arXiv subjects

Carla Farsi

Publications and source records attributed to Carla Farsi.

At least 37 records · Page 2Linked to original sources

arXiv2020

Approximating orbifold spectra using collapsing connected sums

For a closed Riemannian orbifold $O$, we compare the spectra of the Laplacian, acting on functions or differential forms, to the Neumann spectra of the orbifold with boundary given by a domain $U$ in $O$ whose boundary is a smooth manifold. Generalizing results of several authors, we prove that the metric of $O$ can be perturbed to ensure that the first $N$ eigenvalues of $U$ and $O$ are arbitrarily close to one another. This involves a generalization of the Hodge decomposition to the case of orbifolds with manifold boundary. Using these results, we study the behavior of the Laplace spectrum on functions or forms of a connected sum of two Riemannian orbifolds as one orbifold in the pair is collapsed to a point. We show that the limits of the eigenvalues of the connected sum are equal to those of the non-collapsed orbifold in the pair. In doing so, we prove the existence of a sequence of orbifolds with singular points whose eigenvalue spectra come arbitrarily close to the spectrum of a manifold, and a sequence of manifolds whose eigenvalue spectra come arbitrarily close to the eigenvalue spectrum of an orbifold with singular points. We also consider the question of prescribing the first part of the spectrum of an orientable orbifold.

math.DG↗

arXiv2019

Classifying spaces and Bredon (co)homology for transitive groupoids

We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.

math.AT↗

arXiv2019

Spectral triples and wavelets for higher-rank graphs

In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph $Λ$, via the infinite path space $Λ^\infty$ of $Λ$. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of $Λ^\infty$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph $Λ$. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are $ζ$-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure $μ$, and show that $μ$ is a rescaled version of the measure $M$ on $Λ^\infty$ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of $L^2(Λ^\infty, M)$ which was constructed by Farsi et al.

math.OA↗

arXiv2018

Ample groupoids: equivalence, homology, and Matui's HK conjecture

We investigate the homology of ample Hausdorff groupoids. We establish that a number of notions of equivalence of groupoids appearing in the literature coincide for ample Hausdorff groupoids, and deduce that they all preserve groupoid homology. We compute the homology of a Deaconu{Renault groupoid associated to k pairwisecommuting local homeomorphisms of a zero-dimensional space, and show that Matui's HK conjecture holds for such a groupoid when k is one or two. We specialise to k-graph groupoids, and show that their homology can be computed in terms of the adjacency matrices, using a chain complex developed by Evans. We show that Matui's HK conjecture holds for the groupoids of single vertex k-graphs which satisfy a mild joint-coprimality condition. We also prove that there is a natural homomorphism from the categorical homology of a k-graph to the homology of its groupoid.

math.OA↗

arXiv2018

Generalized gauge actions on $k$-graph $C^*$-algebras: KMS states and Hausdorff structure

For a finite, strongly connected $k$-graph $Λ$, an Huef, Laca, Raeburn and Sims studied the KMS states associated to the preferred dynamics of the $k$-graph $C^*$-algebra $C^*(Λ)$. They found that these KMS states are determined by the periodicity of $Λ$ and a certain Borel probability measure $M$ on the infinite path space $Λ^\infty$ of $Λ$. Here we consider different dynamics on $C^*(Λ)$, which arise from a functor $y: Λ\to \mathbb{R}_+$ and were first proposed by McNamara in his thesis. We show that the KMS states associated to McNamara's dynamics are again parametrized by the periodicity group of $Λ$ and a family of Borel probability measures on the infinite path space. Indeed, these measures also arise as Hausdorff measures on $Λ^\infty$, and the associated Hausdorff dimension is intimately linked to the inverse temperatures at which KMS states exist. Our construction of the metrics underlying the Hausdorff structure uses the functors $y: Λ\to \mathbb{R}_+$; the stationary $k$-Bratteli diagram associated to $Λ$; and the concept of exponentially self-similar weights on Bratteli diagrams.

math.OA↗

arXiv2018

Purely atomic representations of higher-rank graph C*-algebras

We study purely atomic representations of C*-algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent and we give necessary and sufficient conditions for a purely atomic representation to be irreducible in terms of the associated projection valued measure. We also investigate the relationship between purely atomic representations, monic representations and permutative representations, and we describe when a purely atomic representation admits a decomposition consisting of permutative representations.

math.OA↗

arXiv2018

Spectral triples for higher-rank graph $C^*$-algebras

In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(Λ)$ of a strongly connected finite higher-rank graph $Λ$. We generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras $C^*(Λ)$, and we prove that these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of $Λ$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. In particular, we prove that the wavelet decomposition of Farsi et al. describes the eigenspaces of the Dirac operator of this spectral triple.

math.OA↗

arXiv2018

Monic representations of finite higher-rank graphs

In this paper we define the notion of monic representation for the $C^*$-algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative $C^*$-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the $Λ$-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.

math.OA↗

arXiv2018

Representations of higher-rank graph $C^*$-algebras associated to $Λ$-semibranching function systems

In this paper, we discuss a method of constructing separable representations of the $C^*$-algebras associated to strongly connected row-finite $k$-graphs $Λ$. We begin by giving an alternative characterization of the $Λ$-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to construct a faithful separable representation for any row-finite source-free $k$-graph.

math.OA↗

arXiv2017

Separable representations of higher-rank graphs

In this monograph we undertake a comprehensive study of separable representations (as well as their unitary equivalence classes) of $C^*$-algebras associated to strongly connected finite $k$-graphs $Λ$. We begin with the representations associated to the $Λ$-semibranching function systems introduced by Farsi, Gillaspy, Kang, and Packer in \cite{FGKP}, by giving an alternative characterization of these systems which is more easily verified in examples. We present a variety of such examples, one of which we use to construct a new faithful separable representation of any row-finite source-free $k$-graph. Next, we analyze the monic representations of $C^*$-algebras of finite $k$-graphs. We completely characterize these representations, generalizing results of Dutkay and Jorgensen \cite{dutkay-jorgensen-monic} and Bezuglyi and Jorgensen \cite{bezuglyi-jorgensen} for Cuntz and Cuntz-Krieger algebras respectively. We also describe a universal representation for non-negative monic representations of finite, strongly connected $k$-graphs. To conclude, we characterize the purely atomic and permutative representations of $k$-graph $C^*$-algebras, and discuss the relationship between these representations and the classes of representations introduced earlier.

math.OA↗

arXiv2017

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 $Λ$. Moreover, we prove that these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of $Λ$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, to associate a family of ultrametric Cantor sets to a finite, strongly connected higher-rank graph $Λ$. Then we show that under mild hypotheses, the Pearson-Bellissard spectral triples of such Cantor sets have a regular $ζ$-function, whose abscissa of convergence agrees with the Hausdorff dimension of the Cantor set, and that the measure $μ$ induced by the associated Dixmier trace agrees with the measure $M$ on the infinite path space $Λ^\infty$ of $Λ$ which was introduced by an Huef, Laca, Raeburn, and Sims. Furthermore, we prove that $μ= M$ is a rescaled version of the Hausdorff measure of the ultrametric Cantor set. From work of Julien and Savinien, we know that for $ζ$-regular Pearson-Bellissard spectral triples, the eigenspaces of the associated Laplace-Beltrami operator constitute an orthogonal decomposition of $L^2(Λ^\infty, μ)$; 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 $C^*$-algebras, and prove that the wavelet decomposition of Farsi et al.~describes the eigenspaces of its Dirac operator.

math.OA↗

arXiv2016

Wavelets and spectral triples for fractal representations of Cuntz algebras

In this article we provide an identification between the wavelet decompositions of certain fractal representations of $C^*-$algebras of directed graphs of M. Marcolli and A. Paolucci, and the eigenspaces of Laplacians associated to spectral triples constructed from Cantor fractal sets that are the infinite path spaces of Bratteli diagrams associated to the representations, with a particular emphasis on wavelets for representations of $\mathcal{O}_D$. In particular, in this setting we use results of J. Pearson and J. Bellissard, and A. Julien and J. Savinien, to construct first the spectral triple and then the Laplace Beltrami operator on the associated Cantor set. We then prove that in certain cases, the orthogonal wavelet decomposition and the decomposition via orthogonal eigenspaces match up precisely. We give several explicit examples, including an example related to a Sierpinski fractal, and compute in detail all the eigenvalues and corresponding eigenspaces of the Laplace Beltrami operators for the equal weight case for representations of Cuntz algebras, and in the uneven weight case for certain representations of $\mathcal{O}_2$, and show how the eigenspaces and wavelet subspaces at different levels are related.

math.OA↗

arXiv2016

Twists over étale groupoids and twisted vector bundles

Inspired by recent papers on twisted $K$-theory, we consider in this article the question of when a twist $\mathcal{R}$ over a locally compact Hausdorff groupoid $\mathcal{G}$ (with unit space a CW-complex) admits a twisted vector bundle, and we relate this question to the Brauer group of $\mathcal{G}$. We show that the twists which admit twisted vector bundles give rise to a subgroup of the Brauer group of $\mathcal{G}$. When $\mathcal{G}$ is an étale groupoid, we establish conditions (involving the classifying space $B\mathcal{G}$ of $\mathcal{G}$) which imply that a torsion twist $\mathcal{R}$ over $\mathcal{G}$ admits a twisted vector bundle.

math.OA↗

arXiv2016

Wavelets and graph $C^*$-algebras

Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.

math.OA↗

arXiv2015

Separable representations, KMS states, and wavelets for higher-rank graphs

Let $Λ$ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of $C^*(Λ)$ on certain separable Hilbert spaces of the form $L^2(X,μ)$, by introducing the notion of a $Λ$-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, when $Λ$ is aperiodic, we obtain a faithful representation of $C^*(Λ)$ on $L^2(Λ^\infty, M)$, where $M$ is the Perron-Frobenius probability measure on the infinite path space $Λ^\infty$ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a $Λ$-semibranching function system gives rise to KMS states for $C^*(Λ)$. For the higher-rank graphs of Robertson and Steger, we also obtain a representation of $C^*(Λ)$ on $L^2(X, μ)$, where $X$ is a fractal subspace of $[0,1]$ by embedding $Λ^{\infty}$ into $[0,1]$ as a fractal subset $X$ of $[0,1]$. In this latter case we additionally show that there exists a KMS state for $C^*(Λ)$ whose inverse temperature is equal to the Hausdorff dimension of $X$. Finally, we construct a wavelet system for $L^2(Λ^\infty, M)$ by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.

math.OA↗

arXiv2013

Stratifications of inertia spaces of compact Lie group actions

We study the topology of the inertia space of a smooth $G$-manifold $M$ where $G$ is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differential forms defined on the inertia space with respect to this stratification.

math.DG↗

arXiv2013

Functional equations for orbifold wreath products

We present generating functions for extensions of multiplicative invariants of wreath symmetric products of orbifolds presented as the quotient by the locally free action of a compact, connected Lie group in terms of orbifold sector decompositions. Particularly interesting instances of these product formulas occur for the Euler and Euler--Satake characteristics, which we compute for a class of weighted projective spaces. This generalizes results known for global quotients by finite groups to all closed, effective orbifolds. We also describe a combinatorial approach to extensions of multiplicative invariants using decomposable functors that recovers the formula for the Euler--Satake characteristic of a wreath product of a global quotient orbifold.

math.AT↗

arXiv2013

On Orbifold Criteria for Symplectic Toric Quotients

We introduce the notion of regular symplectomorphism and graded regular symplectomorphism between singular phase spaces. Our main concern is to exhibit examples of unitary torus representations whose symplectic quotients cannot be graded regularly symplectomorphic to the quotient of a symplectic representation of a finite group, while the corresponding GIT quotients are smooth. Additionally, we relate the question of simplicialness of a torus representation to Gaussian elimination.

math.SG↗