Separable representations, KMS states, and wavelets for higher-rank graphs
aa r X i v : . [ m a t h . OA ] M a y Separable representations, KMS states, and wavelets forhigher-rank graphs
Carla Farsi, Elizabeth Gillaspy, Sooran Kang, and Judith PackerSeptember 24, 2018
Abstract
Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representationsof C ∗ (Λ) on certain separable Hilbert spaces of the form L ( X, µ ), by introducing the notion of aΛ-semibranching function system (a generalization of the semibranching function systems studied byMarcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation of C ∗ (Λ)on L (Λ ∞ , M ), where M is the Perron-Frobenius probability measure on the infinite path space Λ ∞ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching functionsystem gives rise to KMS states for C ∗ (Λ). For the higher-rank graphs of Robertson and Steger, wealso obtain a representation of C ∗ (Λ) on L ( X, µ ), where X is a fractal subspace of [0 ,
1] by embeddingΛ ∞ into [0 ,
1] as a fractal subset X of [0 , C ∗ (Λ) whose inverse temperature is equal to the Hausdorff dimension of X . Finally,we construct a wavelet system for L (Λ ∞ , M ) by generalizing the work of Marcolli and Paolucci fromgraphs to higher-rank graphs. Key words and phrases:
Λ-semibranching function systems; separable representations; Cantor-type fractalsubspaces of [0 , C ∗ -algebras of k -graphs. Contents C ∗ -algebras of k -graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Dynamics and KMS states on C ∗ (Λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Hausdorff measure and Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Λ -semibranching function systems and representations of C ∗ (Λ) L (Λ ∞ , M ) Higher-rank graphs (or k -graphs) and their C ∗ -algebras were introduced by Kumjian and Pask in [22] asgeneralizations of Cuntz-Krieger C ∗ -algebras associated to directed graphs (cf. [11, 2, 24]). Building on work1y Robertson and Steger [32, 33], the higher-rank graph C ∗ -algebras of [22], and their twisted counterparts(developed in [25, 26, 37]) share many of the important properties of graph C ∗ -algebras, including Cuntz-Krieger uniqueness theorems and realizations as groupoid C ∗ -algebras. Moreover, many important examplesof C ∗ -algebras (such as noncommutative tori) can be viewed as twisted k -graph C ∗ -algebras [25]. Furtherexamples of k -graph C ∗ -algebras, including specific examples, and relationships with dynamical systemstheory, can be found in recent work of Pask, Raeburn, and collaborators ([29, 30]). Over the years, manytechniques have been developed for analyzing K -groups of (twisted) Cuntz-Krieger k -graph C ∗ -algebras[34, 12, 1] and their primitive ideal spaces [5, 21], as well as for studying KMS states associated to a varietyof dynamical systems associated on them [17, 18, 19, 16].Although several different types of representations of Cuntz-Krieger k -graph C ∗ -algebras have been stud-ied in many of the references cited above, these representations have almost always been on nonseparable ℓ -spaces canonically associated to the underlying higher-rank graphs. Robertson and Steger noted in Re-mark 3.9 of [33] that there exist nontrivial representations of their higher rank Cuntz-Krieger algebras onseparable Hilbert spaces, but these representations do not seem to have been explicitly constructed.One of our main goals in this paper is to describe faithful separable representations of Cuntz-Krieger C ∗ -algebras associated to strongly connected, finite, aperiodic k -graphs (of which the Robertson-Steger al-gebras of [33] are an example): see Theorems 3.5 and 3.8. Additionally, we use one of these representationsto construct a wavelet system on L (Λ ∞ , M ), where M is the Perron-Frobenius Borel probability mea-sure (hereafter referred to as the Perron-Frobenius measure) on the infinite path space Λ ∞ constructed inProposition 8.1 of [19]. We also study the KMS states associated to these representations.To construct our representations, as well as the wavelets mentioned above, we build on the work ofMarcolli and Paolucci ([28]) and Bezuglyi and Jorgensen ([3]). Marcolli and Paolucci use the concept ofa semibranching function system to define representations of the Cuntz-Krieger C ∗ -algebra O A on severaldifferent Euclidean fractal spaces associated to the matrix A , while Bezuglyi and Jorgensen construct repre-sentations of O A on infinite path spaces associated to stationary Bratteli diagrams. Indeed, many of Marcolliand Paolucci’s representations are also on the infinite path space ( X A , µ A ) of the 1-graph with adjacencymatrix A , where µ A is a probability measure on X A associated to the Perron-Frobenius eigenvector of A .The existence of similar measure spaces for higher-rank graphs is well established in the literature.Already in [23], Kumjian and Pask had described a probability measure on the two-sided infinite pathspace of a k -graph for k > , and in [19], an Huef, Laca, Raeburn and Sims detail a Perron-Frobeniusprobability measure on the one-sided infinite path space of a strongly connected finite k -graph. Thus, wehoped that analogues of the semibranching function systems of [28] and [3] would allow us to constructfaithful representations of higher-rank graphs on such measure spaces.In order to construct our separable representations, we first present a generalization of the notion of asemibranching function system that has as part of its associated data a probability measure µ on a (fractal)space X. In almost all of our examples X = Λ ∞ with µ = M the Perron-Frobenius measure introducedin [19] (cf. Definition 2.5 below). In particular, our generalized “Λ-semibranching function systems” aresystems of partially defined shift operators for which the associated Radon-Nikodym derivatives are almosteverywhere non-zero on their domains (see Definition 3.2 for details). From a Λ-semibranching functionsystem, Theorem 3.5 then tells us how to obtain a representation of C ∗ (Λ) on L ( X, µ ), which is faithfulwhen Λ is aperiodic. We give two examples of these representations in Proposition 3.4 (on L (Λ ∞ , M )) andin Theorem 3.8 (on a fractal subspace X of [0 , X by using the vertex adjacencymatrices A , . . . , A k of Λ and thinking of points in [0 ,
1] in their N -adic expansions, where N = | Λ | .In the case k = 1, our results provide a complementary perspective to those of Marcolli and Paolucciin [28], since we do not require (as they do) that the vertex adjacency matrices have entries from { , } .This can be explained by their interpretation of the Cuntz-Krieger algebra O A as being associated to theadjacency matrix A indexed by the edges of a directed graph; whereas we study the vertex adjacency matrices A i indexed by the vertices of a graph or k -graph.We note that Λ-semibranching function systems also provide a template for establishing the existenceof faithful representations of C ∗ (Λ) on other Hilbert spaces. Indeed, examining recent work of Bezuglyiand Jorgensen [3], and Jorgensen and Dutkay [9, 8], we conjecture that the Perron-Frobenius measure ofDefinition 2.5, although a useful and canonical example of a probability measure, is potentially just one ofmany measures that could give faithful representations of Cuntz-Krieger C ∗ -algebras associated to stronglyconnected finite k -graphs. 2n future work [15], we hope to classify Λ–monic and atomic representations of Cuntz-Krieger C ∗ -algebrasassociated to strongly connected k -graphs and higher-rank Bratteli diagrams thus generalizing some of themain results in [3, 9, 8, 7].The Λ-semibranching function systems that we construct can also be used to give an explicit constructionof many of the KMS states on C ∗ (Λ) whose existence was established by an Huef, Laca, Raeburn, and Simsin [19].The definition of a KMS state arises from physics. In this context, a KMS state on the C ∗ -algebra ofobservables of a physical model represents an equilibrium state with respect to a time evolution (representedby an action of R ). KMS states can be characterized by a commutation condition, which makes sense forany C ∗ -algebra A . Recent research (cf. [4, 13, 17, 18, 27, 19]) into the KMS states of abstract C ∗ -algebrashas shown that the KMS states of a C ∗ -algebra A often encode information about important structuralproperties of A .In [19], an Huef, Laca, Raeburn, and Sims provide a complete description of the KMS states of C ∗ (Λ) fora strongly connected finite k -graph Λ. Their description relies on representations of the periodicity group ofΛ. In Section 3.3, we give an explicit construction of many of these KMS states, by using Λ-semibranchingfunction systems instead of the periodicity group. To be precise, Corollary 3.14 provides an alternative proofof part of Theorem 11.1 of [19]. Moreover, when Λ is a Robertson-Steger k -graph in the sense that the vertexmatrices A i of Λ have only { , } entries and A . . . A k has also { , } entries, we construct in Corollary 3.15a KMS β state on C ∗ (Λ) whose inverse temperature β is equal to the the fractal dimension of the subspace X of [0 ,
1] arising from the N -adic representations of infinite paths of Λ.Having established the existence of separable representations of C ∗ (Λ) for strongly connected finite k -graphs Λ, we show in Section 4 how to obtain a wavelet-type decomposition of the Hilbert space L ( X, µ )associated to our Λ-semibranching function systems. As in the work of Jonsson ([20]) and Marcolli andPaolucci ([28]), one motive for constructing these wavelets is that they give methods of constructing differentfunction spaces on the infinite path spaces for the k -graphs being studied, and illustrate how representationsof the C ∗ -algebras corresponding to higher-rank graphs can in turn give information about the path spacesof those graphs.Additionally, as explained in [6], a wavelet decomposition of a Hilbert space associated to a network ordirected graph is quite useful for performing spatial traffic analysis on the network. Higher-rank graphs canbe viewed as quotients of edge-colored directed graphs, and in future work we hope to extend the results ofSection 5.2 of [28] to Λ-semibranching function systems, which we hope will prove useful for spatial trafficanalysis on networks with qualitatively different edges.To describe in further detail our results on wavelets associated to higher-rank graphs, we first recall thatMarcolli and Paolucci, inspired in part by the work of Jonsson in [20], constructed in [28] two differentfamilies of wavelets associated to the Cuntz-Krieger C ∗ -algebra O A . In both [28] and [20], the authorsprovide families of wavelets in L (Λ A , µ ) , where Λ A is the N -adic fractal associated to the matrix A , and µ is the associated Hausdorff measure on Λ A . One of the constructions of Marcolli and Paolucci uses thePerron-Frobenius theory of irreducible matrices, and provides a finite set of functions in L (Λ A , µ ) that canbe shifted around by the generating isometries in O A to provide an orthonormal basis for the orthogonalcomplement of a specified initial space V . Building on this construction, we are able to show that our examples of Λ-semibranching function systemsgive rise to a wavelet system in L (Λ ∞ , M ), where M is the Perron-Frobenius measure on Λ ∞ . As in [28],rather than being dilated and translated, the wavelets are shifted by the partial isometries on L (Λ ∞ , M )which generate the separable representation of C ∗ (Λ); see Theorem 4.2 for details. We begin in Section 2 by reviewing the basic concepts we will rely on throughout this paper: (stronglyconnected) higher-rank graphs, their associated Cuntz-Krieger C ∗ -algebras, and their KMS states. We alsopresent several useful facts from Perron-Frobenius theory. In this section we also quickly review the basicproperties of Hausdorff measures, as we will need these for one of our examples in Section 3.2. In Section3.1 we present our definition of a Λ-semibranching function system on a finite k -graph, and establish ourmain result (Theorem 3.5), namely, that such a Λ-semibranching function system always gives rise to arepresentation of C ∗ (Λ) on a separable Hilbert space L ( X, µ ), which is faithful when Λ is aperiodic. We3lso present in Proposition 3.4 our main example of Λ-semibranching function systems, together with afractal interpretation of it, see Theorem 3.8. We describe how these Λ-semibranching function systems giverise to KMS states on C ∗ (Λ) in Section 3.3, and show in Section 4 how they give us a wavelet decompositionof L (Λ ∞ , M ) (see Theorem 4.2). This work was partially supported by a grant from the Simons Foundation (
Let k ∈ N with k ≥
1. We write e , . . . , e k for the generators of N k . A higher-rank graph, or k -graph, is acountable category Λ equipped with a functor d : Λ → N k satisfying the factorization property : for everymorphism λ ∈ Λ and m, n ∈ N k with d ( λ ) = m + n , there exist unique morphisms µ, ν ∈ Λ such that λ = µν and d ( µ ) = m , d ( ν ) = n . We will often call morphisms λ ∈ Λ elements or (finite) paths in Λ, in keepingwith our understanding of k -graphs as higher-dimensional generalizations of directed graphs. The elementsin Λ are the identity morphisms, and we call them vertices. We write r, s : Λ → Λ for the range and sourcemaps in Λ. For v, w ∈ Λ and n ∈ N k , we writeΛ n := { λ ∈ Λ : d ( λ ) = n } and v Λ w := { λ ∈ Λ : r ( λ ) = v, s ( λ ) = w } . Also for µ, ν ∈ Λ, we writeΛ min ( µ, ν ) = { ( η, ζ ) ∈ Λ × Λ : µη = νζ, d ( µη ) = d ( µ ) ∨ d ( ν ) } . We say that Λ is finite if Λ n is finite for all n ∈ N k and say that Λ has no sources if v Λ n = ∅ for all v ∈ Λ and n ∈ N k ; this is equivalent to saying that v Λ e i = ∅ for all v ∈ Λ and all e i . Definition 2.1.
1. We say that Λ is strongly connected if, for all v, w ∈ Λ , v Λ w = ∅ .2. For 1 ≤ i ≤ k , let A i be the matrix of M Λ ( N ) with entries A i ( v, w ) = | v Λ e i w | , the number of pathsfrom w to v with degree e i ; we call the A i the vertex adjacency matrices of Λ, or more simply the “vertex matrices” for Λ.We note that if Λ is strongly connected, then Λ has no sources by Lemma 2.1 of [19]. Also, the factorizationproperty of Λ implies that A i A j = A j A i .The following definition comes from [19] Section 3. Definition 2.2. ([19] Section 3) Let A = { A , ..., A k } be a family of nonzero commuting N × N matrices.We say that A = { A , ..., A k } is irreducible if for every ( s, t ) ∈ N there exists n = ( n , . . . , n k ) ∈ N k (depending on ( s, t )) such that A n ....A n k k ( s, t ) > . To justify the next definition, note that if the family of matrices A = { A , . . . , A k } is irreducible in thesense of Definition 2.2, then the proof of Proposition 3.1 of [19] implies that the matrices A i have a uniquecommon unimodular (i.e., of ℓ -norm one) eigenvector with positive entries x Λ , such that A i x Λ = ρ ( A i ) x Λ . Here ρ ( A i ) denotes the spectral radius of A i .It follows from the fact that commuting matrices have the same eigenspaces that x Λ is also the uniqueunimodular eigenvector with positive entries for the product matrix A · · · A k . Definition 2.3.
Let Λ be a strongly connected k -graph with vertex matrices A , . . . , A k .4. We write x Λ for the unique common unimodular Perron-Frobenius eigenvector of the vertex matrices A i . Note that in particular x Λ has all of its entries positive. We will call x Λ the Perron-Frobeniuseigenvector of Λ.2. We call eigenvalues associated to x Λ the Perron-Frobenius eigenvalues . As observed above, the Perron-Frobenius eigenvalue for A i is the spectral radius ρ ( A i ) of A i . Proposition 2.4. (Lemma 4.1 of [19]) A finite k -graph Λ with vertex matrices A , ..., A k is strongly con-nected if and only if A = { A , ..., A k } is an irreducible family of matrices.Also note that Definition 2.2 of an irreducible family is the same as the definition of irreducibility givenby Robertson and Steger (c.f. [35] page 94). Let Λ be a finite k -graph with no sources. To discuss the infinite path space Λ ∞ , consider the setΩ k := { ( p, q ) ∈ N k × N k : p ≤ q } . We make Ω k into a k -graph as follows. Let Ω k = N k , and define r, s : Ω k → N k by r ( p, q ) := p and s ( p, q ) := q . We define composition by ( p, q )( q, m ) = ( p, m ) and degree by d ( p, q ) = q − p . Then Ω k is a k -graph with no sources. As in Definition 2.1 of [22], an infinite path in a k -graph Λ is a k -graph morphism x : Ω k → Λ. We write Λ ∞ for the collection of all infinite paths and call it the infinite path space of Λ. Foreach p ∈ N k , we define σ p : Λ ∞ → Λ ∞ by σ p ( x )( m, n ) = x ( m + p, n + p ) for x ∈ Λ ∞ . For λ ∈ Λ we define Z ( λ ) = { x ∈ Λ ∞ : x (0 , d ( λ )) = λ } and we call it a cylinder set . It is shown in [22] that the cylinder sets { Z ( λ ) } are a basis for the topology on Λ ∞ . Note that Λ ∞ is compact if and only if Λ is finite.We say that a k -graph Λ is aperiodic if for each v ∈ Λ , there exists x ∈ Z ( v ) such that for all m = n ∈ N k ,we have σ m ( x ) = σ n ( x ).The following definition can be found originally in Proposition 8.1 of [19]. Definition 2.5.
Let Λ be a strongly connected finite k -graph with the vertex matrices A i . Define a measure M on Λ ∞ by M ( Z ( λ )) = ρ (Λ) − d ( λ ) x Λ s ( λ ) for all λ ∈ Λ , (1)where ρ (Λ) = ( ρ ( A ) , . . . , ρ ( A k )) and x Λ is the unimodular Perron-Frobenius eigenvector of Λ of Definition2.3. We will call M the Perron-Frobenius measure on Λ ∞ .Proposition 8.1 of [19] establishes that M is the unique Borel probability measure on Λ ∞ that satisfies M ( Z ( λ )) = ρ (Λ) − d ( λ ) M ( Z ( s ( λ )) for all λ ∈ Λ . C ∗ -algebras of k -graphs Definition 2.6.
Let Λ be a finite k -graph with no sources. A Cuntz-Kriger Λ -family is a collection { t λ : λ ∈ Λ } of partial isometries in a C ∗ -algebra A such that(CK1) { t v : v ∈ Λ } is a family of mutually orthogonal projections,(CK2) t µ t λ = t µλ whenever s ( µ ) = r ( λ ),(CK3) t ∗ µ t µ = t s ( µ ) for all µ , and(CK4) for all v ∈ Λ and n ∈ N k , we have t v = X λ ∈ v Λ n t λ t ∗ λ . µ, ν ∈ Λ t ∗ µ t ν = X ( η,ζ ) ∈ Λ min ( µ,ν ) t η t ∗ ζ , (2)where we interpret empty sums as zero.The Cuntz-Krieger C ∗ -algebra C ∗ (Λ) associated to Λ is generated by a universal Cuntz-Krieger Λ-family { s λ : λ ∈ Λ } , and we can show that C ∗ (Λ) = span { s µ s ∗ ν : µ, ν ∈ Λ , s ( µ ) = s ( ν ) } . The universal property gives a gauge action γ of T k on C ∗ (Λ) such that γ z ( s λ ) = z d ( λ ) s λ , where z n = Q ki =1 z n i i for z = ( z , . . . , z k ) ∈ T k and n ∈ Z k . C ∗ (Λ) Suppose (
A, α ) is a dynamical system consisting of an action α of R on a C ∗ -algebra A . We say that a ∈ A is analytic for α if the function t α t ( a ) is the restriction to R of an analytic function z α z ( a ) definedon C . A state φ on A is a KMS state at the inverse temperature β (or a KMS β state of the system ( A, α )) if φ ( ab ) = φ ( bα iβ ( a )) (3)for all analytic elements a, b . According to Proposition 8.12.3. of [31], it suffices to check the KMS conditionon a set of analytic elements which span a dense subspace of A .Let Λ be a finite k -graph with no sources with the vertex matrices A i . Let r ∈ (0 , ∞ ) k and define α r : R → Aut( C ∗ (Λ)) in terms of the gauge action by α r = γ e itr . Then for µ ∈ Λ, we have α rt ( t µ t ∗ ν ) = e itr · ( d ( µ ) − d ( ν )) t µ t ∗ ν is the restriction of the analytic function z e izr · ( d ( µ ) − d ( ν )) t µ t ∗ ν . Thus it suffices to check the KMS condition(3) on the elements t µ t ∗ ν .We are particularly interested in the dynamics α r with r = ln ρ (Λ) = (ln ρ ( A ) , . . . , ln ρ ( A k )) ∈ (0 , ∞ ) k on C ∗ (Λ); this dynamics is called the preferred dynamics . Here, we review the definition of Hausdorff outer measure and Hausdorff dimension for subsets of R n . Moredetails and proofs of these facts can be found in the book by K. Falconer [14], or Chapter 6 of the book byG. Edgar [10], for example.
Definition 2.7.
Fix n, s, δ with n ∈ N , ≤ s < ∞ , and δ > . For each subset E ⊂ R n , let H sδ ( E ) = inf { X j ≥ [diam( A j )] s } , where the infimum is taken over all countable collections of subsets { A j } j ≥ of R n such that E ⊂ ∪ j ≥ A j and diam( A j ) < δ, ∀ j ≥ . This function H sδ is called the Hausdorff outer measure on R n associated to s and δ. One sees that for E ⊂ R n , s ≥ < δ < δ we have H sδ ( E ) ≤ H sδ ( E )since one is taking the infimum over a larger family of coverings for larger δ. efinition 2.8. For 0 ≤ s < ∞ , and E ⊂ R n , define H s ( E ) = lim δ → H sδ ( E ) = sup { H sδ ( E ) : δ > } . One verifies the Carath´eodory criterion, i.e. if E and E are subsets of R n with d ( E , E ) = inf {k x − y k : x ∈ E , y ∈ E } = ǫ > , then H s ( E ∪ E ) = H s ( E ) + H s ( E ) , so that H s is also an outer measure on R n , called the Hausdorff outer measure associated to s ≥ . Theorem 2.9.
Fix s ∈ [0 , ∞ ) . Define the family of measurable sets for H s , denoted by M ( H s ) , by M ( H s ) = { E ⊂ R n | H s ( A ) = H s ( A ∩ E ) + H s ( A − E ) , ∀ A ⊂ R n } . Then:(i) The Borel sets in R n , B ( R n ) , satisfy B ( R n ) ⊂ M ( H s ) . (ii) If E ∈ M ( H s ) and H s ( E ) < ∞ , then there exists an F σ -set F ⊂ R n such that F ⊂ E and H s ( E − F ) =0 . (iii) For E ∈ M ( H s ) , there exists a G δ -set G with E ⊂ G and H s ( E ) = H s ( G ) . (iii) For all t ∈ R n , and E ∈ M ( H s ) , H s ( E ) = H s ( E + t ) . (iv) If d > is a positive scalar, and E ∈ M ( H s ) , then H s ( d · E )) = d sn H s ( E ) . The following proposition is a consequence of the above theorem.
Proposition 2.10.
Fix E ⊂ R n and define the outer measure H s for s ∈ [0 , ∞ ) as above. Then:(a) H s ( E ) = 0 for all s > n. (b) If H s ( E ) < ∞ for some s ∈ [0 , ∞ ) , then H t ( E ) = 0 for all t > s. (c) If H s ( E ) > for some s > , then H t ( E ) = ∞ for all t ∈ [0 , s ) . It follows from the above proposition that for fixed E ⊂ R n , if there exists s ∈ [0 , ∞ ) such that0 < H s ( E ) < ∞ , then, for all t < s , H t ( E ) = ∞ , and for all t > s , H t ( E ) = 0 . We note by the proposition above if such anumber s exists for E, we must have s ≤ n. Definition 2.11.
Let E ⊂ R n . We say that E has Hausdorff dimension s ∈ [0 , ∞ ) if E ∈ M ( H s ) and0 < H s ( E ) < ∞ . The measure H s restricted to subsets of E that are contained in M ( H s ) is called the Hausdorff measure on E. Example . Fix n, d ∈ N , d ≥ . Let B ⊆ { , , · · · , d − } n and for i ∈ B define τ i : [0 , n → [0 , n by τ i ( x ) = x + id , where vectors are added component-wise. It is well-known that there is a unique compact set F d,B ⊂ [0 , n satisfying F d,B = G i ∈ B τ i ( F d,B ) . The Hausdorff dimension of F d,B is log | B | log d . If B = { , , · · · , d − } n , the set F d,B is just the unit cube [0 , n with dimension n. If n = 1 , d = 3 , and B = { , } ⊆ { , , } , this construction gives the standard Cantorset. 7 Λ -semibranching function systems and representations of C ∗ (Λ) In this section, we show how to construct a representation of C ∗ (Λ) out of a generalized semibranchingfunction system, which we call a Λ-semibranching function system (see Definition 3.2). In particular, whenΛ is a strongly connected, aperiodic, finite k -graph, this construction enables us to represent C ∗ (Λ) faithfullyon L (Λ ∞ , M ) where M is the Perron-Frobenius measure of Definition 2.5. Λ -semibranching function systems We begin by recalling from [28] the definition of a semibranching function system. See also [3].
Definition 3.1. [28, Definition 2.1] Let (
X, µ ) be a measure space, and let I be a finite index set such that | I | = N . Suppose that, for each i ∈ I , we have a measurable map σ i : D i → X , for some measurable subsets D i ⊂ X . The family { σ i } is a semibranching function system if the following holds.(a) There exists a corresponding family { R i } Ni =1 of subsets of X with the property that µ ( X \ ∪ i R i ) = 0 , µ ( R i ∩ R j ) = 0 for i = j, where R i = σ i ( D i ).(b) There is a Radon-Nikodym derivative Φ σ i = d ( µ ◦ σ i ) dµ with Φ σ i > µ -almost everywhere on D i .A measurable map σ : X → X is called a coding map for the family { σ i } if σ ◦ σ i ( x ) = x for all x ∈ D i .The fact that σ ◦ σ i = id save on a subset of D i of measure zero implies that we also have σ i ◦ σ = id | R i off a set of measure 0. To see this, let y ∈ R i be arbitrary, and write y = σ i ( x ) for some x ∈ D i . Then,unless x is in the set M of measure 0 on which σ ◦ σ i = id , we have σ ( y ) = x , and thus σ i ◦ σ ( y ) = σ i ( x ) = y. Since µ ( M ) = 0 and σ i is measurable, µ ( σ i ( M )) = 0, so the above equality holds for almost all y ∈ R i .In [28], the authors show how a finite directed graph associated to an irreducible { , } -matrix givesrise to a semibranching function system and thence to a representation of the Cuntz-Krieger C ∗ -algebraassociated to the directed graph. Additionally, in [3] the authors consider monic representations of Cuntz-Krieger C ∗ -algebras associated to semibranching function systems, and their equivalence classes. Our goalin this paper is to generalize some of these constructions to obtain representations of finite higher-rank graphCuntz-Krieger C ∗ -algebras; to do this we need the following generalization of Definition 3.1. Definition 3.2.
Let Λ be a finite k -graph and let ( X, µ ) be a measure space. A Λ -semibranching functionsystem on (
X, µ ) is a collection { D λ } λ ∈ Λ of measurable subsets of X , together with a family of prefixingmaps { 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, with coding map τ m .(b) If v ∈ Λ , then τ v = id , and µ ( D v ) > 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 measure 0), 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 implies that the codingmaps pairwise commute.) 8 emark . Note that Condition (a) of Definition 3.2 implies that Λ must be a finite k -graph, since therequirement that { τ λ : d ( λ ) = m } forms a semibranching function system implies that | Λ m | is finite, for each m ∈ N k .We also observe that if k = 1, Definition 3.2 is a stronger requirement than Definition 3.1. In particular,if ( X, µ ) admits a semibranching function system where the index set I corresponds to the edges of a directedgraph Λ, there is no obvious way to define a prefixing map τ v for a vertex v of Λ; and if we instead use thevertices of Λ as our index set I for a semibranching function system, as in [28], we need not have σ v = id for each vertex v of Λ. Proposition 3.4.
Let Λ be a strongly connected finite k -graph. The measure space (Λ ∞ , M ) , together withthe prefixing maps { σ λ : Z ( s ( λ )) → Z ( λ ) } λ ∈ Λ given by σ λ ( x ) = λx and the coding maps { σ m } m ∈ N k given by σ m ( x ) = x ( m, ∞ ) , forms a Λ -semibranching function system.Proof. We first note that if m, n ∈ N k and x ∈ Λ ∞ , applying the factorization property to x (0 , m + n ) tellsus that σ m ◦ σ n = σ m + n . Thus, Condition (d) of Definition 3.2 holds. To see Condition (b), note that if v ∈ Λ then σ v = id | Z ( v ) ; and M ( Z ( v )) = x Λ v > v ∈ Λ . For (c), recall that if v ∈ Λ , then for any n ∈ N k , we have Z ( v ) = ∪ λ ∈ v Λ n Z ( λ ) , so Z ( λ ) = R λ ⊆ Z ( v ). If s ( ν ) = r ( λ ) = v , then D ν = Z ( v ), so R λ ⊆ D ν as desired whenever s ( ν ) = r ( λ ).Similarly, the factorization property implies that if s ( ν ) = r ( λ ) then σ ν ◦ σ λ = σ νλ .It only remains to check Condition (a), namely, that for each m ∈ N k , the family { σ λ } d ( λ )= m forms asemibranching function system on (Λ ∞ , M ). Observe that for fixed m ∈ N k , ∪ λ ∈ Λ m Z ( λ ) = Λ ∞ : if x ∈ Λ ∞ then x ∈ Z ( x (0 , m )) , and x (0 , m ) = λ for a unique λ ∈ Λ m . Thus, M (Λ ∞ \ ∪ λ ∈ Λ m σ λ ( D λ )) = M ( ∅ ) = 0 , so Condition (a) of Definition 3.1 is satisfied.Since Λ is finite, for any m ∈ N k the set of paths of degree m is finite; since Λ is source-free the set isnonempty. Note that if d ( λ ) = m , then σ m ◦ σ λ = id | Z ( s ( λ )) , so σ m is indeed a coding map for { σ λ } d ( λ )= m .Thus, it merely remains to check that the Radon-Nikodym derivative Φ λ = d ( M ◦ σ m ) dM is strictly positivealmost everywhere on D λ = Z ( s ( λ )).Let λ, ν ∈ Λ. We compute M ◦ σ λ ( Z ( ν )) = M ( Z ( λν )) = δ s ( λ ) ,r ( ν ) ρ (Λ) − d ( λν ) x Λ s ( ν ) and M ( Z ( ν )) = ρ (Λ) − d ( ν ) x Λ s ( ν ) . Thus, M ◦ σ λ ( Z ( ν )) M ( Z ( ν )) = δ s ( λ ) ,r ( ν ) ρ (Λ) − d ( λν ) x Λ s ( ν ) ρ (Λ) − d ( ν ) x Λ s ( ν ) = δ s ( λ ) ,r ( ν ) ρ (Λ) − d ( λ ) . Since this value is the same for any ν ∈ s ( λ )Λ, it follows that Φ λ is constant on D λ : for any x ∈ Z ( s ( λ )) wehave Φ λ ( x ) = ρ (Λ) − d ( λ ) > ρ (Λ) ∈ (0 , ∞ ) k . Thus, we have a Λ-semibranching function system on Λ ∞ as claimed. Theorem 3.5.
Let Λ be a finite k -graph with no sources and suppose that we have a Λ -semibranchingfunction system on a measure space ( X, µ ) . For each λ ∈ Λ , define S λ ∈ B ( L ( X, µ )) by S λ ξ ( x ) = χ R λ ( x )(Φ τ λ ( τ d ( λ ) ( x ))) − / ξ ( τ d ( λ ) ( x )) . Then the operators { S λ } λ ∈ Λ generate a representation of C ∗ (Λ) . If Λ is aperiodic then this representationis faithful. roof. We begin by computing ( S λ ) ∗ : h ( S λ ) ∗ ξ, ζ i = h ξ, S λ ζ i = Z X ξ ( x ) S λ ζ ( x ) dµ ( x )= Z R λ ξ ( x )Φ τ λ ( τ d ( λ ) ( x )) − / ζ ( τ d ( λ ) ( x )) dµ ( x )= Z D λ ξ ( τ λ y )Φ τ λ ( y ) / ζ ( y ) dµ ( y ) . So the adjoint of S λ is given by ( S ∗ λ ξ )( x ) = χ D λ ( x )(Φ τ λ ( x )) / ξ ( τ λ x ) . A straightforward computation, using the fact that τ λ ◦ τ d ( λ ) = id a.e., will show that S λ S ∗ λ S λ = S λ asoperators on L ( X, µ ). Similarly, the hypothesis that τ v = τ d ( v ) = id | D v for v ∈ Λ implies that S v ξ ( x ) = χ D v ( x ) ξ ( x ) , so S v is a projection for all v ∈ Λ .The fact that the projections S v are mutually orthogonal follows from the hypothesis that { τ v : v ∈ Λ } forms a Λ-semibranching function system, and consequently that µ ( D v ∩ D w ) = 0 if w = v ∈ Λ . To beprecise, fix v, w ∈ Λ ; then compute S v S w ξ ( x ) = ( ( S w ξ )( x ) if x ∈ D v ( ξ ( x ) if v = w v = w , then S v S w = 0. So { S v : v ∈ Λ } consists of mutually orthogonal projections, which gives(CK1).To check (CK2), fix λ, ν ∈ Λ and compute S λ S ν ξ ( x ) = ( (Φ τ λ ( τ d ( λ ) ( x ))) − / ( S ν ξ )( τ d ( λ ) x ) if x ∈ R λ ,0 otherwise= (Φ τ λ ( τ d ( λ ) ( x ))) − / (Φ τ ν ( τ d ( ν ) ( τ d ( λ ) ( x )))) − / ξ ( τ d ( ν )+ d ( λ ) ( x ))if x ∈ R λ , τ d ( λ ) x ∈ R ν ,0 otherwiseNote that τ d ( λ ) x ∈ D λ = D s ( λ ) = R s ( λ ) , and that R ν ⊆ D r ( ν ) = R r ( ν ) by Condition (c) of Definition 3.2.Therefore, if S λ S ν ξ ( x ) is to be nonzero, we must have τ d ( λ ) x ∈ R s ( λ ) ∩ R r ( ν ) , so this intersection is nonempty.In order for S λ S ν ξ to be a nonzero element of L ( X, µ ), then, we must have µ ( R s ( λ ) ∩ R r ( ν ) ) = 0. Condition(a) of Definition 3.1 then tells us that we must have s ( λ ) = r ( ν ) . Thus, as we check via the computationsthat follow that S λ S ν = S λν , we will assume that s ( λ ) = r ( ν ).Note that if x ∈ R λ , then x = τ λ y for some y ∈ D λ , and if τ d ( λ ) x = y ∈ R ν , then y = τ ν ( z ) for some z ∈ D ν . Since s ( λ ) = r ( ν ), we can now use Definition 3.2(c) to conclude that x = τ λ τ ν ( z ) = τ λν z. We claim that if x ∈ R λν we have:Φ τ λ ( τ d ( λ ) ( x ))Φ τ ν ( τ d ( ν ) ( τ d ( λ ) ( x ))) = Φ τ λν ( τ d ( λν ) ( x )) . (4)To see this, we compute that for a.e. x ∈ R λ ,Φ τ λ ( τ d ( λ ) ( x )) = d ( µ ◦ τ λ )( τ d ( λ ) ( x )) dµ ( τ d ( λ ) ( x )) = dµ ( x ) dµ ( τ d ( λ ) ( x )) , τ λ τ d ( λ ) = id | R λ a.e. Similarly, if τ d ( λ ) ( x ) ∈ R ν ,Φ τ ν ( τ d ( ν ) ( τ d ( λ ) ( x ))) = d ( µ ◦ τ ν )( τ d ( ν ) ( τ d ( λ ) x )) dµ ( τ d ( ν ) ( τ d ( λ ) ( x ))) = dµ ( τ d ( λ ) ( x )) dµ ( τ d ( ν )+ d ( λ ) ( x )) . Thus, for a.e. x such that x ∈ R λ , τ d ( λ ) ( x ) ∈ R ν ,Φ τ λ ( τ d ( λ ) ( x ))Φ τ ν ( τ d ( ν ) ( τ d ( λ ) ( x ))) = dµ ( x ) dµ ( τ d ( λ ) ( x )) dµ ( τ d ( λ ) ( x )) dµ ( τ d ( ν )+ d ( λ ) ( x ))= dµ ( x ) dµ ( τ d ( ν )+ d ( λ ) ( x ))= dµ ( τ λν ( τ d ( λν ) ( x )) dµ ( τ d ( λν ) ( x ))= Φ τ λν ( τ d ( λν ) ( x ))whenever x ∈ R λν ⊆ R λ . It remains to show that R λν = { x ∈ R λ : τ d ( λ ) ( x ) ∈ R ν } . (5)If x ∈ R λ and τ d ( λ ) ( x ) ∈ R ν , we have τ d ( λ ) ( x ) = τ ν y for some y ∈ D ν , so for almost all such x we have y = τ d ( ν ) ◦ τ d ( λ ) x = τ d ( λν ) ( x ) ⇒ τ λν ( y ) = x. In other words, { x ∈ R λ : τ d ( λ ) ( x ) ∈ R ν } ⊆ R λν .Now, suppose x ∈ R λν ⊆ R λ , so x = τ λ y = τ λν z . Then Condition (d) of Definition 3.2 implies that z = τ d ( λν ) ( x ) = τ d ( ν )+ d ( λ ) ( x ) = τ d ( ν ) ◦ τ d ( λ ) ( x ) = τ d ( ν ) ( y )almost everywhere, so τ d ( ν ) ( y ) = z ∈ D λν = D ν for a.e. y = τ d ( λ ) ( x ). Consequently, for almost all x ∈ R λν ⊆ R λ , we have τ d ( λ ) ( x ) ∈ R ν , so Equation (5) (and thus also Equation (4)) is true up to a set ofmeasure 0.Then the claim implies S λ S ν ξ ( x ) = ( (Φ τ λν ( τ d ( λν ) ( x ))) − / ξ ( τ d ( λν ) ( x )) if x ∈ R λν S λν ξ ( x ) . Thus (CK2) holds.For (CK3), fix λ ∈ Λ. Then a straightforward computation implies that S ∗ λ S λ ξ ( x ) = χ D λ ( x ) ξ ( x ) = ( S s ( λ ) ) ξ ( x ) . This gives (CK3).Finally, we check (CK4): fix v ∈ Λ and n ∈ N k . Let λ ∈ v Λ n . If x R v = D v ⊇ R λ , then S λ S ∗ λ ξ ( x ) = 0 = S v ξ ( x ), and hence ( P λ ∈ v Λ n S λ S ∗ λ ) ξ ( x ) = 0 = S v ξ ( x ). So suppose that x ∈ D v . Wecompute S λ S ∗ λ ξ ( x ) = ( (Φ τ λ ( τ d ( λ ) ( x ))) − / ( S ∗ λ ξ )( τ d ( λ ) x ) if x ∈ R λ ,0 otherwise= ( ξ ( x ) if x ∈ R λ P λ ∈ v Λ n S λ S ∗ λ ξ ( x ) = P λ ∈ v Λ n χ R λ ( x ) ξ ( x ) . We claim that X λ ∈ v Λ n χ R λ = χ D v . (6)11o see this, recall that Condition (a) of Definition 3.1 implies that X = ∪ d ( λ )= n R λ , so in particular,there exists a set S ⊆ Λ n such that D v ⊆ ∪ λ ∈ S R λ . Suppose that λ ∈ S but r ( λ ) = v . If x ∈ R λ , Condition(c) of Definition 3.2 tells us that x ∈ D r ( λ ) , so the fact that the sets { D w : w ∈ Λ } form a Λ-semibranchingfunction system with D w = R w for all w ∈ Λ implies that µ ( D v ∩ D r ( λ ) ) = 0 unless v = r ( λ ). So,we lose nothing by removing from S any λ with r ( λ ) = v , allowing us to assume that S ⊆ v Λ n . Thus D v ⊆ ∪ λ ∈ v Λ n R λ .Condition (c) of Definition 3.2 now tells us that R λ ⊆ D v ∀ λ ∈ v Λ. (Thus ∪ λ ∈ v Λ n R λ ⊆ D v , and hence D v = ∪ λ ∈ v Λ n R λ ). Thus Equation (6) holds.It follows that P λ ∈ v Λ n S λ S ∗ λ = S v , so (CK4) holds.Since { S λ : λ ∈ Λ } is a Cuntz-Krieger Λ-family on L ( X, µ ), the universal property of C ∗ (Λ) gives arepresentation π S : C ∗ (Λ) → B ( L (Λ ∞ , µ )). For any v ∈ Λ , we have S v ξ ( x ) = χ D v ( x ) ξ ( x ), and hence S v isnonzero for each v ∈ Λ . Thus, by Theorem 4.6 of [21], π S is faithful whenever Λ is aperiodic. Corollary 3.6.
Let Λ be a strongly connected, aperiodic, finite k -graph. Then we have a faithful represen-tation of C ∗ (Λ) on L (Λ ∞ , M ) . Λ -semibranching function systems In this section, we strengthen slightly our working hypothesis that Λ is a strongly connected finite k -graph.To be precise, we also assume throughout this section that the vertex matrices A , . . . , A k are { , } matricesand that the product A := A · · · A k is also a { , } matrix. Under these hypotheses, we can construct Λ-semibranching function systems (and hence, by Theorem 3.5, representations of C ∗ (Λ)) on L ( X ) for severalfractal subspaces X of [0 , n . We describe two such constructions in this section.The reason for the stronger hypotheses is to be able to apply Theorem 2.17 of [28] to A . Although thestatement of Theorem 2.17 requires A to be irreducible, we observe that the proof only requires A to havea unique unimodular eigenvector with positive entries. Our hypothesis that Λ is strongly connected impliesthat the Perron-Frobenius eigenvector of Λ is the unique Perron-Frobenius eigenvector for A , see Definitions2.2 and 2.3.Moreover, many examples of strongly connected finite k -graphs also satisfy the hypotheses of this section.One such example (for k = 2) comes from [30]. Example . (Ledrappier’s example; [29], [30]) The skeleton of this 2-graph is given on page 1622 of [30].The vertex matrices are A = , A = Note also that the Perron-Frobenius eigenvalues ρ ( A ), ρ ( A ), and ρ ( A A ) respectively of A , A , and A A are given by ρ ( A ) = ρ ( A ) = 2 , ρ ( A A ) = 4 . Let Λ be a k -graph satisfying our newly strengthened hypotheses. For our first example of a fractalrepresentation of C ∗ (Λ), we will construct a Λ-semibranching function system on a Cantor-type fractalsubspace X of [0 ,
1] for any k -graph Λ that satisfies the conditions specified at the beginning of this section.We begin by describing the construction of X and an embedding of Λ ∞ into X .Observe that to any infinite path x ∈ Λ ∞ , we can associate a unique sequence of edges f x , f x , . . . , f xk , f x , . . . , f xk , f x , . . . , such that x = f x f x · · · f xk f x · · · and d ( f xij ) = e i for all j . In other words, the decomposition x = f x f x · · · f xk f x · · · is a “rainbow decomposition,” with edges of color i occurring in the nk + i th spotfor each n ∈ N .By our hypothesis that the vertex matrices of Λ are 0-1 matrices, we know that given any v, w ∈ Λ ,there is at most one edge of shape i with source v and range w . This implies that the string of edges f x f x · · · f xk f x · · · can be replaced by a unique string of vertices v x , v x , . . . , v xk , v x , . . . , where A i ( v xij , v x ( i +1) j ) = 1 ∀ j, ∀ ≤ i ≤ k − A k ( v xkj , v x j +1) ) = 1 ∀ j. v x ) for this sequence of vertices.Write N = | Λ | . Henceforth we will assume we have chosen a labeling of Λ by the integers 0 , , . . . , N − ∞ → [0 , ,
1] in their N -adicexpansions: Ψ( x ) = 0 .v x v x . . . v xk v x v x . . . = X ≤ i ≤ k,j ∈ N v xij N i + k ( j − . The fact that each path x ∈ Λ ∞ is associated to a unique string of vertices v x , v x , . . . , v xk , v x , . . . impliesthat Ψ is injective; the image of Ψ in [0 ,
1] is a fractal-type subspace X .Since Ψ : Λ ∞ → [0 ,
1] is injective, we can use Ψ to transfer the prefixing and coding maps from Proposition3.4 to X = Ψ(Λ ∞ ), obtaining a new family of prefixing and coding maps τ m := Ψ ◦ σ m ◦ Ψ − , τ λ := Ψ ◦ σ λ ◦ Ψ − on X . If we further define a measure µ on X by µ ( E ) = M (Ψ − ( E )) , then the maps { τ m , τ λ } form aΛ-semibranching function system on ( X, µ ). It follows then from Theorem 3.5 that we have a representationof C ∗ (Λ) on L ( X, µ ).The following is a generalization of Theorem 2.17 of [28].
Theorem 3.8.
Let Λ be a strongly connected finite k -graph with { , } vertex matrices A , . . . , A k , such thatthe product A := A · · · A k is also a { , } matrix. Let X = Ψ(Λ ∞ ) and let µ be the measure on X given by µ ( E ) = M (Ψ − ( E )) , where M is the measure on Λ ∞ given by Equation 1. Then µ = H s ( X ) H s , where H s is the Hausdorff measure on X associated to its Hausdorff dimension s . Moreover N ks = ρ ( A ) , s = 1 k ln ρ ( A )ln N where N = | Λ | and ρ ( A ) denotes the Perron-Frobenius eigenvalue of A .Remark . Theorem 3.8 can be rephrased, equivalently, by saying that the two Hilbert spaces L (Λ ∞ , M )and L ( X, H s ( X ) H s ) are isometric when the hypotheses of the theorem are satisfied.The proof of Theorem 3.8 requires a number of preliminary steps. We begin by showing that X has finiteand nonzero Hausdorff measure. Proposition 3.10.
Let Λ be a strongly connected finite k -graph with { , } vertex matrices A , ..., A k , suchthat A ...A k is also a { , } matrix. Write N = | Λ | and write points x ∈ [0 , in their N -adic expansions: x = 0 .x x . . . x k x x . . . x k x . . . . Let X = { x ∈ [0 ,
1] : ∀ j, ∀ i < k, A i ( x ij , x ( i +1) j ) = 1 and A k ( x kj , x j +1) ) = 1 } . Then X has finite, positive Hausdorff dimension. To prove Proposition 3.10, we will need the following Lemma.
Lemma 3.11.
Let X be as in Proposition 3.10 and define φ : X → [0 , by φ ( x ) = φ (0 .x x . . . x k x . . . x k x . . . ) = 0 .x x x · · · . Then φ is 1-H¨older continuous.Proof. Recall that φ is 1-H¨older continuous if N ( φ ) := sup x = y | φ ( x ) − φ ( y ) || x − y | < ∞ . Thus, suppose that x = y . Define J = { j ∈ N : x j = y j } ; without loss of generality, we may assume J = ∅ (if J = ∅ then | φ ( x ) − φ ( y ) | = 0).For each j ∈ J , let α j = | x j − y j | . Observe that | φ ( x ) − φ ( y ) | = P j ∈ J α j N j , whereas | x − y | ≥ X j ∈ J α j N ( j − k +1 ≥ α j N ( j − k +1 j ∈ J . Consequently, N ( φ ) = X j ∈ J α j N j | x − y | ≤ X j ∈ J α j N j α j N ( j − k +1 = N ( j − k +1 α j X j ∈ J α j N j ≤ N ( j − k +1 α j < ∞ , since P j ∈ J α j N j ≤ P n ∈ N N − N n = 1 . Proof of Proposition 3.10.
Write A = A A · · · A k . Observe that if x = 0 .x x . . . x k x x . . . x k x . . . ∈ X, then the sequence x , x , x , · · · satisfies A ( x j , x j +1) ) = 1 ∀ j ∈ N . Our hypotheses imply that A is a { , } matrix with a unique positive unimodular eigenvector, so the proofof Theorem 2.17 of [28] says that if we define φ : X → [0 ,
1] by φ ( x ) = 0 .x x x · · · , as in Lemma 3.11,then φ ( X ) is the space Λ A ⊆ [0 ,
1] studied in Theorem 2.17 of [28]. Thus, Part (3) of this theorem tells usthat φ ( X ) has finite, positive Hausdorff dimension.Since φ is 1-H¨older continuous by Lemma 3.11, Proposition 2.2 of [36] tells us that the Hausdorff dimensionof X is bounded below by that of φ ( X ). In particular, X has nonzero Hausdorff dimension.To see that the Hausdorff dimension of X is finite, recall that the Hausdorff dimension of X is δ X = inf { d ≥ C d ( X ) = 0 } , where C d ( X ) = inf { X i ∈ I r di : there is a cover { U i } i ∈ I of X with the radius of U i being r i } . Thus, if we can show that there exists d ≥ C d ( X ) = 0, this will establish that X has finiteHausdorff dimension.Fix d > ℓ ∈ N be arbitrary. Consider the collection I ℓ of finite N -adic numbers in [0 , y ∈ I ℓ has an N -adic expansion of length ℓ . We define a cover { U y } y ∈ I ℓ of X by U y = { x ∈ X : the first ℓ terms of x are given by y } . Notice that there are at most N ℓ nonempty sets U y , and that the diameter of the set U y is strictly less than1 /N ℓ . Thus, if d > C d ( X ) ≤ inf ℓ ∈ N N ℓ N dℓ = inf ℓ N ( d − ℓ = 0 . In other words, δ X ≤ < ∞ , so the Hausdorff dimension of X is finite and nonzero.Having established the existence of a finite, nonzero Hausdorff measure H s on X , we now show that H s ( X ) H s = µ by using the uniqueness of the Perron-Frobenius measure M of Definition 2.5. Proof of Theorem 3.8.
Recall from Proposition 8.1 of [19] that M is the unique Borel probability measureon Λ ∞ such that M ( Z ( λ )) = ρ (Λ) − d ( λ ) M ( Z ( s ( λ ))) for all λ ∈ Λ. By construction, our rescaled Hausdorffmeasure H s ( X ) H s is a probability measure. Thus, if we show that H s satisfies H s (Ψ( Z ( λ ))) = ρ (Λ) − d ( λ ) H s (Ψ( Z ( s ( λ )))) , (7)this will establish that H s ( X ) H s = µ .We begin by proving Equation (7) when d ( λ ) = ( m, m, . . . , m ) for some m ∈ N . In this case, if we abusenotation by writing Ψ( λ ) = 0 .y y · · · y k y · · · y km y m +1) , we have Ψ( Z ( λ )) = { x ∈ X : x ij = y ij ∀ ≤ i ≤ k, ≤ j ≤ m + 1 } . Z ( s ( λ ))) = { x ∈ X : x = y m +1) } = { ( x − Ψ( λ )) N km + y N : x ∈ Ψ( Z ( λ )) } . Thus, the scaling- and translation-invariance of the Hausdorff measure (Theorem 2.9) implies that H s (Ψ( Z ( s ( λ )))) = N ( km ) s H s (Ψ( Z ( λ ))) . (8)Recall that for any vertex v , we can write Z ( v ) as a disjoint union Z ( v ) = ∪ λ ∈ v Λ (1 ,..., Z ( λ ). Moreover,we can identify each such path λ with the unique string of vertices ( v λ ) = vv v · · · v k such that the edge f i from v i to v i − has shape e i and λ = f · · · f k . By abuse of notation, we will also write λ = vv v · · · v k .Now, the uniqueness of the string of vertices ( v λ ), combined with the fact that the vertex matrices A i are0-1 matrices, implies that Z ( v ) = ∪ v i ∈ Λ A ( v, v ) A ( v , v ) · · · A k ( v k − , v k ) Z ( vv · · · v k ) , and hence Ψ( Z ( v )) = ∪ v i ∈ Λ A ( v, v ) · · · A k ( v k − , v k )Ψ( Z ( vv · · · v k )) . The fact that this union is disjoint means that H s (Ψ( Z ( v )) = X v i ∈ Λ A ( v, v ) · · · A k ( v k − , v k ) H s (Ψ( Z ( vv · · · v k )))= X v i A ( v, v ) · · · A k ( v k − , v k ) N − ks H s (Ψ( Z ( v k )))= X v k ∈ Λ A A · · · A k ( v, v k ) N − ks H s (Ψ( Z ( v k ))) . In other words, the vector ( H s (Ψ( Z ( v ))) v ∈ Λ is an eigenvector for the product A = A · · · A k , with eigenvalue N ks .On the other hand, we know from Proposition 3.1 of [19] that x Λ is the unique common unimodu-lar Perron-Frobenius eigenvector for the vertex matrices A i . Since commuting matrices have the sameeigenspaces, and ( H s (Ψ( Z ( v ))) v ∈ Λ is an eigenvector for A , it follows that ( H s (Ψ( Z ( v ))) v ∈ Λ is an eigen-vector for each vertex matrix A i . Thus, assuming that H s (Ψ( Z ( v ))) is nonzero for all vertices v ∈ Λ ,the uniqueness of x Λ tells us that ( H s (Ψ( Z ( v ))) v ∈ Λ must be a scalar multiple of x Λ , and moreover that N ks = ρ (Λ) (1 ,..., . Inserting this into Equation (8) tells us that if d ( λ ) = ( m, . . . , m ), then H s (Ψ( Z ( λ ))) = ρ (Λ) − ( m,...,m ) H s (Ψ( Z ( s ( λ )))) . To see that H s (Ψ( Z ( v ))) is nonzero, we observe that (in the notation of Lemma 3.11) φ (Ψ( Z ( v ))) = { x ∈ [0 ,
1] : x = 0 .v . . . } = Ψ( Z ( v )) . Let h be the Hausdorff measure on φ ( X ) associated to its Hausdorff dimension (which we know is finite byLemma 3.11); then Theorem 2.17(2) of [28] tells us that h (Ψ( Z ( v ))) = x Λ v . In particular, h (Ψ( Z ( v ))) is finiteand nonzero. Now, the uniqueness of the nonzero Hausdorff measure of a set (Proposition 2.10) tells us that x Λ v = h ( φ (Ψ( Z ( v ))) = h (Ψ( Z ( v ))) ⇒ H s (Ψ( Z ( v )) = h (Ψ( Z ( v ))) = x Λ v , and x Λ v is nonzero for all v ∈ Λ by definition.Having thus established Equation (7) when d ( λ ) = ( m, m, . . . , m ), we now proceed to the general case.Let λ ∈ Λ be arbitrary, and write d ( λ ) = ( m , . . . , m k ). Let m = max { m i } and let n = ( m, m, . . . , m ) − d ( µ ).Define C λ = { ν ∈ Λ : r ( ν ) = s ( λ ) , d ( ν ) = n } .
15n words, C λ consists of the paths ν such that the product λν is defined and d ( λν ) = d ( ν ) + d ( λ ) = ( m, m, . . . , m ) . Moreover, Z ( λ ) can be written as a disjoint union Z ( λ ) = ∪ ν ∈ C λ Z ( λν ). Thus, H s (Ψ( Z ( λ ))) = X ν ∈ C λ H s (Ψ( Z ( λν ))) = X ν ∈ C λ ρ (Λ) − ( m,...,m ) H s (Ψ( Z ( s ( ν )))) . For each path ν ∈ C λ , write ν = f ν f ν · · · f ν | d ( ν ) | , where | d ( f νi ) | = 1 ∀ i , and we list all the color-1 edgesfirst, then all the color-2 edges, etc. Notice that for each ν ∈ C λ , there will be m − m edges of color 1, then m − m edges of color 2, and so on. Write s ( λ ) v ν v ν · · · v ν m − m ) v ν · · · v ν m − m ) · · · v νk ( m − m k ) for the unique sequence of vertices associated to ν in this decomposition.Since ( H s (Ψ( Z ( v )))) v ∈ Λ is a scalar multiple of x Λ and x Λ is an eigenvector for each vertex matrix A i ,with eigenvalue r ( A i ) the Perron-Frobenius eigenvalue of A i , we also have that ( H s (Ψ( Z ( v )))) v ∈ Λ is aneigenvector for each vertex matrix A i , with eigenvalue r ( A i ). Thus, H s (Ψ( Z ( s ( λ ))) == ρ (Λ) − ( m,...,m )+ d ( λ ) X v ij ∈ Λ A ( s ( λ ) , v ) A ( v , v ) · · · A ( v m − m − , v m − m ) ) × A ( v m − m ) , v ) A ( v , v ) · · · A ( v m − m − , v m − m ) ) · · ·× · · · A k ( v k ( m − m k − , v k ( m − m k ) ) H s (Ψ( Z ( v k ( m − m k ) )))= X ν ∈ C λ ρ (Λ) − d ( ν ) H s (Ψ( Z ( s ( ν )))) , since ν ∈ C λ precisely when ν is associated to a string of vertices v v · · · v m − m ) · · · v k ( m − m k ) such that the massive matrix product above is nonzero.It now follows that H s (Ψ( Z ( s ( λ )))) = ρ (Λ) − ( m,...,m )+ d ( λ ) X ν ∈ C λ H s (Ψ( Z ( s ( ν ))))= ρ (Λ) − ( m,...,m )+ d ( λ ) ρ (Λ) ( m,...,m ) H s (Ψ( Z ( λ )))= ρ (Λ) d ( λ ) H s (Ψ( Z ( λ ))) . In other words, H s satisfies the scaling property (7) for all λ ∈ Λ. It follows that the Hausdorff measure H s and the measure µ are multiples of each other, as claimed.By a similar construction, one can also produce a Λ-semibranching function system on a Sierpinski-typefractal set in [0 , k +1 , in analogy with the construction in Section 2.6 of [28]. Thus, we also obtain arepresentation of C ∗ (Λ) on a fractal subspace of R k +1 .As a corollary of the above proof, we obtain that in many cases, M is the unique measure on Λ ∞ thatadmits a Λ-semibranching function system. Corollary 3.12.
Let Λ be a strongly connected finite k -graph, and suppose that we have a Λ -semibranchingfunction system on the probability space (Λ ∞ , µ ) for some measure µ on Λ ∞ . Suppose that there exists C ∈ (0 , ∞ ) k such that the Radon-Nikodym derivative is given by Φ τ λ = C − d ( λ ) for all λ ∈ Λ . Then µ = M ,where M is the Perron-Frobenius measure. roof. As in the proof of Theorem 3.8, the uniqueness of the measure M means that it will suffice to showthat µ satisfies Equation (7) with µ replacing H s ◦ Ψ. Our hypothesis that the Radon-Nikodym derivativeΦ τ λ is given by C − d ( λ ) implies that µ ( Z ( λ )) = C − d ( λ ) µ ( Z ( s ( λ ))) . One now uses this fact to observe that ( µ ( Z ( v ))) v ∈ Λ is an eigenvector for each of the vertex matrices A i of Λ,with eigenvalue C e i . Then the uniqueness of the Perron-Frobenius eigenvector implies that ( µ ( Z ( v ))) v ∈ Λ = x Λ , and that C = ρ (Λ). As indicated above, the uniqueness of M now implies that M = µ . Λ -semibranching function systems In this section we show how to obtain KMS states on C ∗ (Λ) from certain Λ-semibranching function systems.As we show in Theorem 3.13 below, we obtain KMS states on C ∗ (Λ) whenever the Radon-Nikodym derivativeΦ τ λ of the Λ-semibranching function system has a special form and the dynamics on C ∗ (Λ) are given interms of the Radon-Nikodym derivative. We apply Theorem 3.13 to our two main examples discussed inSection 3.2. In Corollary 3.14, we show that the measure space (Λ ∞ , M ) given in (1) with the preferreddynamics α r defines a KMS state at the inverse temperature β = 1. In Corollary 3.16, we show that theCantor-type fractal measure space ( X, µ ) associated to the Hausdorff measure defines a KMS state at theinverse temperature β = s , where s is the Hausdorff dimension. Moreover we show in Corollary 3.16 howto apply the ideas of Theorem 3.13 to obtain KMS states associated to a semibranching function system.Note that our KMS states provide a concrete realization of certain of the KMS states whose existence wasestablished in [18, 19, 16]. Theorem 3.13.
Let Λ be a finite k -graph with no sources, and suppose that we have a Λ -semibranchingfunction system on the probability space ( X, µ ) . Suppose that there exists ω ∈ (0 , ∞ ) and C ∈ (0 , ∞ ) k suchthat the Radon-Nikodym derivative is given by Φ τ λ = C − ωd ( λ ) for all λ ∈ Λ . Let α be the dynamics on C ∗ (Λ) given by α t = γ C it , where γ is the gauge action on C ∗ (Λ) . Then the measure µ on X defines a KMS state φ of the system ( C ∗ (Λ) , α ) at the inverse temperature β = ω by φ ( s λ s ∗ ν ) = δ λ,ν µ ( R λ ) . (9) .Proof. Since µ is a probability measure, φ is positive and φ (1) = 1. Thus φ is a state on C ∗ (Λ).To see that φ is a KMS state, first we compute, for λ ∈ Λ µ ( R λ ) = Z D λ d ( µ ◦ τ λ ) dµ dµ = Z D λ Φ τ λ dµ = Φ τ λ µ ( D λ ) = C − ωd ( λ ) µ ( R s ( λ ) ) (10)So φ satisfies the following: φ ( s λ s ∗ ν ) = δ λ,ν C − ωd ( λ ) φ ( s s ( λ ) ) for all λ, ν ∈ Λ . Then Proposition 3.1(b) of [18] implies that φ is a KMS state of ( C ∗ (Λ) , α ) at the inverse temperature β = ω .Note that by Corollary 3.12, every Λ-semibranching system on Λ ∞ with constant Radon-Nikodym deriva-tive of the form specified in Theorem 3.13 is endowed with the Perron-Frobenius measure M , hence if X = Λ ∞ , Theorem 3.13 specializes to the Λ-semibranching system of Proposition 3.4.The dynamics defined on C ∗ (Λ) in Theorem 3.13 is very similar to the preferred dynamics α r on C ∗ (Λ),where r = ln ρ (Λ). (See Section 2.4). Thus Proposition 3.4 with the preferred dynamics gives an alternateproof of the first statement of Theorem 11.1 of [19]. 17 orollary 3.14. ([19] Theorem 11.1) Let Λ be a strongly connected finite k -graph. Let α r be the preferreddynamics given by α rt = γ ρ (Λ) it . Then there is a unique KMS state φ of the system ( C ∗ (Λ) , α r ) at the inverse temperature β = 1 , given by φ ( s µ s ∗ ν ) = δ µ,ν M ( Z ( µ )) Proof.
Proposition 3.4 shows that the measure space (Λ ∞ , M ) with the prefixing maps { σ λ : Z ( s ( λ )) → Z ( λ ) } and the coding maps { σ m : Λ ∞ → Λ ∞ } forms a Λ-semibranching function system. Also it shows thatΦ σ λ = ρ (Λ) − d ( λ ) for λ ∈ Λ. Since ρ (Λ) ∈ (0 , ∞ ) k , Theorem 3.13 implies that the state φ on C ∗ (Λ) definedby φ ( s µ s ∗ ν ) = δ µ,ν M ( Z ( µ ))is a KMS state of ( C ∗ (Λ) , α r ).The uniqueness of φ follows from Theorem 11.1 of [19].We now apply Theorem 3.13 to our Cantor-type fractal subspace X with the measure µ associated tothe Hausdorff measure given in Theorem 3.8. Corollary 3.15.
Let Λ be a strongly connected finite k -graph with { , } vertex matrices A , ..., A k , suchthat A · · · A k is also a { , } matrix, as in Section 3.2. Let X = Ψ(Λ ∞ ) be the Cantor-type fractal subspaceof [0 , with the probability measure µ = H s ( X ) H s given in Theorem 3.8, where H s is the Hausdorff measureon X associated to its Hausdorff dimension s . Define the dynamics α ′ on C ∗ (Λ) by α ′ t = γ ρ (Λ) its , where γ is the gauge action on C ∗ (Λ) . Then the measure µ on X defines a KMS state of the system ( C ∗ (Λ) , α ′ ) at the inverse temperature β = s .Proof. As shown in Theorem 3.8, the map Ψ : Λ ∞ → X is an isometry with respect to the measures µ, M .Thus, since we know from Proposition 3.4 that the Radon-Nikodym derivatives on (Λ ∞ , M ) are given byΦ σ λ = ρ (Λ) − d ( λ ) , it follows that the Radon-Nikodym derivatives on ( X, µ ) are also of the formΦ τ λ = ρ (Λ) − d ( λ ) for all λ ∈ Λ.Then Corollary 3.14 implies that the formula given by φ ( s µ s ∗ ν ) = δ µ,ν M ( R µ )defines a KMS state φ of ( C ∗ (Λ) , α r ) at the inverse temperature β = 1, where α r is the preferred dynamics.Since α ′ t = α rts − , Lemma 2.1 of [16] implies that φ is a KMS state of ( C ∗ (Λ) , α ′ ) at the inverse temperature β = s , which givesthe desired result.We can also use the idea of Theorem 3.13 to construct KMS states associated to semibranching functionsystems with constant Radon-Nikodym derivatives. Corollary 3.16.
Let I be a finite index set. Let ( X, µ ) be a probability measure space that gives a semi-branching function system with prefixing maps { σ i : D i → R i } i ∈ I and coding map σ . Let A = ( A ij ) be a { , } -matrix satisfying χ D i = X j A ij χ R j . Let O A be the associated Cuntz-Krieger algebra on L ( X, µ ) as established in Proposition 2.5 of [28]. Supposethat there exists ω ∈ (0 , ∞ ) and C ∈ (0 , ∞ ) such that the Radon-Nikodym derivative is given by Φ σ i = C − ω for all i ∈ I . Let α be the dynamics on O A given by α t = γ C it , here γ is the gauge action on O A . Then the measure µ on X defines a KMS state φ of the system ( O A , α ) at the inverse temperature β = ω , by φ ( S i S ∗ j ) = δ i,j µ ( R i ) for i, j ∈ I . (11) Proof.
Recall that, if N = | I | , the Cuntz-Krieger algebra O A is the universal C ∗ -algebra generated by N partial isometries S i such that S ∗ i S i = X j A ij S j S ∗ j and X i S i S ∗ i = 1 . Moreover, Proposition 2.5 of [28] establishes that, if we define T i ∈ B ( L ( X, µ )) by T i ξ ( x ) = χ R i ( x ) (Φ σ i ( σ ( x ))) − / ξ ( σ ( x )) , then the operators T i generate a representation of O A .Since µ is a probability measure, φ is positive and φ (1) = 1. Thus φ is a state on O A .For each i ∈ I , we compute µ ( R i ) = Z D i Φ σ i dµ = C − ω µ ( D i ) . (12)To see that φ is a KMS state of ( O A , α ) at the inverse temperature β = ω , it suffices to show that φ ( S ∗ i S i ) = C ω φ ( S i S ∗ i ) for each i ∈ I .Using (2.8) and (2.9) of [28], and (12), we obtain φ ( S ∗ i S i ) = X j A ij φ ( S j S ∗ j ) = X j A ij µ ( R j ) = µ ( D i ) = C ω µ ( R i ) = C ω φ ( S i S ∗ i ) . Thus φ is a KMS state of ( O A , α ) at the inverse temperature β = ω . Remark . When we apply Corollary 3.16 to the representation of O A on L (Λ A , µ A ), for the Cantor setΛ A described in (2.1) of [28], we recover the KMS state of Corollary 2.20 in [28]. L (Λ ∞ , M ) We now proceed to construct an orthonormal decomposition of L (Λ ∞ , M ), which we call a wavelet decom-position , following Section 3 of [28]. Instead of obtaining our wavelets by scaling and translating a basicfamily of wavelet functions, our wavelet decomposition is constructed by applying (some of) the operators S λ of Theorem 3.5 to a basic family of functions in L (Λ ∞ , M ).While we can use the same procedure to obtain a family of orthonormal functions in L ( X, µ ) wheneverwe have a Λ-semibranching function system on (
X, µ ), we cannot establish in general that this orthonormaldecomposition densely spans L ( X, µ ) – we have no analogue of Lemma 4.1 for general Λ-semibranchingfunction systems. Moreover, by Corollary 3.12, every Λ-semibranching system on Λ ∞ with constant Radon-Nykodim derivative is endowed with the Perron-Frobenius measure. Thus, in this section, we restrict our-selves to the case of (Λ ∞ , M ). We also note that our proofs in this section follow the same ideas found inthe proof of Theorem 3.2 of [28].For a path λ ∈ Λ, let Θ λ denote the characteristic function of Z ( λ ) ⊆ Λ ∞ . Recall that M is the uniqueBorel probability measure on Λ ∞ satisfying (1). Lemma 4.1.
Let Λ be a strongly connected k -graph. Then the span of the set S := { Θ λ : d ( λ ) = ( n, n, . . . , n ) for some n ∈ N } is dense in L (Λ ∞ , M ) . roof. Let µ ∈ Λ. We will show that we can write Θ µ as a linear combination of functions from S .Suppose d ( µ ) = ( m , . . . , m k ). Let m = max { m i } and let n = ( m, m, . . . , m ) − d ( µ ). Let C µ = { λ ∈ Λ : r ( λ ) = s ( µ ) , d ( λ ) = n } . In words, C µ consists of the paths that we could append to µ such that µλ ∈ S : if λ ∈ C µ then the product µλ is defined and d ( µλ ) = d ( µ ) + d ( λ ) = ( m, m, . . . , m ) . Observe that, if λ, λ ′ ∈ C µ and µλ = µλ ′ , the factorization property tells us that λ = λ ′ . Similarly,since d ( µλ ) = d ( µλ ′ ) = ( m, . . . , m ), if x ∈ Z ( µλ ) ∩ Z ( µλ ′ ) then the fact that x (0 , ( m, . . . , m )) is well definedimplies that x (0 , ( m, . . . , m )) = µλ = µλ ′ ⇒ λ = λ ′ . It follows that if λ, λ ′ ∈ C µ , then Z ( µλ ) ∩ Z ( µλ ′ ) = ∅ . Since every infinite path x ∈ Z ( µ ) has a well-defined“first segment” of shape ( m, . . . , m ) – namely x (0 , ( m, . . . , m )) – every x ∈ Z ( µ ) must live in Z ( µλ ) forprecisely one λ ∈ C µ . Thus, we can write Z ( µ ) as a disjoint union, Z ( µ ) = ∪ λ ∈ C µ Z ( µλ ) . It follows that Θ µ = P λ ∈ C µ Θ µλ , so the span of functions in S includes the characteristic functions ofcylinder sets. Since the cylinder sets Z ( µ ) form a basis for the topology on Λ ∞ with respect to which M isa Borel measure, it follows that the span of S is dense in L (Λ ∞ , M ) as claimed.Since the span of the functions in S is dense in L (Λ ∞ , M ), we will show how to decompose span S asan orthogonal direct sum, span S = V , Λ ⊕ ∞ M j =0 W j, Λ , where we can construct W j, Λ for each j > W , Λ and (some of) the operators S λ discussed in Section 3.1. The construction of W , Λ is similar to that given in Section 3 of [28] for the caseof a directed graph.We begin by setting V ,λ equal to the subspace spanned by the functions { Θ v : v ∈ Λ } . Indeed thefunctions { Θ v : v ∈ Λ } form an orthogonal set in L (Λ ∞ , M ) , whose span includes those functions that areconstant on Λ ∞ : Z Λ ∞ Θ v Θ w dM = δ v,w M ( Z ( v ))= δ v,w x Λ v , and X v ∈ Λ Θ v ( x ) ≡ . Thus, the set { √ x Λ v Θ v : v ∈ Λ } is an orthonormal set in S. We define V , Λ := span { p x Λ v Θ v : v ∈ Λ } . To construct W , Λ , let v ∈ Λ be arbitrary. Let D v = { λ : d ( λ ) = (1 , . . . ,
1) and r ( λ ) = v } , and write d v for | D v | (note that by our hypothesis that Λ is a finite k -graph we have d v < ∞ ).Define an inner product on C D v by h ~v, ~w i = X λ ∈ D v v λ w λ ρ (Λ) ( − ,..., − x Λ s ( λ ) (13)20nd let { c m,v } d v − m =1 be an orthonormal basis for the orthogonal complement of (1 , . . . , ∈ C D v with respectto this inner product.For each pair ( m, v ) with m ≤ d v − v a vertex in Λ , define f m,v = X λ ∈ D v c m,vλ Θ λ . Note that by our definition of the measure M on Λ ∞ , since the vectors c m,v are orthogonal to (1 , . . . ,
1) inthe inner product (13), we have Z Λ ∞ f m,v dM = X λ ∈ D v c m,vλ M ( Z ( λ ))= X λ ∈ D v c m,vλ ρ (Λ) ( − ,..., − x Λ s ( λ ) = 0for each ( m, v ). Moreover, the arguments of Lemma 4.1 tell us that Θ λ Θ λ ′ = δ λ,λ ′ Θ λ for any λ, λ ′ with d ( λ ) = d ( λ ′ ) = (1 , . . . , λ ∈ D v , λ ′ ∈ D v ′ for v = v ′ , we have Θ λ Θ λ ′ = 0. It followsthat Z Λ ∞ f m,v f m ′ ,v ′ dM = δ v,v ′ X λ ∈ D v c m,vλ c m ′ ,vλ M ( Z ( λ ))= δ v,v ′ δ m,m ′ since the vectors { c m,v } form an orthonormal set with respect to the inner product (13). Thus, the functions { f m,v } are an orthonormal set in L (Λ ∞ , M ). We define W , Λ := span { f m,v : v ∈ Λ , ≤ m ≤ d v } . Note that V is orthogonal to W , Λ . To see this, let g ∈ V be arbitrary, so g = P v ∈ Λ g v Θ v with g v ∈ C for all v . Then Z Λ ∞ f m,v ′ ( x ) g ( x ) dM = δ v ′ ,v g v X λ c m,vλ M ( Z ( λ ))= 0 , since P λ c m,vλ M ( Z ( λ )) = 0 for all fixed v, m . Thus, g is orthogonal to every basis element f m,v of W , Λ .The basis { f m,v : v ∈ Λ , ≤ m ≤ d v } for W , Λ is the analogue for k -graphs of the graph wavelets of [28]. As the following Theorem shows, by shifting these functions using the operators S λ of Theorem3.5, we obtain an orthonormal basis for L (Λ ∞ , M ), and thus k -graph wavelets associated to a separablerepresentation of C ∗ (Λ). Theorem 4.2.
Let Λ be a strongly connected finite k -graph. For each fixed j ∈ N + and v ∈ Λ , let C j,v := { λ ∈ Λ : s ( λ ) = v, d ( λ ) = ( j, j, . . . , j ) } , and let S λ be the operator on L (Λ ∞ , M ) described in Theorem 3.5. Then { S λ f m,v : v ∈ Λ , λ ∈ C j,v , ≤ m ≤ d v } is an orthonormal set, and moreover, if λ ∈ C j,v , µ ∈ C i,v ′ for < i < j, we have Z Λ ∞ S λ f m,v S µ f m ′ ,v ′ dM = 0 ∀ m, m ′ . It follows that defining W j, Λ := span { S λ f m,v : v ∈ Λ , λ ∈ C j,v , ≤ m ≤ d v } , or j ≥ , we obtain an orthonormal decomposition L (Λ ∞ , M ) = span S = V ⊕ ∞ M j =0 W j, Λ . Proof.
We first observe that if s ( λ ) = v , then S λ f m,v = X µ ∈ D v c m,vµ ρ (Λ) d ( λ ) / Θ λµ , because the Radon-Nikodym derivatives Φ σ λ are constant on Z ( s ( λ )) for each λ ∈ Λ, thanks to Proposition3.4. In particular, if d ( λ ) = 0 then S λ f m,v = f m,v . Thus, if d ( λ ) = d ( λ ′ ) = ( j, . . . , j ), the factorizationproperty and the fact that d ( λµ ) = d ( λ ′ µ ′ ) = ( j + 1 , . . . , j + 1) for every µ ∈ D s ( λ ) , µ ′ ∈ D s ( λ ′ ) implies thatΘ λµ Θ λ ′ µ ′ = δ λ,λ ′ δ µ,µ ′ ∀ µ ∈ D s ( λ ) , µ ′ ∈ D s ( λ ′ ) . In particular, S λ f m,v S λ ′ f m ′ ,v ′ = 0 unless λ = λ ′ (and hence v = v ′ ). Moreover, Z Λ ∞ S λ f m,v S λ f m ′ ,v dM = X µ ∈ D v c m,vµ c m ′ ,vµ ρ (Λ) d ( λ ) M ( Z ( λµ ))= X µ ∈ D v c m,vµ c m ′ ,vµ ρ (Λ) − d ( µ ) x Λ s ( µ ) = δ m,m ′ , by the definition of the vectors c m,vµ , since d ( µ ) = (1 , , . . . ,
1) for each µ ∈ D v .Now, suppose λ ∈ C ,v . Observe that S λ f m,v f m ′ ,v ′ is nonzero only when v ′ = r ( λ ), and in this case wehave Z Λ ∞ S λ f m,v f m ′ ,v ′ dM = X µ ∈ D v c m,vµ c m ′ ,v ′ λ ρ (Λ) d ( λ ) / M ( Z ( λµ ))= c m ′ ,v ′ λ ρ (Λ) − d ( λ ) / X µ ∈ D v c m,vµ ρ (Λ) − d ( µ ) x Λ s ( µ ) = 0 . Thus, W , Λ is orthogonal to W , Λ .In more generality, suppose that λ ∈ C j,v , λ ′ ∈ C i,v ′ , j > i ≥
1. We observe that S λ f m,v S λ ′ f m ′ ,v ′ isnonzero only when λ = λ ′ ν with ν ∈ C j − i,v , so we have S λ f m,v S λ ′ f m ′ ,v ′ = S λ ′ ( S ν f m,v ) S λ ′ f m ′ ,v ′ . Consequently, Z Λ ∞ S λ f m,v S λ ′ f m ′ ,v ′ dM = Z Λ ∞ S λ ′ ( S ν f m,v ) S λ ′ f m ′ ,v ′ dM = Z Λ ∞ ( S ν f m,v ) S ∗ λ ′ S λ ′ f m ′ ,v ′ dM = Z Λ ∞ ( S ν f m,v ) f m ′ ,v ′ dM = X µ ∈ D v c m,vµ c m ′ ,v ′ ν ρ (Λ) d ( ν ) / M ( Z ( νµ ))= c m ′ ,v ′ ν ρ (Λ) − d ( ν ) / X µ ∈ D v c m,vµ ρ (Λ) − d ( µ ) x Λ s ( µ ) = 0 . Thus, the sets W j, Λ are mutually orthogonal as claimed.22n the setting of Theorem 3.8, where the infinite path space Λ ∞ embeds as a fractal subset X of [0 , , following [20] and [28], the polynomials of degree m P m restricted to Λ ∞ will form a subspace of dimension m + 1 of L ( X, µ ). For every integer m ≥ V , Λ ,m to be those functions in L ( X, µ )which, when restricted to the cylinder sets Z ( v ) , v ∈ Λ , are a polynomial of degree ≤ m. Such functionscan be thought of as generalized m -splines. Then as in Section 3 of [28], it is possible to form generalizedmultiresolution analyses from the space V , Λ ,m by using linear algebra and the partial isometries S λ for λ ∈ C j,v . References [1] S. Allen, D. Pask and A. Sims,
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E-mail address : [email protected], [email protected], [email protected] Sooran Kang : Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054,New Zealand.
E-mail address , [email protected]@maths.otago.ac.nz