arXiv subjects

Elizabeth Gillaspy

Publications and source records attributed to Elizabeth Gillaspy.

29 records · Page 2Linked to original sources

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.

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arXiv2017

Cohomology for small categories: $k$-graphs and groupoids

Given a higher-rank graph $Λ$, we investigate the relationship between the cohomology of $Λ$ and the cohomology of the associated groupoid $G_Λ$. We define an exact functor between the abelian category of right modules over a higher-rank graph $Λ$ and the category of $G_Λ$-sheaves, where $G_Λ$ is the path groupoid of $Λ$. We use this functor to construct compatible homomorphisms from both the cohomology of $Λ$ with coefficients in a right $Λ$-module, and the continuous cocycle cohomology of $G_Λ$ with values in the corresponding $G_Λ$-sheaf, into the sheaf cohomology of $G_Λ$.

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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.

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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.

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arXiv2016

Irreducible representations of nilpotent groups generate classifiable C*-algebras

We show that C*-algebras generated by irreducible representations of finitely generated nilpotent groups satisfy the universal coefficient theorem of Rosenberg and Schochet. This result combines with previous work to show that these algebras are classifiable by their Elliott invariants within the class of unital, simple, separable, nuclear C*-algebras with finite nuclear dimension that satisfy the universal coefficient theorem. We also show that these C*-algebras are central cutdowns of twisted group C*-algebras with homotopically trivial cocycles.

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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.

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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.

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arXiv2015

$K$-theory and homotopies of 2-cocycles on higher-rank graphs

This paper continues our investigation into the question of when a homotopy $ω= \{ω_t\}_{t \in [0,1]}$ of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an isomorphism of the $K$-theory groups of the twisted groupoid $C^*$-algebras: $K_*(C^*(\mathcal{G}, ω_0)) \cong K_*(C^*(\mathcal{G}, ω_1)).$ In particular, we build on work by Kumjian, Pask, and Sims to show that if $\mathcal{G} = \mathcal{G}_Λ$ is the infinite path groupoid associated to a row-finite higher-rank graph $Λ$ with no sources, and $\{c_t\}_{t \in [0,1]}$ is a homotopy of 2-cocycles on $Λ$, then $K_*(C^*(\mathcal{G}_Λ, σ_{c_0})) \cong K_*(C^*(\mathcal{G}_Λ, σ_{c_1})),$ where $σ_{c_t}$ denotes the 2-cocycle on $\mathcal{G}_Λ$ associated to the 2-cocycle $c_t$ on $Λ$. We also prove a technical result (Theorem 3.3), namely that a homotopy of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an upper semi-continuous $C^*$-bundle.

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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.

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arXiv2014

$K$-theory and homotopies of 2-cocycles on group bundles

This paper continues the author's program to investigate the question of when a homotopy of 2-cocycles $Ω= \{ω_t\}_{t \in [0,1]}$ on a locally compact Hausdorff groupoid $\mathcal{G}$ induces an isomorphism of the $K$-theory groups of the twisted groupoid $C^*$-algebras: $K_*(C^*(\mathcal{G}, ω_0)) \cong K_*(C^*(\mathcal{G}, ω_1)).$ Building on our earlier work, we show that if $π: \mathcal{G} \to M$ is a locally trivial bundle of amenable groups over a locally compact Hausdorff space $M$, a homotopy $Ω= \{ω_t\}_{t \in [0,1]}$ of 2-cocycles on $\mathcal{G} $ gives rise to an isomorphism $K_*(C^*(\mathcal{G}, ω_0)) \cong K_*(C^*(\mathcal{G}, ω_1)).$

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arXiv2014

$K$-theory and homotopies of 2-cocycles on transformation groups

This paper constitutes a first step in the author's program to investigate the question of when a homotopy of 2-cocycles $ω= \{ω_t\}_{t \in [0,1]}$ on a locally compact Hausdorff groupoid $\mathcal{G}$ induces an isomorphism of the $K$-theory groups of the reduced twisted groupoid $C^*$-algebras: $K_*(C^*_r(\mathcal{G}, ω_0)) \cong K_*(C^*_r(\mathcal{G}, ω_1)).$ Generalizing work of Echterhoff, Lück, Phillips, and Walters from 2010, we show that if $\mathcal{G} = G \ltimes X$ is a second countable locally compact transformation group, then whenever $G$ satisfies the Baum-Connes conjecture with coefficients, a homotopy $ω= \{ω_t\}_{t \in [0,1]}$ of 2-cocycles on $G \ltimes X$ gives rise to an isomorphism $K_*(C^*_r(G \ltimes X, ω_0)) \cong K_*(C^*_r(G \ltimes X, ω_1)).$

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