λ-projective systems. Proposition 4.6 tells usthat all irreducible Λ-projective representations arise from Λ-semibranching function systemswith jointly ergodic coding maps. From this, we deduce Theorem 4.8, which states that onlycofinal k -graphs admit irreducible Λ-semibranching representations. Finally, Proposition4.13 describes how the functions f λ associated to an irreducible Λ-projective representationmust interact. We also present a variety of examples, indicating the insufficiency of thesenecessary conditions to guarantee irreducibility. Proposition 4.6.
Let Λ be a row-finite k -graph with no sources. Suppose that we have a Λ -projective system on ( X, µ ) , and let { T λ : λ ∈ Λ } be the associated representation of C ∗ (Λ) on L ( X, µ ) . If the representation { T λ : λ ∈ Λ } is irreducible, then the coding maps τ n arejointly ergodic with respect to the measure µ .Proof. Suppose the representation generated by { T λ } λ is irreducible, and suppose A ⊆ X satisfies µ ( A ∆( τ n ) − ( A )) = 0 for all n . Then both ( τ n ) − ( A ) \ A and A \ ( τ n ) − ( A ) havemeasure zero, for any n ∈ N k , and consequently M χ A ◦ τ n = M χ ( τn ) − A ) = M χ ( τn ) − A ) ∩ A = M χ A . It now follows from the definition of the operators T λ that, for all λ ∈ Λ, M χ A commuteswith T λ and T ∗ λ . To see this, recall that T λ M χ A ( f ) = f λ · ( χ A ◦ τ d ( λ ) ) · ( f ◦ τ d ( λ ) )while M χ A T λ ( f ) = χ A · f λ · ( f ◦ τ d ( λ ) ), so since f λ is supported on R λ it suffices to show that χ A ∩ R λ = χ R λ ( χ A ◦ τ d ( λ ) ). Observe that χ R λ ( χ A ◦ τ d ( λ ) ) = χ τ λ ( A ) , and consequently it suffices to show that µ (( A ∩ R λ )∆ τ λ ( A )) = 0.By definition, ( A ∩ R λ )∆ τ λ ( A ) = { τ λ ( z ) ∈ A : z A } ∪ { τ λ ( z ) A : z ∈ A } . Ourhypothesis that A is invariant implies that0 = µ ( A ∆( τ d ( λ ) ) − ( A )) = µ (cid:0) { y ∈ A : y = τ η ( z ) for any z ∈ A, d ( η ) = d ( λ ) } ∪ { y A : τ d ( λ ) ( y ) ∈ A } (cid:1) Restricting the sets described in the previous equation to R λ will not increase their measure,so we also have0 = µ (cid:0) { y ∈ A ∩ R λ : y = τ λ ( z ) for any z ∈ A } ∪ { y ∈ R λ \ A : τ d ( λ ) ( y ) ∈ A } (cid:1) = µ ( { τ λ ( z ) ∈ A : z A } ∪ { τ λ ( z ) A : z ∈ A } )26y using the fact that τ d ( λ ) ◦ τ λ = id | D s ( λ ) a.e. and hence τ λ is injective a.e.. We concludethat T λ commutes with M χ A as claimed.To see that T ∗ λ also commutes with M χ A , we use the fact that T λ M χ A = M χ A T λ . Thus,for any f, g ∈ L ( X, µ ), h T λ M χ A f, g i = h M χ A T λ f, g i = h f, T ∗ λ M χ A g i as M χ A is self-adjoint. On the other hand, we also have h T λ M χ A f, g i = h f, M χ A T ∗ λ g i , so forall f, g ∈ L ( X, µ ) , h f, ( T ∗ λ M χ A − M χ A T ∗ λ ) g i = 0 . It follows that T ∗ λ M χ A − M χ A T ∗ λ = 0, so M χ A commutes with T ∗ λ as well, for all λ ∈ Λ.As the representation is irreducible, this implies that M χ A ∈ C C M χ X . Since M χ A isa projection, it follows that either µ ( A ) = 0 (if M χ A = 0) or µ ( A c ) = 0 (if M χ A = 1 = M χ X ).Thus, the operators τ n are jointly ergodic with respect to the measure µ .Next, we give an example that the converse of the above proposition does not hold ingeneral. Example . Let E be the graph with one vertex v and one edge e . We will build asemibranching function system for E on the set { , } with counting measure. Let R e = { , } = D v and define the prefixing map f e : D v → R e by f e ( x ) = x + 1 mod 2. Then { D v , R e , f e } is a branching system in the sense of [27, Definition 3.6]. By [27, Proposition 3.12and Remark 3.11], this branching system gives rise to a semibranching function system. Let π be the induced representation of C ∗ (Λ) on ℓ ( { , } ). By Proposition 3.9 this representationis not monic. Hence, Theorem 5.5 implies that π is not irreducible. Indeed ℓ { } and ℓ { } are invariant subspaces of ℓ { , } . On the other hand, there are no nonempty proper subsetsof { , } which are invariant by all of the powers of the coding map. Theorem 4.8.
Let Λ be a row-finite k -graph with no sources. Suppose Λ is not cofinal(Definition 2.4). Then there are no irreducible representations of C ∗ (Λ) arising from Λ -semibranching function systems.Proof. Let x ∈ Λ ∞ and v ∈ Λ be obtained from the fact that Λ is not cofinal. Given aΛ-semibranching function system { τ n , τ λ } on ( X, µ ), let A denote the smallest set whichcontains R r ( x ) and is invariant with respect to all the coding maps τ n . Observe that µ ( A ∩ R v ) = 0 so, since µ ( R v ) > µ ( A ) ≥ µ ( R r ( x ) ) >
0, it follows from Proposition 4.6 thatany representation associated to a Λ-semibranching function system will be reducible.
Remark . However, Λ-semibranching function systems on non-cofinal k -graphs can giverise to irreducible representations if we restrict our attention to a minimal µ -invariant subset;see Definition 4.17 and Theorem 4.20 below.We now conclude this subsection by applying the work of Carlsen et al. [8] on irreduciblerepresentations of k -graphs to the setting of Λ-projective representations. We remark thatwhen the following definitions occur in Carlsen et al Section 4, they assume Λ is a maximaltail; we have not invoked this hypothesis as it merely implies that the relation λ ∼ ν ⇔ ( λ, ν ) periodic is an equivalence relation, which is irrelevant for our purposes.27 efinition 4.10. Let Λ be a k -graph. A pair ( λ, ν ) ∈ Λ × Λ is a periodic pair if s ( λ ) = s ( ν )and λx = νx for all x ∈ Z ( s ( λ )). We setPer(Λ) := { d ( λ ) − d ( ν ) : ( λ, ν ) periodic } . One can check that Per(Λ) is a subgroup of Z k .Following [8], we set H Per (Λ) to be the set of vertices which realize all of Per(Λ): that is, H Per (Λ) = { v ∈ Λ : if r ( λ ) = v and ∃ m ∈ N k : d ( λ ) − m ∈ Per(Λ) , then ∃ µ ∈ v Λ m s.t. ( λ, µ ) periodic } . Observe that if ( λ, ν ) is a periodic pair then σ d ( λ ) = σ d ( ν ) on Z ( λ ) = Z ( ν ). However, weneed not have τ d ( λ ) = τ d ( µ ) on an arbitrary Λ-semibranching function system. Example . Consider the following 1-graph Λ: v v e g f Define a Λ-semibranching function system on [0 ,
1] (equipped with Lebesgue measure) bysetting D v = (0 , /
2) and D v = (1 / , τ e ( x ) = τ g ( x ) = x , τ f ( x ) = 32 − x. The coding map is then given by τ ( x ) = ( x, x ∈ (0 , / − x, x ∈ (1 / , . One easily checks that the hypotheses of a Λ-semibranching function system are satisfied.Moreover, ( f, v ) is a periodic pair, but τ = id on R f ∩ R v = (1 / , Proposition 4.12.
In any Λ -semibranching function system, if ( λ, ν ) is a periodic pair,then µ ( R λ ∆ R ν ) = 0 . Proof.
Suppose x ∈ R λ . By definition, x = τ λ ( y ) for some y ∈ D s ( λ ) = D s ( ν ) . Let m =( d ( λ ) ∨ d ( ν )) − d ( λ ); by the definition of a Λ-semibranching function system, we can write D s ( λ ) = G η ∈ s ( λ )Λ m R η modulo sets of measure zero. Therefore, without loss of generality, we may assume y = τ η ( z )for some η ∈ s ( λ )Λ m and some z ∈ D s ( η ) .The fact that ( λ, ν ) is a periodic pair implies, in particular, that since d ( λη ) = d ( λ ) ∨ d ( ν ) ≥ d ( ν ), we have λη = ν ˜ η for some ˜ η ∈ s ( ν )Λ. Consequently, since x = τ λ ( y ) = τ λη ( z ),we must have x = τ ν ˜ η (˜ z ) for some ˜ z ∈ R ˜ η . It follows that x = τ ν ( τ ˜ η (˜ z )) ∈ R ν , and so almostall x ∈ R λ are also in R ν , as claimed. 28 roposition 4.13. Let Λ be a row-finite, source-free k -graph and let { T λ } λ ∈ Λ be a Λ -projective representation of C ∗ (Λ) on L ( X, µ ) , with associated functions f λ defined on R λ .If { T λ } λ ∈ Λ is irreducible and for every periodic pair ( λ, ν ) we have τ d ( λ ) = τ d ( ν ) on R λ ∩ R ν ,then there exists z ∈ T k such that for every periodic pair ( λ, ν ) with r ( λ ) = r ( ν ) ∈ H P er (Λ) ,we have f λ f ν = z d ( λ ) − d ( ν ) on R λ ∩ R ν . Proof.
Suppose { T λ } λ is irreducible. As T v = M χ Rv is nonzero for each v ∈ Λ , , the maximaltail T invoked in [8, Theorem 5.3(2)] is given in our case by T = Λ . Therefore, [8, Theorem5.3(2)] implies that there exists z ∈ T k (which implements a character of the periodicitygroup Per(Λ)) such that T λ = z d ( λ ) − d ( ν ) T ν for all ( λ, ν ) as in the statement of the proposition.Moreover, if ( λ, ν ) is periodic, then by Proposition 4.12, R λ = R ν a.e. and therefore f λ , f ν are defined and nonzero on R λ ∩ R ν . From our hypothesis that τ d ( λ ) = τ d ( ν ) on this domain,it follows that T λ = z d ( λ ) − d ( ν ) T ν ⇔ f λ f ν = z d ( λ ) − d ( ν ) on R Λ ∩ R ν . However, the converse of this proposition fails. In the discussion of the following example,we will interpret the 0th power of a loop λ to mean the vertex r ( λ ). Example . Consider the 1-graph Λ from Example 4.11. Observe that in this case Λ ∞ = { e ∞ , { e n gf ∞ } n ∈ N , f ∞ } , and that for all m, n ∈ N , Z ( f ) = Z ( f m ) = { f ∞ } ; Z ( e n ) = { e ∞ , { e m gf ∞ } m ≥ n } ; Z ( e n g ) = Z ( e n gf m ) = { e n gf ∞ } . Therefore, every infinite path save for e ∞ constitutes an clopen set, and so the topology andthe Borel σ -algebra of Λ ∞ is precisely the power set of Λ ∞ . Moreover, the set of periodicpairs is { ( g i f m , g i f n ) : i ∈ { , } , m, n ∈ N } , so Per(Λ) = Z . However, the set H Per (Λ) ofvertices with maximal periodicity is { v } , since the paths e n all have source v but do notappear in any periodic pair.Define a measure µ on Λ ∞ by µ ( Z ( f )) = 1 / µ ( Z ( e n g )) = 12 n +2 ; µ ( Z ( e n )) = 14 + 12 n +1 . Observe that this measure gives µ ( { e ∞ } ) = 1 /
4. Furthermore, ( σ n ) − ( { e ∞ } ) = { e ∞ } for all n , so µ ( σ − ( { e ∞ } )∆ { e ∞ } ) = 0, but neither { e ∞ } nor Λ ∞ \{ e ∞ } has measure zero. Theorem3.12(b) of [18] (or Proposition 4.6 above) now implies that any Λ-projective representationassociated to (Λ ∞ , µ ) will fail to be irreducible.We now compute the Radon–Nikodym derivatives associated to the Λ-semibranchingfunction system on (Λ ∞ , µ ) given by the usual coding and prefixing maps. First, the factthat the cylinder sets Z ( g ) = Z ( gf m ) and Z ( f ) = Z ( f m ) consist of single points makes iteasy to compute thatΦ g = Φ gf m = 1 for all m ∈ N ; Φ f = Φ f m = 1 ∀ m ∈ N . e is not constant on its domain Z ( v ):Φ e ( e ∞ ) = d ( µ ◦ σ e ) dµ ( e ∞ ) = 1 , but Φ e ( e n gf ∞ ) = 12 . In particular, if (for each finite path λ ) we set f λ = Φ − / λ on its domain Z ( λ ), then for( λ, ν ) = ( f n , f m ) periodic we have f λ f ν = f f n f f m = 1 = 1 d ( λ ) − d ( µ ) . Thus, the conclusion of Proposition 4.13 holds but the hypothesis fails.We conclude by observing that the irreducible representation π [ f ∞ ] , that Carlsen et al. [8]associate to the maximal tail { v , v } , the element 1 ∈ T k , and the cofinal path f ∞ is, up torescaling, our Λ-projective representation on Λ ∞ \ e ∞ . In this subsection we show that, although the necessary conditions for irreducibility describedin the previous subsection are not necessarily sufficient, for Λ-projective representations onthe infinite path space of a k -graph the situation is different. For example, we obtaina converse to Proposition 4.6 in Theorem 4.15 below (with E = X = Λ ∞ ). If Λ hasno sinks, then we obtain an alternative sufficient condition for irreducibility in Theorem4.20. Finally, we identify another sufficient condition in Proposition 4.26 by relating theΛ-projective representation of C ∗ (Λ) on Λ ∞ to a representation of a related 1-graph. Avariety of examples indicate that each of these sufficient conditions has a different domainof applicability. Theorem 4.15.
Let Λ be a row-finite k -graph with no sources. Suppose that the infinite pathspace Λ ∞ admits a Λ -projective system on (Λ ∞ , µ ) , for some Radon measure µ with standardprefixing maps { σ λ : λ ∈ Λ } , coding maps { σ n : n ∈ N k } , and functions { f λ : λ ∈ Λ } satisfying Conditions (a) and (b) of Definition 2.9, with the measure of all cylinder setsfinite. Let { T λ : λ ∈ Λ } be the operators given by Equation (7) of Proposition 2.10. Let E ⊆ Λ ∞ satisfy the hypotheses of Proposition 2.15, so that the restriction { T Eλ } λ ∈ Λ of the Λ -projective representation to L ( E, µ ) is again a Λ -projective representation. Then(a) The commutant of the operators { T Eλ : λ ∈ Λ } consists of multiplication operators byfunctions h with h ◦ σ n = h , µ E -a.e., for all n ∈ N k .(b) If σ n is ergodic with respect to µ E for some n ∈ N k , then { T Eλ } λ ∈ Λ is irreducible.(c) If { σ n } is a jointly ergodic family with respect to µ E , then { T Eλ } λ ∈ Λ is irreducible.Proof. First we prove (a). As established in Proposition 4.5, if T ∈ B ( L (Λ ∞ , µ E )) is anoperator in the commutant of the Λ-projective representation { T Eλ } λ , then T = M h for some h ∈ L ∞ ( E, µ ). We will show that h ◦ σ n = h a.e. for all n ∈ N k . By Proposition 4.5, thiscompletes the proof of (a). 30ince T commutes with T Eλ for all λ ∈ Λ, for any f ∈ L ( E, µ ) we have T λ T f = T T λ f ,and consequently h f λ | E ( f ◦ σ n ) = ( h ◦ σ n ) f λ | E ( f ◦ σ n ) whenever d ( λ ) = n, f ∈ L ( E, µ ) . Fix λ ∈ Λ n and consider f = χ Z ( s ( λ )) ∩ E , which is a nonzero element of L ( E, µ ) by hypothesis.The definition of a Λ-projective system and the above equation combine to reveal that h ◦ σ n = h , µ E -a.e. on Z ( λ ). Since Λ ∞ = F λ ∈ Λ n Z ( λ ) for any n ∈ N k , it follows that h ◦ σ n = h a.e. on E .For (b), choose T = M h in the commutant of the representation { T Eλ } λ , so that h = h ◦ σ n for all n . Ergodicity of one of the coding maps σ n implies, by item (a) of Lemma 4.4, that h is constant, and so { T Eλ } λ is irreducible.Finally, (c) follows as (b) above, using item (b) of Lemma 4.4 this time. Remark . One can take E = Λ ∞ in the previous Theorem; in this case, combiningTheorem 4.15 with Proposition 4.6, we have a necessary and sufficient characterization ofthe irreducibility of a Λ-projective representation on Λ ∞ .We obtain another sufficient condition for the irreducibility of the restriction to L ( E, µ )of the representation { T λ } λ ∈ Λ if, instead of requiring that µ ( E ∩ Z ( v )) be nonzero for allvertices v , we ask that E satisfy the following definition. Recall that a µ -measurable set E isinvariant with respect to a function T if µ ( E ∆ T − ( E )) = 0, and note that if E is invariantwith respect to T , then µ ( E ∩ T − ( E )) = µ ( E ). Definition 4.17.
Fix a Radon measure µ on Λ ∞ . A Borel subset of non-zero measure A ⊆ Λ ∞ is µ -invariant if A is invariant with respect to the standard coding maps σ n , for all n ∈ N k , i.e., µ ( A ∆( σ n ) − ( A )) = 0, ∀ n . A µ -invariant subset E ⊆ Λ ∞ is minimal µ -invariant if there is no µ -invariant subset A of E with µ ( A ∆ E ) = 0.To obtain our next results, we restrict our attention to higher rank graphs with no sinks – that is, for every 1 ≤ i ≤ k , and for every v ∈ Λ , we have Λ e i v = ∅ . If Λ has no sinksthen, for every n , the coding map σ n : Λ ∞ → Λ ∞ is surjective. Lemma 4.18.
Suppose Λ is a row-finite, source-free higher-rank graph with no sinks.(a) Let A ⊆ Λ ∞ , with µ ( A ) > . If µ (( σ n ) − ( A )∆ A ) = 0 for some n , then µ ( A ∆ σ n ( A )) =0 .(b) If E ⊆ Λ ∞ is µ -invariant, µ ( E ) > , and A ⊂ E satisfies µ (( σ n | E ) − ( A )∆ A ) = 0 forsome n , then µ ( A ∆ σ n ( A )) = 0 .(c) If E is minimal µ -invariant then { σ n } is a jointly ergodic family in E , that is, if A ⊂ E , with µ ( A ∆( σ n | E ) − ( A )) = 0 for all n , then either µ ( A ) = 0 or µ ( E \ A ) = 0 .Proof. (a) By definition A ∆ σ n ( A ) = { x ∈ A : x = σ n ( y ) for all y ∈ A } ∪ { x ∈ A c : x = σ n ( y ) for some y ∈ A } σ n ) − ( A )∆ A = { x ∈ A : σ n ( x ) A } ∪ { x ∈ A c : σ n ( x ) ∈ A } . Assuming that Λhas no sinks, each of the sets in the description of A ∆ σ n ( A ) above is contained in theimage under σ n of one of the sets making up ( σ n ) − ( A )∆ A. To be precise, σ n ( { x ∈ A : σ n ( x ) A } ) = { y ∈ A c : y = σ n ( x ) for some x ∈ A } and, since each coding map is surjective by hypothesis, σ n ( { x ∈ A c : σ n ( x ) ∈ A } ) ⊇ { y ∈ A : y = σ n ( x ) for any x ∈ A } . By Lemma 2.13, the claim follows.(b) Let E ⊆ Λ ∞ , with µ ( E ) >
0, be µ -invariant. By item (a), µ ( E ∆ σ n ( E )) = 0 for all n .Let A ⊂ E be such that µ (( σ n | E ) − ( A )∆ A ) = 0 for some n . Notice that( σ n | E ) − ( A )∆ A = { x ∈ A : σ n ( x ) A } ∪ { x ∈ E \ A : σ n ( x ) ∈ A } , and that A ∆ σ n ( A ) ⊆ E a.e.. Proceeding as in item (a), we conclude that A ∆ σ n ( A ) ⊆ σ n (cid:0) ( σ n | E ) − ( A )∆ A (cid:1) a.e. , where the inclusion σ n ( { x ∈ E \ A : σ n ( x ) ∈ A } ) ⊇ { y ∈ A : y = σ n ( x ) for any x ∈ A } a.e.follows from A ⊆ E and µ ( E ∆ σ n ( E )) = 0. The proof now follows as in the proof ofitem (a), taking B := ( σ n | E ) − ( A )∆ A and C := A ∆ σ n ( A ).(c) Let E be a minimal µ -invariant set, and let A ⊂ E be such that µ ( A ) > µ ( A ∆( σ n | E ) − ( A )) = 0 for all n . Consequently, µ ( { x ∈ A : σ n ( x ) A } ) = 0 and µ ( { x ∈ E \ A : σ n ( x ) ∈ A } ) = 0. If A is not µ -invariant, then there exists n ∈ N k suchthat µ ( A ∆( σ n ) − ( A )) >
0, and consequently µ ( { x ∈ A c : σ n ( x ) ∈ A } ) >
0. As E is µ -invariant, µ ( { x ∈ E c : σ n ( x ) ∈ A ⊆ E } ) = 0. We conclude that µ ( { x ∈ E \ A : σ n ( x ) ∈ A } ) >
0, which contradicts the fact that µ ( A ∆( σ n | E ) − ( A )) = 0. In other words, A must be µ -invariant. By the minimality of E it follows that µ ( E \ A ) = µ ( E ∆ A ) = 0,as desired. Remark . Lemma 4.18 need not hold for k -graphs with sinks. For a simple example,consider the graph E with vertices v , v and edges e , e such that s ( e ) = r ( e ) = s ( e ) = v and r ( e ) = v . Then Λ ∞ = { e ∞ , e e ∞ } . Let µ be a measure on Λ ∞ satisfying µ ( { e ∞ } ) > µ ( { e e ∞ } ) = 0. Then none of the conclusions of the above lemma hold. For example, forany n = 0, { e e ∞ } ∆ σ n { e e ∞ } = Λ ∞ has positive measure, while ( σ n ) − ( { e e ∞ } )∆ { e e ∞ } = { e e ∞ } has measure zero. Theorem 4.20.
Let Λ be a row-finite, source-free k -graph with no sinks, and µ a Radonmeasure on Λ ∞ which gives rise to a Λ -projective system. Suppose E ⊆ Λ ∞ is a minimal µ -invariant subset. Then the restriction of { T λ : λ ∈ Λ } to L ( E, µ ) gives an irreduciblerepresentation of C ∗ (Λ) . roof. We first show that T λ ( L ( E, µ )) ⊆ L ( E, µ ). By Lemma 4.18, if µ (( σ n ) − ( E )∆ E ) = 0for all n , then µ ( E ∆ σ n ( E )) = 0 for all n as well. If we fix f ∈ L ( E, µ ) and λ ∈ Λ, thenCondition (a) of Definition 2.9 implies that Z E | T λ ( f ) | dµ = Z E | f λ | · (cid:12)(cid:12) f ◦ σ d ( λ ) (cid:12)(cid:12) dµ = Z E (cid:12)(cid:12) f ◦ σ d ( λ ) (cid:12)(cid:12) d ( µ ◦ ( σ λ ) − ) dµ dµ = Z E (cid:12)(cid:12) f ◦ σ d ( λ ) (cid:12)(cid:12) d ( µ ◦ ( σ λ ) − ) = Z ( σ λ ) − ( E ) | f | dµ ≤ Z σ d ( λ ) ( E ) | f | dµ = Z E | f | dµ < ∞ , where the second line holds because ( σ λ ) − ( E ) ⊆ σ d ( λ ) ( E ). It follows that T λ ( L ( E, µ )) ⊆ L ( E, µ ) as claimed.Now, suppose T commutes with T λ | L ( E,µ ) for all λ ∈ Λ. As T λ | L ( E,µ ) T ∗ λ | L ( E,µ ) = M χ Z ( λ ) ∩ E , a careful read of the proof of Theorem 4.15(a) reveals that, in this case as well, wecan conclude that T = M h for some h ∈ L ∞ ( E, µ ). Thus, if f ∈ L ( E, µ ) we have h · f λ · ( f ◦ σ d ( λ ) ) = f λ · ( h ◦ σ d ( λ ) ) · ( f ◦ σ d ( λ ) ) (25)as functions in L ( E, µ ). The fact that f λ is nonzero precisely on Z ( λ ) means that bothsides of Equation (25) are supported on Z ( λ ) ∩ E .If µ ( E ∩ Z ( λ )) = 0, the fact that E is invariant with respect to σ d ( λ ) implies that µ ( E ∩ Z ( λ )) = µ ( E ∩ Z ( s ( λ ))), and so f = χ Z ( s ( λ )) is a nonzero element of L ( E, µ ). Consequently,Equation (25) implies that h = h ◦ σ d ( λ ) , µ E -a.e. on Z ( λ ), whenever µ E ( Z ( λ )) = 0.Recall that, for any n ∈ N k , we have E = F λ ∈ Λ n Z ( λ ) ∩ E . It now follows that h = h ◦ σ n as functions in L ( E, µ ) for all n ∈ N k . So, h is jointly invariant with respect to { σ n } . Since,by Lemma 4.18, the maps { σ n } are jointly ergodic in E , it follows from Lemma 4.4 that h is constant, that is, h ∈ C χ E . Thus the restriction of { T λ : λ ∈ Λ } to L ( E, µ ) is indeed anirreducible representation.
Remark . If a monic representation of a sink-free, source-free, row-finite k -graph hasan atom, then Theorem 4.20 implies that the restriction of the representation to the orbitof this atom is irreducible. We will study representations with atoms in detail in the nextsection. For non-atomic measures, a minimal invariant set should be seen as an analogue ofan orbit of an atom.The following example shows that Theorem 4.20 may apply when Theorem 4.15 doesnot. Indeed, this example shows that Λ-semibranching function systems may give rise toirreducible representations even when Λ is not cofinal, if we can apply Theorem 4.20. Example . Consider the following 1-graph Λ: v wue g fh k µ on Λ ∞ by µ ( Z ( e n )) = n +1 , µ ( Z ( e n g )) = n +3 = µ ( Z ( e n h )) , µ ( Z ( u )) = µ ( Z ( w )) = 1 /
4. (Recall that for a loop λ , we denote λ = s ( λ ).) One easily checks thatthe usual prefixing and coding maps { σ n , σ λ } make (Λ ∞ , µ ) a Λ-semibranching functionsystem. Observe that each infinite path in Λ ∞ except for e ∞ has nonzero µ -measure. Set E = { e n gf ∞ : n ∈ N } ∪ { f ∞ } . Since E ∩ Z ( u ) = ∅ , E does not satisfy the hypotheses ofProposition 2.15 (and consequently of Theorem 4.15). However, E is µ -invariant, because E = σ − ( E ). In fact, E is minimal µ -invariant: if A ⊆ E and µ ( A ∆ E ) = µ ( E \ A ) = 0, thenthere is an infinite path x ∈ E \ A . Since each infinite path in E has nontrivial measure, and E is µ -invariant, in order to have µ ( A ∆ σ − ( A )) = 0 we must have A = σ − ( A ). Consequently, A cannot contain any path of the form λσ n ( x ), for λ a finite path in Λ and n ∈ N . As everypath in E is of this form, we conclude that the only proper µ -invariant subset A of E is A = ∅ , so E is minimal µ -invariant.Theorem 4.20 now implies that any Λ-projective representation of C ∗ (Λ) on L (Λ ∞ , µ )will restrict to an irreducible representation on L ( E, µ ).Our final sufficient condition for the irreducibility of a Λ-projective representation relieson the link between k -graphs and directed graphs which we now describe.If Λ is a row-finite k -graph, for each 1 ≤ i ≤ k , write A i for the Λ × Λ (infinite)matrix with entry A i ( v, w ) = | v Λ e i w | . Fix J = ( j , . . . , j k ), with all j s ∈ N > , and define A := A J = A j . . . A j k k . Then A can also be interpreted as the matrix of a row-finite 1-graphwhich we will call Λ A . Note that every finite path λ A ∈ Λ A can be viewed as a finite path˜ λ A ∈ Λ of degree | λ A | J by using the “diagonal construction” (see [21] for the case j r = 1 , ∀ r ;the general case is similar). More formally, there is a functor Φ A between the path categoriesof the 1-graph Λ A and the k -graph Λ defined byΦ A ( v A ) := ˜ v A , Φ A ( λ A ) := ˜ λ A . ∀ v A ∈ Λ A , λ A ∈ Λ A Now recall from [23, Lemma 5.1] (cf. also [21, Proposition 2.10]) that the infinite pathspace Λ ∞ of Λ is homeomorphic and Borel isomorphic to the infinite path space Λ ∞ A of theΛ A : Proposition 4.23.
There is a canonical homeomorphism Ψ A between Λ ∞ A and Λ ∞ inducedby Φ A . Moreover, Ψ A also induces an isomorphism of Borel structures.Proof. The result follows from the observation that the cylinder sets of degree nJ , n ∈ N ,generate the topology in Λ ∞ . This observation was established for row-finite source-free k -graphs in the references given above, and one easily checks that the proofs given therework for all source-free row-finite higher-rank graphs.Moreover, since the Cuntz-Krieger relations for C ∗ (Λ A ) are satisfied by { S ˜ λ A : λ A ∈ Λ A } , the universal property of graph C ∗ -algebras, see [4], implies that there exists a ∗ -homomorphism ϕ A : C ∗ (Λ A ) → C ∗ (Λ) such that ϕ A ( S λ A ) = S ˜ λ A . Now, let π : C ∗ (Λ) →B (cid:0) L (Λ ∞ , µ π ) (cid:1) be a monic representation of C ∗ (Λ). Proposition 4.24.
The composition π A := π ◦ ϕ A is a monic representation of C ∗ (Λ A ).34 roof. By Proposition 4.23, we know that L (Λ ∞ , µ π ) ∼ = L (Λ ∞ A , µ π ). Moreover, if σ λ , σ n denote the standard coding and prefixing maps for Λ, and we denote by τ λ , τ n the codingmaps for Λ A , we have τ n ◦ Ψ A = σ nJ , τ λ ◦ Ψ A = σ ˜ λ . Therefore, by applying Theorem 3.6 we conclude that π A is monic.In addition, Proposition 4.23 implies the following result. Proposition 4.25.
The measure µ π A on Λ ∞ A , obtained by integrating against the monicvector the projection-valued measure of Theorem 3.1, is the same as the measure µ π , whenwe identify Λ ∞ and Λ ∞ A using Proposition 4.23. In particular the two coding maps satisfy σ A = σ ( j ,j ...,j k )Λ . Now recall from Theorem 4.15 that the representation π (resp π A ) is irreducible if, andonly if, the coding maps σ n , n ∈ N k (resp. σ mA , m ∈ N ) are jointly ergodic with respectto the measure µ π (resp. µ π A ); here, additionally, µ π = µ π A . We thus obtain the followingresult. Proposition 4.26.
Assume, with hypotheses as above, that the representation π A is irre-ducible. Then π is irreducible.Proof. By Theorem 4.15, it is enough to check that if B ⊂ Λ ∞ satisfies µ π (( σ n ) − ( B )∆ B ) =0 for all n ∈ N k , then either µ π ( B ) = 0 or µ π (Λ ∞ \ B ) = 0. For this, let B ⊂ Λ ∞ ∼ = Λ ∞ A , with( σ n ) − ( B ) = B for all n ∈ N k ; then in particular µ π A (( σ mA ) − ( B )∆ B ) = 0 for all m ∈ N .Since π A is irreducible, Theorem 4.15 now implies that µ π A ( B ) = 0 or µ π A (Λ ∞ A \ B ) = 0,which also implies µ π ( B ) = 0 or µ π (Λ ∞ \ B ) = 0. In this subsection we explore a few more results related with the irreducibility of repre-sentations arising from Λ-semibranching function systems. Our first result regards the de-composition of a representation as a direct sum of irreducible representations. For purelyatomic representations this follows by restricting the representation to the orbits, see [19,Corollary 3.4] and [15, Proposition 4.3]. So we will focus on the case of measures on Λ ∞ without atoms (since measures with atoms induce purely atomic representations, see Theo-rem 5.3). In this more general setting, we will need an extra hypothesis to obtain the desireddecomposition (see Definition 4.27 below). The section concludes with Proposition 4.29,which relates the question of when two Λ-semibranching representations are disjoint to thesupports of the associated measures. Definition 4.27.
Fix a Radon measure µ on Λ ∞ . Let E ⊆ Λ ∞ be a µ -invariant Borel subsetof non-zero measure. We say that E is ε -approximately ergodic if there exists ε such that if B ⊂ E is invariant and 0 < µ ( B ) < µ ( E ), then ε < µ ( B ) < µ ( E ) − ε .35 roposition 4.28. Let Λ be a row-finite, source-free k -graph with no sinks, and µ a Radonmeasure on Λ ∞ which gives rise to a Λ -projective system. Suppose that there exists an ε > such that every µ -invariant subset of non-zero measure is ε -approximately ergodic and that µ (Λ ∞ ) < ∞ . Then the representation { T λ : λ ∈ Λ } splits as a direct sum of irreduciblerepresentations.Proof. Our goal in this proof is to show that Λ ∞ = ⊔ E i , where each E i is a minimal µ -invariant set. Once we have this, the result follows from L (Λ ∞ ) = L L ( E i ) and Theo-rem 4.20.If the representation given by { T λ : λ ∈ Λ } is irreducible, we are done. Suppose itis not. Then, applying Theorem 4.15(c) to the set E = Λ ∞ , we see that there exists a µ − invariant Borel subset B ⊆ Λ ∞ such that 0 < µ ( B ) < µ (Λ ∞ ). Notice that Λ ∞ \ B isalso µ -invariant and 0 < µ (Λ ∞ \ B ) < µ (Λ ∞ ). By hypothesis, both B and Λ ∞ \ B are ε -approximately ergodic, so ε < µ ( B ) < µ (Λ ∞ ) − ε . If B is a minimal µ -invariant set, we let E := B . If not, there is a µ -invariant Borel subset B of B such that 0 < µ ( B ) < µ ( B ), B \ B is µ -invariant and 0 < µ ( B \ B ) < µ ( B ). By hypothesis, both B and B \ B are ε -approximately ergodic. If B is minimal µ -invariant we let E := B . If not, then ε < µ ( B ) < µ ( B ) − ε < µ (Λ ∞ ) − ε and proceed inductively. As µ (Λ ∞ ) is finite, thisprocess must terminate; that is, one of the B n ’s must be a minimal µ -invariant set, hence E is defined.Now, suppose that E = B N . Notice that µ ( E ) > ε and, furthermore, Λ ∞ \ E =(Λ ∞ \ B ) ∪ ( B \ B ) ∪ . . . ∪ ( B N − \ B N ), which is µ -invariant and has non-zero measure.Applying the same procedure described in the paragraph above, starting with Λ ∞ \ E insteadof Λ ∞ , we obtain a set E such that ε < µ ( E ) and E ⊆ Λ ∞ \ E . Proceeding inductively,we define the E n .Finally, since µ (Λ ∞ ) < ∞ , µ ( E n ) > ε for every n , and the E n are disjoint, we obtain thedesired decomposition into minimal µ -invariant sets, that is, Λ ∞ = ⊔ E n .In [18, Theorem 3.10] the authors prove that for finite k -graphs, and for representationsof C ∗ (Λ) arising from Λ-projective systems, on Λ ∞ , the task of checking when two repre-sentations are equivalent reduces to a measure-theoretical problem. To be precise, in thefinite setting, two Λ-semibranching representations which arise from measures µ, µ ′ on Λ ∞ are disjoint if, and only if, µ, µ ′ are mutually singular. As with Theorem 4.15, it is interestingto know if this result holds for general measure spaces. In fact, as before, the result onlyholds partially, as we show below. Proposition 4.29. (Cf. [14, Theorem 2.12] and [18, Theorem 3.10] ) Let Λ be a row-finite k -graph with no sources. Suppose that L ( X, µ ) and L ( X, µ ′ ) are two Λ -semibranchingfunction systems as in Definitions 2.6 and 2.7, with σ -finite measures µ, µ ′ , identical D λ , R λ , and coding maps { τ n : n ∈ N k } . Choose nonnegative functions f λ , f ′ λ satisfying Definition2.9 and let { T λ : λ ∈ Λ } and { T ′ λ : λ ∈ Λ } be the associated representations of C ∗ (Λ) as inProposition 2.10. If the representations { T λ : λ ∈ Λ } and { T ′ λ : λ ∈ Λ } are disjoint, then thetwo measures µ, µ ′ are mutually singular.Conversely, if µ, µ ′ are finite measures which are mutually singular and X = Λ ∞ , thenthe associated Λ -projective representations are disjoint. roof. We first observe that [18, Proposition 3.6], although stated for finite k -graphs only,still holds for row-finite k -graphs.Assume that the representations { T λ } λ , { T ′ λ } λ , are disjoint. Following [18, Proposition3.6], decompose dµ ′ = h dµ + dν , where ν is supported on B and µ is supported on A = X \ B ,the sets A, B are invariant under the prefixing and coding maps, and h ≥
0. Since A isinvariant under the prefixing map τ n for all n , L ( A, µ ′ ) is an invariant subspace for therepresentation { T ′ λ : λ ∈ Λ } .Define the operator W on L ( X, µ ′ ) by W ( f ) = f · h if f ∈ L ( A, µ ′ ), and W ( f ) = 0 onthe orthogonal complement of L ( A, µ ′ ) ⊆ L ( X, µ ′ ). To check that W is intertwining, werecall from [18, Proposition 3.6(d)] and the non-negativity condition on { f λ } and { f ′ λ } that f λ · ( h ◦ τ d ( λ ) = f ′ λ · h on A . We consequently obtain the following equalities for f ∈ L ( A, µ ′ ): T λ W ( f ) = f λ ( h ◦ τ d ( λ ) )( f ◦ τ d ( λ ) ) = f ′ λ h ( f ◦ τ d ( λ ) ) = W T ′ λ ( f ) . If f ∈ L ( A, µ ′ ) ⊥ then W ( f ) = T ′ λ ( f ) = 0 and so the above equality holds on L ( X, µ ′ ).Since W intertwines the representations { T λ } λ ∈ Λ , { T ′ λ } λ ∈ Λ of C ∗ (Λ), we must have W = 0;hence h = 0, so µ, µ ′ are mutually singular.The proof of the final statement follows verbatim as in [18, Theorem 3.10], so we refrainfrom presenting it here. Example . For a simple example that the converse of the above proposition does not holdwhen X = Λ ∞ , let E be the graph with vertices v , v and edges e , e such that s ( e i ) = v i , i = 1 , r ( e ) = v and r ( e ) = v . As with Example 4.7, we first define a branching systemin the sense of [27], which then gives us a semibranching function system. Let X = { , , , } and define R e = D v = { , } and R e = D v = { , } . Let f e : D s ( e ) → R e be given by f e ( x ) = x − f e : D s ( e ) → R e be given by f e ( x ) = x + 2. Now we define two mutuallysingular measures. Let µ (2 i + 1) = 1 and µ (2 i + 2) = 0 i = 0 ,
1, and µ (2 i + 1) = 0 and µ (2 i + 2) = 1 i = 0 ,
1. Notice that both measures are non-zero on the sets D v i , i = 1 ,
2. So,following [27, Proposition 3.12 and Remark 3.11] we get two semibranching function systems.Let π and π be the induced representation on ℓ ( X, µ ) and ℓ ( X, µ ) respectively. Theyare clearly unitary equivalent, but the measures µ and µ are mutually singular. C ∗ (Λ) The study of (irreducible) purely atomic representations of C ∗ (Λ) was initiated in [19]. Pre-viously to this, in the algebraic context, irreducible representations of Leavitt path algebras(Chen modules) were studied in [10]. Many results in [19] (restricted to 1-graphs) and[10] can be put in correspondence, as is usual in the development of the theory of graphalgebras. The most natural candidate for the analytical counterpart of a purely algebraic(semi)-branching system is to equip the (semi)-branching system with counting measure (see[7, 30, 31] for some of the algebraic results regarding representations arising from branch-ing systems). However, it is not immediately obvious that a representation arising from aΛ-semibranching function system on a measure space ( X, µ ) which has an atom is purelyatomic in the sense of [19]. We prove in Theorem 5.3 below that this is indeed the case, ifthe associated Λ-projective representation is irreducible. Our results in this section (when37estricted to the 1-graph case) therefore clarify and strengthen the parallel between the al-gebraic and analytical settings. In particular, we show in Theorem 5.5 that any irreducibleΛ-projective representation on an atomic measure space must be monic. Finally, we finishthe paper with an application of our results in the context of Naimark’s problem for k -graph C ∗ -algebras.We begin our analysis by recalling the definition of purely atomic representations. Definition 5.1. [19, Definition 3.1]Let Λ be a row-finite k -graph with no sources. A representation { t λ } λ ∈ Λ of C ∗ (Λ) on aHilbert space H is called purely atomic if there exists a Borel subset Ω ⊂ Λ ∞ such that theprojection valued measure P defined on the Borel sets of Λ ∞ as in Proposition 3.1 satisfies(a) P (Λ ∞ \ Ω) = 0 H , (b) P ( { ω } ) = 0 H for all ω ∈ Ω,(c) L ω ∈ Ω P ( { ω } ) = Id H , where the sum on the left-hand side of (c) converges in the strong operator topology.Thus, a representation of C ∗ (Λ) is said to be purely atomic if the corresponding projection-valued measure is purely atomic on the Borel σ -algebra B o (Λ ∞ ) of Λ ∞ .Let π be a representation of C ∗ (Λ) arising from a Λ-semibranching function system on( X, µ ). In the results below, it will be key to find a “coding” of X in the path space Λ ∞ .This will be accomplished by defining a map from a “large” subset of X to the path spaceΛ ∞ . The ideas presented here generalize the constructions in [10, 32] to the k -graph setting.Given a Λ-semibranching function system on ( X, µ ), define Y ⊆ X by discarding a setof measure zero, so that Y = G v ∈ Λ ( Y ∩ R v ) , Y ∩ R λ ⊆ Y ∩ D r ( λ ) ∀ λ ∈ Λ , and for all v ∈ Λ and all n ∈ N k , Y ∩ R v = G λ ∈ v Λ n Y ∩ R λ . We define a map φ from Y to Λ ∞ as follows. Pick y ∈ Y ; then y ∈ R λ for a unique λ ∈ Λ (1 ,..., . In fact, for any n ∈ N , we have y ∈ R λ n for a unique λ n ∈ Λ ( n,...,n ) , and for each k < n we have a decomposition of λ n as λ n = λ k λ n,k , where d ( λ n,k ) = ( n − k, n − k, . . . , n − k ).Thus, [37, Remark 2.2] implies that the sequence ( λ n ) n ∈ N determines a unique path in Λ ∞ ;this is φ ( y ). Notice moreover that the prefixing and coding maps on Y are taken via φ tothe standard prefixing and coding maps σ n , σ λ on Λ ∞ . Remark . The map φ defined above does not need to be onto. The same example used in[32] in the algebraic context (for algebras associated to ultragraphs) also applies here. Moreprecisely, let E be the graph with one vertex u and two edges, say e and e . Let R e and R e be two infinite countable disjoint sets, X = D u = R e ∪ R e with counting measure, andlet τ e i : R e i → D u be any bijection, for i ∈ { , } . Then X is countable, but Λ ∞ is not.38e can now show that an irreducible representation arising from a Λ-semibranchingfunction system on ( X, µ ), where µ has an atom, must be purely atomic. The orbit of aninfinite path was defined in Equation (5). Theorem 5.3.
Let Λ be a row-finite source-free k -graph and π be an irreducible represen-tation of C ∗ (Λ) arising from a Λ -semibranching function system on ( X, µ ) , where µ has anatom, say y ∈ X . Then π is purely atomic and the associated projection valued measure issupported on the orbit of φ ( y ) .Proof. Recall that the projection valued-measure P associated to π assigns to any Borel set A ⊆ Λ ∞ a projection P ( A ) ∈ B ( L ( X, µ )), see Proposition 3.1. Also recall that by Remark2.12, on cylinder sets we have t λ t ∗ λ = M χ Rλ . Let y ∈ X be an atom for ( X, µ ) and let γ = φ ( y ). By [19, Proposition 3.8] it is enough to prove that { γ } is an atom of P – that is,that P ( { γ } ) = 0. We will show that P ( { γ } )( δ y ) = 0.Observe first that P ( Z ( γ n ))( δ y ) = t γ n t ∗ γ n ( δ y ) = M χ Rγn ( δ y ) = δ y , where γ n are the initialsegments of γ defined in the construction of φ ( y ). Moreover, for any n , P ( Z ( r ( γ )) \ Z ( γ n ))( δ y ) = X γ n = λ ∈ r ( γ )Λ ( n,...,n ) P ( Z ( λ ))( δ y ) = X γ n = λ ∈ r ( γ )Λ ( n,...,n ) M χ Rλ δ y = 0 . By the regularity of the projection-valued measure P , it follows that P ( Z ( r ( γ )) \{ γ } )( δ y ) = 0.However, since P ( Z ( r ( γ )))( δ y ) = δ y , the finite additivity of P implies that P ( { γ } ) δ y = δ y . As δ y ∈ L ( Y, µ ) is nonzero, we conclude that { γ } is an atom of P .We now state a row-finite version of [19, Theorem 3.12]. Since the proof of this theoremis analogous to the finite case proof given in [19] we refrain from presenting it here. We onlyremark that the forward direction of the proof in [19, Theorem 3.12] uses Theorems 3.13and 4.2 in [18], for which we have row-finite versions, see Theorem 4.15 and Theorem 3.6respectively. Theorem 5.4. [19, Theorem 3.12] Let Λ be a row-finite k -graph with no sources. Let { t λ : λ ∈ Λ } be a purely atomic representation of C ∗ (Λ) on a separable Hilbert space H.Suppose that t λ t ∗ λ = 0 for all λ ∈ Λ . Then the representation is monic if, and only if,for every atom x ∈ Λ ∞ , P ( { x } ) is one-dimensional. Moreover, in this case the associatedmeasure µ arising from the monic representation (see Equation (22) ) is atomic. Using the above theorem we will show that any representation arising from a Λ-semibranchingfunction system on (
X, µ ), where µ has an atom, is monic and so if such a representationis irreducible we conclude that it is unitarily equivalent to a representation arising from aΛ-semibranching function system on the path space with the standard coding and prefixingmaps (see Theorem 3.6). This is the final step to complete the analytical correspondencewith the results in [10, 32] for 1-graphs. 39 heorem 5.5. Let Λ be a row-finite source-free k -graph and π be an irreducible represen-tation of C ∗ (Λ) arising from a Λ -semibranching function system on ( X, µ ) , where µ has anatom. Then π is monic.Proof. Suppose that y is an atom for µ . Then, by Theorem 5.3, π is purely atomic and,furthermore, π is supported on the orbit of φ ( y ). Since π is irreducible, by [19, Proposi-tion 3.10(c)], we have that dim[Range( P ( { φ ( y ) } )] = 1; this is also true for the atoms in theorbit of φ ( y ), see [19, Proposition 3.3]. Now apply Theorem 5.4.We finish the paper with an application of our results in the context of Naimark’s problemfor k -graph C ∗ -algebras. Naimark’s problem asks whether a C ∗ -algebra that has only one irreducible ∗ -representationup to unitary equivalence is isomorphic to the C ∗ -algebra of compact operators on some (notnecessarily separable) Hilbert space, see [41]. Recently, Suri and Tomforde showed that thisproblem has a positive answer for AF graph C ∗ -algebras as well as for C ∗ -algebras of graphsin which each vertex emits a countable number of edges, see [46]. It is therefore interestingto obtain a combinatorial description of graphs for which the associated C ∗ -algebra is AFand has a unique irreducible representation up to unitary equivalence. This is partially donein [46]. In fact, in [46, Proposition 3.5] the authors show that if E is a directed graph suchthat C ∗ ( E ) is AF and has a unique irreducible representation up to unitary equivalence,then one of two distinct possibilities must occur: Either(1) E has exactly one source and no infinite paths; or(2) E has no source and E contains an infinite path α := e e . . . with r − ( r ( e i )) = { e i } for all i ∈ N .We will show next that the converse of [46, Proposition 3.5] does not hold, that is, wewill build a graph such that the associated C ∗ -algebra is AF and satisfies item (2) above,but such that there are two non-equivalent irreducible representations. Example . Let E be the graph with vertices { v i } , i = 0 , , , . . . and { w i } , i = 1 , , . . . ,and edges e i and f i such that r ( e ) = r ( f ) = v , s ( e i ) = v i and s ( f i ) = w i for i = 1 , , . . . ,and r ( e i ) = v i − , r ( f i ) = w i − for i = 2 , , . . . . Proposition 5.7.
Let E be the graph of Example 5.6. Then C ∗ ( E ) is AF and satisfiessatisfies item (2) above, but there exist two non-equivalent irreducible representations of C ∗ ( E ) .Proof. Since E has no loops, C ∗ ( E ) is AF (see [38]).Let π be the Λ-semibranching representation of C ∗ ( E ) associated to (Λ ∞ , µ ), with thestandard coding and prefixing maps and counting measure µ . Let p be the path e e e . . . and denote by [ p ] the orbit of p . Similarly, let q be the path f f f . . . and denote by [ q ] the In [46], the authors use the reverse convention regarding ranges and sources of edges. This means thatthe statement that follows has been changed. q . By Theorem 5.3, both the restriction of π to [ p ], and its restriction to [ q ], arepurely atomic. Moreover, [19, Theorem 3.10(c)] implies that both of these restrictions areirreducible. However, they are disjoint (hence not equivalent) by [19, Theorem 3.10(a)].For k -graphs (with k > C ∗ (Λ) having only oneirreducible ∗ -representation up to unitary equivalence, then C ∗ (Λ) must be simple by [46,Lemma 2.4] (which also holds in this case). Moreover, if C ∗ (Λ) is separable (this for examplehappens if Λ is finite) then, by Rosenberg’s theorem (see [46, Page 487] and [44, Theorem 4]), C ∗ (Λ) must be isomorphic to the compact operators. Furthermore, since C ∗ (Λ) is AF, by[17, Corollary 3.8] Λ does not contain generalized cycles with an entrance. On the otherhand, if Λ is finite and Λ contains no cycles then, by [17, Theorem 5.2], C ∗ (Λ) is a matrixalgebra (because it is simple). Finally, if Λ is a locally convex k-graph such that Λ is finiteand Λ contains no cycles, then Λ contains a unique source, and C ∗ (Λ) = M Λ v ( C ), see [17,Corollary 5.7]. References [1] W. Arveson,
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E-mail address : [email protected] Elizabeth Gillaspy : Department of Mathematics, University of Montana, 32Campus Drive
E-mail address : [email protected] Daniel Gonc¸alves : Departamento de Matem´atica - UFSC - Florian´opolis - SC -88040-900, Brazil
E-mail address : [email protected]@gmail.com