arXiv subjects

Elizabeth Gillaspy

Publications and source records attributed to Elizabeth Gillaspy.

At least 19 recordsLinked to original sources

arXiv2026

Functoriality of real crossed product K-theory spectral sequences with respect to group homomorphisms

Spectral sequences are a key tool for computing the K-theory of a crossed product C$^*$-algebra. However, the impact of a group homomorphism $\Omega\colon G \to H$ on such a spectral sequence was unknown until quite recently, even when $G = \mathbb Z^\ell$, $H = \mathbb Z^{k}.$ Recent work [Mil25] of the fourth-named author in the complex case establishes that ABC spectral sequences are functorial with respect to group homomorphisms. In this paper, we obtain the analogous result for real K-theory and for united K-theory. Specifically, we first show that the ABC spectral sequence approximates KO$_*(G \ltimes_r A)$ with the group homology H$_p(G;KO_q(A))$ when $G$ is a torsion-free discrete group satisfying the Baum--Connes conjecture with coefficients in $A$. Then, for a homomorphism $\Omega \colon G \to H$ of such groups with amenable kernel, and a real $H$-C$^*$-algebra $A$, we show moreover that the map in K-theory induced by the $*$-homomorphism $G \ltimes_r A \to H \ltimes_r A$ is approximated by the natural map in group homology.

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arXiv2025

Relating insplittings of 2-graphs and of textile systems

The graphical operation of insplitting is key to understanding conjugacy of shifts of finite type (SFTs) in both one and two dimensions. In this paper, we consider two approaches to studying 2-dimensional SFTs: textile systems and rank-2 graphs. Nasu's textile systems describe all two-sided 2D SFTs up to conjugacy, whereas the 2-graphs (higher-rank graphs of rank 2) introduced by Kumjian and Pask yield associated C*-algebras. Both models have a naturally-associated notion of insplitting. We show that these notions do not coincide, raising the question of whether insplitting a 2-graph induces a conjugacy of the associated one-sided 2-dimensional SFTs. Our first main result shows how to reconstruct 2-graph insplitting using textile-system insplits and inversions, and consequently proves that 2-graph insplitting induces a conjugacy of dynamical systems. We also present several other facets of the relationship between 2-graph insplitting and textile-system insplitting. Incorporating an insplit of the bottom graph of the textile system turns out to be key to this relationship. By articulating the connection between operator-algebraic and dynamical notions of insplitting in two dimensions, this article lays the groundwork for a C*-algebraic framework for classifying one-sided conjugacy in higher-dimensional SFTs.

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arXiv2025

Products, crossed products, and Zappa--Sz\'ep products for $k$-graphs

We use the lens of Zappa--Sz\'ep decomposition to examine the relationship between directed graph products and $k$-graph products. There are many examples of higher-rank graphs, or $k$-graphs, whose underlying directed graph may be factored as a product, but the $k$-graph itself is not a product. In such examples, we establish that the Zappa--Sz\'ep structure of the $k$-graph gives rise to "actions'' of the underlying directed factors on each other. Although these "actions'' are in general poorly behaved, if one of them is trivial (or trivial up to isomorphism), we obtain a crossed-product-like structure on the $k$-graph. We provide examples where this crossed-product structure is visible in the associated $C^*$-algebra, and we characterize those $k$-graphs whose Zappa--Sz\'ep induced actions are trivial up to isomorphism.

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arXiv2024

Homotopy of product systems and K-theory of Cuntz-Nica-Pimsner algebras

We introduce the notion of a homotopy of product systems, and show that the Cuntz-Nica-Pimsner algebras of homotopic product systems over N^k have isomorphic K-theory. As an application, we give a new proof that the K-theory of a 2-graph C*-algebra is independent of the factorisation rules, and we further show that the K-theory of any twisted k-graph C*-algebra is independent of the twisting 2-cocycle. We also explore applications to K-theory for the C*-algebras of single-vertex k-graphs, reducing the question of whether the $K$-theory is independent of the factorisation rules to a question about path-connectedness of the space of solutions to an equation of Yang-Baxter type.

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arXiv2023

Nontraditional models of $Γ$-Cartan pairs

This paper explores the tension between multiple models and rigidity for groupoid $C^*$-algebras. We begin by identifying $Γ$-Cartan subalgebras $D$ inside twisted groupoid $C^*$-algebras $C^*_r(G, ω)$, using similar techniques to those developed in [DGN$^+$20]. When $D \not= C_0(G^{(0)})$, [BFPR21, Theorem 4.19] then gives another groupoid $H$, and a twist $Σ$ over $H$, so that $D \cong C_0(H^{(0)})$ and $C^*_r(G, ω) \cong C^*_r(H; Σ)$. However, there is a close relationship between $G$ and $H$. In addition to showing how to construct $H$ and $Σ$ in terms of $G$ and $ω$, we also show how to reconstruct $G$ from $H$ if we assume the 2-cocycle $ω$ is trivial. This latter construction involves a new type of twisting datum, which may be of independent interest.

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arXiv2023

The Stable Exotic Cuntz Algebras are Higher-Rank Graph Algebras

For each odd integer $n \geq 3$, we construct a rank-3 graph $Λ_n$ with involution $γ_n$ whose real C*-algebra $C^*_\mathbb{R}(Λ_n, γ_n)$ is stably isomorphic to the exotic Cuntz algebra $\mathcal E_n^\mathbb{R}$. This construction is optimal, as we prove that a rank-2 graph with involution $(Λ,γ)$ can never satisfy $C^*_\mathbb{R}(Λ, γ)\sim_{ME} \mathcal E_n^\mathbb{R}$, and the first author reached the same conclusion in previous work. Our construction relies on a rank-1 graph with involution $(Λ, γ)$ whose real C*-algebra $C^*_\mathbb{R}(Λ, γ)$ is stably isomorphic to the suspension $ S \mathbb{R}$. In the Appendix, we show that the i-fold suspension $S^i \mathbb{R}$ is stably isomorphic to a graph algebra iff $-2 \leq i \leq 1$.

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arXiv2022

$K$-theory for real $k$-graph $C^*$-algebras

We initiate the study of real $C^*$-algebras associated to higher-rank graphs $Λ$, with a focus on their $K$-theory. Following Kasparov and Evans, we identify a spectral sequence which computes the $\mathcal{CR}$ $K$-theory of $C^*_{\mathbb R} (Λ, γ)$ for any involution $γ$ on $Λ$, and show that the $E^2$ page of this spectral sequence can be straightforwardly computed from the combinatorial data of the $k$-graph $Λ$ and the involution $γ$. We provide a complete description of $K^{CR}(C^*_{\mathbb R}(Λ, γ))$ for several examples of higher-rank graphs $Λ$ with involution.

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arXiv2021

Irreducibility and monicity for representations of $k$-graph $C^*$-algebras

The representations of a $k$-graph $C^*$-algebra $C^*(Λ)$ which arise from $Λ$-semibranching function systems are closely linked to the dynamics of the $k$-graph $Λ$. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. We also explore the relationship between irreducible $Λ$-semibranching representations and purely atomic representations of $C^*(Λ)$. Throughout the paper, we work in the setting of row-finite source-free $k$-graphs; this paper constitutes the first analysis of $Λ$-semibranching representations at this level of generality.

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arXiv2020

Moves on $k$-graphs preserving Morita equivalence

We initiate the program of extending to higher-rank graphs ($k$-graphs) the geometric classification of directed graph $C^*$-algebras, as completed in the 2016 paper of Eilers, Restorff, Ruiz, and Sorensen [ERRS16]. To be precise, we identify four "moves," or modifications, one can perform on a $k$-graph $Λ$, which leave invariant the Morita equivalence class of its $C^*$-algebra $C^*(Λ)$. These moves -- insplitting, delay, sink deletion, and reduction -- are inspired by the moves for directed graphs described by Sorensen [S\o13] and Bates-Pask [BP04]. Because of this, our perspective on $k$-graphs focuses on the underlying directed graph. We consequently include two new results, Theorem 2.3 and Lemma 2.9, about the relationship between a $k$-graph and its underlying directed graph.

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arXiv2020

Cartan subalgebras for non-principal twisted groupoid $C^*$-algebras

Renault proved in 2008 that if $G$ is a topologically principal groupoid, then $C_0(G^{(0)})$ is a Cartan subalgebra in $C^*_r(G, Σ)$ for any twist $Σ$ over $G$. However, there are many groupoids which are not topologically principal, yet their (twisted) $C^*$-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle $c$ on $G$ and a subgroupoid $S \subseteq G$ under which $C^*_r(S, c)$ is Cartan in $C^*_r(G, c)$. When $G$ is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.

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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.

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arXiv2019

Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph

We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph $Λ$, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we construct, explicitly and in both directions, maps on the level of (co-)chain complexes that implement said isomorphism. Along the way, we extend the definition of cubical (co-)homology to allow arbitrary coefficient modules.

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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.

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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.

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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.

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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.

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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.

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arXiv2018

Finite decomposition rank for virtually nilpotent groups

We show that inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras, extending results of the first author on solvable virtually nilpotent groups. We use this result to show that the decomposition rank of the group C*-algebra of a finitely generated virtually nilpotent group $G$ is bounded by $2\cdot h(G)!-1$, where $h(G)$ is the Hirsch length of $G.$ This extends and sharpens results of the first and third authors on finitely generated nilpotent groups. It then follows that if a C*-algebra generated by an irreducible representation of a virtually nilpotent group satisfies the universal coefficient theorem, it is classified by its Elliott invariant.

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