Wavelets and spectral triples for higher-rank graphs
In this paper, we present two new ways to associate a spectral triple to a higher-rank graph . Moreover, we prove that these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary -Bratteli diagrams, to associate a family of ultrametric Cantor sets to a finite, strongly connected higher-rank graph . Then we show that under mild hypotheses, the Pearson-Bellissard spectral triples of such Cantor sets have a regular -function, whose abscissa of convergence agrees with the Hausdorff dimension of the Cantor set, and that the measure induced by the associated Dixmier trace agrees with the measure on the infinite path space of which was introduced by an Huef, Laca, Raeburn, and Sims. Furthermore, we prove that is a rescaled version of the Hausdorff measure of the ultrametric Cantor set.
From work of Julien and Savinien, we know that for -regular Pearson-Bellissard spectral triples, the eigenspaces of the associated Laplace-Beltrami operator constitute an orthogonal decomposition of ; we show that this orthogonal decomposition refines the wavelet decomposition of Farsi et al. In addition, we generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph -algebras, and prove that the wavelet decomposition of Farsi et al. describes the eigenspaces of its Dirac operator.
2010 Mathematics Subject Classification: 46L05, 46L87, 58J42.
Key words and phrases: Spectral triple, wavelets, higher-rank graph, Laplace-Beltrami operator, Hausdorff measure, -function, Dixmier trace, -Bratteli diagram, ultrametric Cantor set.
- 1 Introduction
- 2 Higher-rank graphs and ultrametric Cantor sets
- 3 Spectral triples and Hausdorff dimension for ultrametric higher-rank graph Cantor sets
- 4 Eigenvectors of Laplace-Beltrami operators and the wavelets of 
- 5 Consani-Marcolli spectral triple for strongly connected higher-rank graphs
Both spectral triples and wavelets are algebraic structures which encode geometrical information. However, to our knowledge our earlier paper  was the first to highlight a connection between wavelets and spectral triples. In this paper, we expand the correspondence established in  between wavelets and spectral triples for the Cuntz algebras to the setting of higher-rank graphs. We also introduce a new spectral triple for the -algebra of a higher-rank graph , and establish its compatibility with the wavelet decomposition of .
The objective of the initial work on wavelet analysis, pioneered by Mallat , Meyer , and Daubechies  in the late 1980’s, was to identify orthonormal bases or frames for which behaved well under compression algorithms, for the purpose of signal or image storage. A few years later, Jonsson  and Strichartz  began to study orthonormal bases of more general Hilbert spaces which arise from dilations and translations of finite sets of functions in . When is a fractal space, Jonsson and Strichartz’ wavelets reflect the self-similar structure of ; these fractal wavelets were the inspiration for the wavelet decompositions associated to graphs  and higher-rank graphs  which are the wavelets we focus on in this paper.
Wavelet analysis has many applications in various areas of mathematics, physics and engineering. For example, it has been used to study -adic spectral analysis , pseudodifferential operators and dynamics on ultrametric spaces [35, 34], and the theory of quantum gravity [17, 2].
The idea of a spectral triple was introduced by Connes in  as the noncommutative generalization of a compact Riemannian manifold. A spectral triple consists of a representation of a pre--algebra on Hilbert space , together with a Dirac-type operator on , which satisfy certain commutation relations. In the case when is the algebra of smooth functions on a compact spin manifold, Connes showed  that the algebraic structure of the associated spectral triple suffices to reconstruct the Riemannian metric on .
In addition to spin manifolds, Connes studied spectral triples for the triadic Cantor set and Julia set in . Shortly thereafter, Lapidus  suggested studying spectral triples where is a commutative algebra of functions on a fractal space , and investigating which aspects of the geometry of are recovered from the spectral triple. Such spectral triples usually recover the fractal dimension of , and in particularly nice cases [47, 9, 23, 41] they also recover the metric structure of the fractal space.
As we show in Theorem 4.2 below, the spectral triples of Pearson and Bellissard  are closely related to the wavelets of [43, 20] (see Equation (1) below). Here, is the algebra of Lipschitz continuous functions on an ultrametric Cantor set constructed from a weighted tree. Since the wavelet decompositions of [43, 20] involve the infinite-path space of a graph or higher-rank graph , we first explain in Section 2 below how to construct a family of ultrametric Cantor sets from a given higher-rank graph . A -dimensional generalization of directed graphs, higher-rank graphs (also called -graphs) were introduced by Kumjian and Pask in  to provide computable, combinatorial examples of -algebras. The combinatorial character of -graph -algebras has facilitated the analysis of their structural properties, such as simplicity and ideal structure [48, 49, 15, 33, 7], quasidiagonality  and KMS states [28, 27, 26]. In particular, results such as [52, 6, 5, 45] show that higher-rank graphs often provide concrete examples of -algebras which are relevant to Elliott’s classification program for simple separable nuclear -algebras.
Having laid the groundwork in Section 2 for associating a Pearson-Bellissard spectral triple to a -graph, we study the -function and Dixmier trace of this spectral triple in Section 3. Inspired by Julien and Savinien , we use Bratteli diagrams (more precisely, the stationary -Bratteli diagrams which we introduce in Definition 2.5) to facilitate this analysis. Theorem 3.8 establishes that the -function of the spectral triple associated to the ultrametric Cantor set has abscissa of convergence ; we later show in Corollary 3.13 that is also the Hausdorff dimension of . Weaker analogues of both of these results were obtained in ; we relied heavily on the Bratteli diagram structure to obtain our stronger results.
The structure of the stationary -Bratteli diagrams also enables us to prove (in Theorem 3.9) that in our setting, the Pearson-Bellissard spectral triples satisfy the crucial hypothesis of -regularity invoked in . This implies the existence, for each , of a Dixmier trace on , and an associated probability measure (also denoted ) on . Corollary 3.10 shows that the probability measures agree with the Borel probability measure on which was identified in Proposition 8.1 of  and which we used in  to construct a wavelet decomposition of . Section 3 concludes with Theorem 3.14, which proves that, after rescaling, the Hausdorff measure of also agrees with and .
Section 4 presents the promised connection between the Pearson-Bellissard spectral triples and the wavelet decomposition of from . In , four of the authors of the current paper constructed an orthogonal decomposition
where each subspace111The subspaces denoted in this paper by were labeled for in Theorem 4.2 of . is constructed from by means of generalized “scaling and translation” operators which reflect the (higher-rank) graph structure of . The geometric structure of this orthogonal decomposition led the authors of  to label it a wavelet decomposition, following Marcolli and Paolucci , Jonsson  and Strichartz .
As Julien and Savinien show in Theorem 4.3 of , the Pearson-Bellissard spectral triples give rise to another orthogonal decomposition of . To be precise, given an ultrametric Cantor set whose associated Pearson-Bellissard spectral triple is -regular, write for the associated Dixmier trace. Pearson and Bellissard constructed in  an associated family of Laplace-Beltrami operators on . Theorem 4.3 of  shows that the eigenspaces of are independent of and they form an orthogonal decomposition of ; moreover, when arises from a Bratteli diagram, the eigenspaces of are labeled by the finite paths in the Bratteli diagram.
Theorem 4.2 of the current paper shows that these two orthogonal decompositions are compatible. More precisely, it proves that
so the eigenspaces of the Laplace-Beltrami operators refine the wavelet decomposition of . The remainder of Section 4 presents some variations of the wavelet decomposition of  which are also related to the Pearson-Bellissard spectral triples and their associated Laplace-Beltrami operators.
Inspired by this close relationship between spectral triples and wavelets on the infinite path space of a -graph, one might (naturally) ask whether there is a connection between the wavelets of  or  and any of the known noncommutative spectral triples [46, 13, 21, 22] associated to graph -algebras. Of the spectral triples listed above, only the one given by Consani and Marcolli in  uses for the Hilbert space.
We conclude this paper in Section 5 by establishing a link between Consani-Marcolli type spectral triples for graph or -graph -algebras and ’s wavelet decomposition of . More precisely, given a -graph , we construct in Theorem 5.4 a spectral triple , where is a dense (noncommutative) subalgebra of . Theorem 5.5 then establishes that the eigenspaces of the Dirac operator of this spectral triple agree with the wavelet decomposition of .
We would like to thank Palle Jorgensen for helpful discussions about Bratteli diagrams associated to higher-rank graphs. E.G. was partially supported by the SFB 878 “Groups, Geometry, and Actions” of the Westfälische-Wilhelms-Universität Münster. J.P. was partially supported by a grant from the Simons Foundation (#316981).
2 Higher-rank graphs and ultrametric Cantor sets
In this section, we review the basic definitions and results that we will need about directed graphs, higher-rank graphs, (weighted/stationary) Bratteli diagrams, infinite path spaces, and (ultrametric) Cantor sets. Throughout this article, will denote the non-negative integers.
2.1 Bratteli diagram
A directed graph is given by a quadruple , where is the set of vertices of the graph, is the set of edges, and denote the range and source of each edge. A vertex in a directed graph is a sink if we say is a source if .
 A Bratteli diagram is a directed graph with vertex set , and edge set , where consists of edges whose source vertex lies in and whose range vertex lies in , and and are finite sets for all .
For a Bratteli diagram , define a sequence of adjacency matrices of for , where
where by we denote the cardinality of the set . A Bratteli diagram is stationary if are the same for all . We say that is a finite path of if there exists such that for , and in that case the length of , denoted by , is .
In the literature, Bratteli diagrams traditionally have and ; our edges point the other direction for consistency with the standard conventions for higher-rank graphs and their -algebras.
It is also common in the literature to require and to call this vertex the root of the Bratteli diagram; we will NOT invoke this hypothesis in this paper.
Given a Bratteli diagram , denote by the set of all of its infinite paths:
For each finite path in with , and , define the cylinder set by
The collection of all cylinder sets forms a compact open sub-basis for a locally compact Hausdorff topology on and cylinder sets are clopen; we will always consider with this topology.
The following proposition will tell us when is a Cantor set; that is, a totally disconnected, compact, perfect topological space.
(Lemma 6.4. of ) Let be a Bratteli diagram such that has no sinks outside of , and no sources. Then is a totally disconnected compact Haudorff space, and the following statements are equivalent:
The infinite path space of is a Cantor set;
For each infinite path in and each there is an infinite path with
For each and each there is and such that there is a path from to and
2.2 Higher-rank graphs and stationary -Bratteli diagrams
Let be matrices with non-negative integer entries. The stationary -Bratteli diagram associated to the matrices , which we will call , is given by a filtered set of vertices and a filtered set of edges , where the edges in go from to , such that:
For each , consists of vertices, which we will label .
When , there are edges whose range is the vertex of and whose source is the vertex of .
In other words, the matrix determines the edges with source in and range in ; then the matrix determines the edges with source in and range in ; etc. The matrix determines the edges with source in and range in , and the matrix determines the edges with range in and source in .
Just as a directed graph has an associated adjacency matrix which also describes a stationary Bratteli diagram , the higher-dimensional generalizations of directed graphs known as higher-rank graphs or -graphs give us commuting matrices and hence a stationary -Bratteli diagram.
We use the standard terminology and notation for higher-rank graphs, which we review below for the reader’s convenience.
 A -graph is a countable small category equipped with a degree functor222We view as a category with one object, namely , and with composition of morphisms given by addition. satisfying the factorization property: whenever is a morphism in such that , there are unique morphisms such that , and .
We use the arrows-only picture of category theory; thus, means that is a morphism in . For , we write
When , is the set of objects of , which we also refer to as the vertices of .
Let identify the range and source of each morphism, respectively. For a vertex, we define
We say that is finite if for all , and we say is source-free or has no sources if for all and .
For , write for the th standard basis vector of , and define a matrix by
We call the th adjacency matrix of . Note that the factorization property implies that the matrices commute.
Despite their formal definition as a category, it is often useful to think of -graphs as -dimensional generalizations of directed graphs. In this interpretation, is the set of “edges of color ” in . The factorization property implies that each can be written as a concatenation of edges in the following sense: A morphism with can be thought of as a -dimensional hyper-rectangle of dimension . Any minimal-length lattice path in through the rectangle lying between 0 and corresponds to a choice of how to order the edges making up , and hence to a unique decomposition or “factorization” of . For example, the lattice path given by walking in straight lines from to to to , and so on, corresponds to the factorization of into edges of color 1, then edges of color 2, then edges of color 3, etc.
For any directed graph , the category of its finite paths is a 1-graph; the degree functor takes a finite path to its length . Example 2.7 below gives a less trivial example of a -graph. The -graphs of Example 2.7 are also fundamental to the definition of the space of infinite paths in a -graph.
For , let be the small category with
If we define by , then is a -graph with degree functor .
Let be a -graph. An infinite path of is a -graph morphism
we write for the set of infinite paths in . For each , we have a map given by
for and .
Given , we often write for the terminal vertex of . This convention means that an infinite path has a range but not a source.
We equip with the topology generated by the sub-basis of compact open sets, where
Note that we use the same notation for a cylinder set of and a cylinder set of in Definition 2.3 since is homeomorphic to for a finite, source-free -graph . See the details in Proposition 2.10. Remark 2.5 of  establishes that, with this topology, is a locally compact Hausdorff space.
For any and any with , we write for the unique infinite path such that and . If , the maps and are local homeomorphisms which are mutually inverse:
although the domain of is .
Informally, one should think of as “chopping off” the initial segment of length , and the map as “gluing on” to the front of . By “front” and “initial segment” we mean the range of , since an infinite path has no source.
We can now state precisely the connection between -graphs and stationary -Bratteli diagrams.
Let be a finite, source-free -graph with adjacency matrices . Denote by the stationary -Bratteli diagram associated to the matrices . Then is homeomorphic to .
Fix and write . Then the factorization property for implies that there is a unique sequence
such that with . (See the details in Remark 2.2 and Proposition 2.3 of ). Since there is a unique way to write as a composable sequence of edges with , we have
where the th edge has color . Thus, for each , corresponds to an entry in , and hence
Conversely, given , we construct an associated -graph infinite path as follows. To we associate a sequence of finite paths in , where
is the unique morphism in of degree represented by the sequence of composable edges .
Recall from  Remark 2.2 that a morphism is uniquely determined by . Thus, the sequence determines :
The map is easily checked to be a bijection which is inverse to the map .
Moreover, for any , , and any
with , both of these bijections preserve the cylinder set . In particular, these bijections preserve the “square” cylinder sets associated to paths with for some . (If then we interpret as meaning that is a vertex in .) From the proof of Lemma 4.1 of , any cylinder set can be written as a disjoint union of square cylinder sets, and therefore the square cylinder sets generate the topology on . We deduce that and are homeomorphic, as claimed. ∎
Thanks to Proposition 2.10, we will usually identify the infinite path spaces and , denoting this space by the symbol which is most appropriate for the context. In particular, the Borel structures on and are isomorphic, and so any Borel measure on induces a unique Borel measure on and vice versa.
The bijection of Proposition 2.10 between infinite paths in the -graph and in the associated Bratteli diagram does not extend to finite paths. While any finite path in the Bratteli diagram determines a finite path, or morphism, in , not all morphisms in have a representation in the Bratteli diagram. For example, if is a morphism of degree in a -graph () with , the composition is a morphism in the -graph which cannot be represented as a path on the Bratteli diagram. However, the proof of Proposition 2.10 above establishes that “rainbow” paths in – morphisms of degree for some and – can be represented uniquely as paths of length in the Bratteli diagram.
2.3 Ultrametrics on
Although the Cantor set is unique up to homeomorphism, different metrics on it can induce quite different geometric structures. In this section, we will focus on Bratteli diagrams for which the infinite path space is a Cantor set. In this setting, we construct ultrametrics on by using weights on . To do so, we first need to introduce some definitions and notation.
A metric on a Cantor set is called an ultrametric if induces the Cantor set topology and satisfies
The inequality of (2) often called the strong triangle inequality.
Let be a Bratteli diagram. Denote by the set of finite paths in with range in . For any , we write
Given two (finite or infinite) paths in , we say is a sub-path of if there is a sequence of edges, with , such that .
For any two infinite paths , we define to be the longest path such that is a sub-path of and . We write when no such path exists.
A weight on a Bratteli diagram is a function such that
For any vertex .
If is a sub-path of , then .
A Bratteli diagram with a weight often called a weighted Bratteli diagram and denoted by .
Observe that the third condition implies that for any path (finite or infinite),
The concept above of a weight was inspired by Definition 2.9 of ; indeed, if one denotes a weight in the sense of  Definition 2.9 by , and defines , then is a weight on in the sense of Definition 2.14 above.
Let be a weighted Bratteli diagram such that is a Cantor set. The function given by
is an ultrametric on .
It is evident from the defining conditions of a weight that is symmetric and satisfies . Since the inequality (2) is stronger than the triangle inequality, once we show that satisfies the ultrametric condition (2) it will follow that is indeed a metric.
To that end, first suppose that ; in other words, and have no common sub-path. This implies that for any , at least one of and must be 1, so
as desired. Now, suppose that . If for all then we are done. On the other hand, if there exists such that , then the maximal common sub-path of and must be longer than that of and . This implies that
consequently, in this case as well we have .
Finally, we observe that the metric topology induced by agrees with the cylinder set topology. To see this, fix and . Then the conditions in Definition 2.14 imply that there is a smallest such that . Then,
so cylinder sets of and open balls induced by the metric agree. (If then we interpret as .) ∎
2.4 Strongly connected higher-rank graphs
When is a finite -graph whose adjacency matrices satisfy some additional properties, there is a natural family of weights on the associated Bratteli diagram which induce ultrametrics on the infinite path space . We describe these additional properties on and the formula of the weights below.
A -graph is strongly connected if, for all , .
In Lemma 4.1 of , an Huef et al. show that a finite -graph is strongly connected if and only if the adjacency matrices of form an irreducible family of matrices. Also, Proposition 3.1 of  implies that if is a finite strongly connected -graph, then there is a unique positive vector such that and for all ,
where denotes the spectral radius of . We call the Perron-Frobenius eigenvector of . Moreover, an Huef et al. constructed a Borel probability measure on in Proposition 8.1 of  when is finite, strongly connected -graph. The measure on is given by
where is the Perron-Frobenius eigenvector of and , and for ,
for , where with and is the Perron-Frobenius eigenvector of .
In the proof that follows, we rely heavily on the identification between and of Proposition 2.10. We also use the observation from Remark 2.11 that every finite path in corresponds to a unique finite path .
Let be a finite, strongly connected -graph with adjacency matrices . Then the infinite path space is a Cantor set whenever .
We let ; it is a matrix whose entries are indexed by , and its spectral radius is . We assume that is not a Cantor set, and will prove that the spectral radius of is at most , hence proving the Proposition.
Since is compact Hausdorff and totally disconnected, but not a Cantor set, it has an isolated point . We write for the increasing sequence of finite paths in which are sub-paths of . If , then and (thinking of as an element of ) with occurrences of . Since is an isolated point, there exists such that for all , . Without loss of generality, we can assume that is a multiple of , so that . For , we write , with and , so that , with occurrences of .
Our hypothesis that is an isolated point implies that for all , is the unique path of degree whose range is . This, in turn, implies that for all , we have is equal to for a single , and otherwise. In other words, if we consider the column vector which is at the vertex and else, we have that
Note that for each with , is the label of the only non-zero entry in row of the matrix . Since each entry in the sequence is completely determined by a finite set of inputs – namely, the previous entry in the sequence, and the entries of the matrices – and the set of vertices is finite, the sequence is eventually periodic. Let be a period for this sequence. Then is also a period, so there exists such that for all we have
We average along one period and define
so is an eigenvector of with eigenvalue , with nonnegative entries.
Since is strongly connected by hypothesis, Lemma 4.1 of  implies that there exists a matrix which is a finite sum of finite products of the matrices and which has positive entries. This matrix commutes with , and therefore
and so is an eigenvector of with eigenvalue . Since is positive and is nonnegative, is positive. Therefore, we can apply Lemma 3.2 of  and conclude that . ∎
The proof of Proposition 2.17 simplifies considerably if we add the hypothesis that each row sum of each adjacency matrix is at least 2. In this case, any finite path in the Bratteli diagram has at least two extensions and . In terms of neighbourhoods, this means that each clopen set contains at least two disjoint non-trivial sets . It is therefore impossible to have a cylinder set consist of a single point. Therefore, there is no isolated point in , and the path space is a Cantor set.
Let be a finite, strongly connected -graph with adjacency matrices . For with , write for some with . For each define by
where is the unimodular Perron-Frobenius eigenvector for . If the spectral radius of satisfies , then is a weight on .
Recall that , and for all ; thus, for any , , and the first condition of Definition 2.14 is satisfied. Since for all and ,
Thus the second condition of Definition 2.14 holds. To see the third condition, we observe that it is enough to show that for any edge of any color with . Note that if for and , so that , then
Here the second equality follows since is an eigenvector for with eigenvalue , and the final inequality holds because and and consequently
Our primary application for the results of this section is the following.
Let be a finite, strongly connected -graph with adjacency matrices and let be the spectral radius for , . Suppose that for all . Let be the associated weighted -stationary Bratteli diagram given in Proposition 2.19. Then the infinite path space is an ultrametric Cantor set with the metric induced by the weight .
3 Spectral triples and Hausdorff dimension for ultrametric higher-rank graph Cantor sets
Proposition 8 of  (also see Proposition 3.1 of ) gives a recipe for constructing an even spectral triple for any ultrametric Cantor set induced by a weighted tree. We begin this section by explaining how this construction works in the case of the ultrametric Cantor sets associated to a finite strongly connected -graph as in the previous section. In Section 3.1, we investigate the -function and Dixmier trace of these spectral triples, and Section 3.2 computes the Hausdorff measure and Hausdorff dimension of the underlying Cantor sets.
To be precise, consider the Cantor set with the ultrametric induced by the weight of Equation (5). (Because of Proposition 2.10, we will identify the infinite path spaces of and of , and use either or to denote this space, depending on the context.) Under additional (but mild) hypotheses, Theorem 3.8 establishes that the -function of the associated spectral triple has abscissa of convergence . After proving in Theorem 3.9 that the Dixmier trace of the spectral triple induces a well-defined measure on , Corollary 3.10 establishes that agrees with the measure introduced in  and used in  to construct a wavelet decomposition of . Finally, Theorem 3.14 shows that in many cases, both and agree with the Hausdorff measure on the ultrametric Cantor set
Analogues of Theorems 3.8 and 3.9 were proved in Section 3 of  for stationary Bratteli diagrams (equivalently, directed graphs) with primitive adjacency matrices. However, even for directed graphs our results in this section are stronger than those of , since in this setting, our hypotheses are equivalent to saying that the adjacency matrix is merely irreducible.
We begin by recalling the definition of a spectral triple.
Given a pre--algebra , a faithful -representation , and an unbounded operator on such that
we say that is an (odd) spectral triple. If has a grading operator – a self-adjoint unitary – such that
we say that is an even spectral triple.
Sometimes the representation is also included in the notation for a spectral triple.
To any spectral triple, even or odd, we associate a -function and Dixmier trace as follows.
The -function associated to a spectral triple is given by