arXiv subjects

Alex Kumjian

Publications and source records attributed to Alex Kumjian.

At least 19 recordsLinked to original sources

arXiv2024

Embeddability of higher-rank graphs in groupoids, and the structure of their C*-algebras

We show that the C*-algebra of a row-finite source-free k-graph is Rieffel-Morita equivalent to a crossed product of an AF algebra by the fundamental group of the k-graph. When the k-graph embeds in its fundamental groupoid, this AF algebra is a Fell algebra; and simple-connectedness of a certain sub-1-graph characterises when this Fell algebra is Rieffel--Morita equivalent to a commutative C*-algebra. We provide a substantial suite of results for determining if a given k-graph embeds in its fundamental groupoid, and provide a large class of examples, arising via work of Cartwright, Robertson, Steger et al. from the theory of $\tilde{A_2}$-groups, that do embed.

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arXiv2021

Pushouts of extensions of groupoids by bundles of abelian groups

We analyse extensions $Σ$ of groupoids $G$ by bundles $A$ of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid $G$ by a given bundle $A$. There is a natural action of $Σ$ on the dual of $A$, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of $A$ with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of $G$ on the dual of $A$. We prove that the full $C^*$-algebra of this twist is isomorphic to the full $C^*$-algebra of $Σ$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.

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arXiv2021

Cocycles on Certain Groupoids Associated to $ \mathbb{N}^{k} $-Actions

We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space, and study generalized Ruelle operators and $ C^{\ast} $-algebras associated to these groupoids. We provide a new characterization of $ 1 $-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle-Perron-Frobenius theory for dynamical systems of several commuting operators ($ k $-Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast} $-algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence-uniqueness results for KMS states associated to the graphs.

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arXiv2020

C*-Algebras of extensions of groupoids by group bundles

Given a normal subgroup bundle $\mathcal A$ of the isotropy bundle of a groupoid $Σ$, we obtain a twisted action of the quotient groupoid $Σ/\mathcal A$ on the bundle of group $C^*$-algebras determined by $\mathcal A$ whose twisted crossed product recovers the groupoid $C^*$-algebra $C^*(Σ)$. Restricting to the case where $\mathcal A$ is abelian, we describe $C^*(Σ)$ as the $C^*$-algebra associated to a $\mathbf T$-groupoid over the tranformation groupoid obtained from the canonical action of $Σ/\mathcal A$ on the Pontryagin dual space of $\mathcal A$. We give some illustrative examples of this result.

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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.

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arXiv2018

The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with Continuous Trace

Given a locally compact abelian group $G$, we give an explicit formula for the Dixmier--Douady invariant of the $C^*$-algebra of the groupoid extension associated to a Čech $2$-cocycle in the sheaf of germs of continuous $G$-valued functions. We then exploit the blow-up construction for groupoids to extend this to some more general central extensions of étale equivalence relations.

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arXiv2017

Obstructions to lifting cocycles on groupoids and the associated $C^*$-algebras

Given a short exact sequence of locally compact abelian groups $0 \to A \to B \to C \to 0$ and a continuous $C$-valued $1$-cocycle $ϕ$ on a locally compact Hausdorff groupoid $Γ$ we construct a twist of $Γ$ by $A$ that is trivial if and only if $ϕ$ lifts. The cocycle determines a strongly continuous action of $\widehat{C}$ into $\operatorname{Aut} C^*(Γ)$ and we prove that the $C^*$-algebra of the twist is isomorphic to the induced algebra of this action if $Γ$ is amenable. We apply our results to a groupoid determined by a locally finite cover of a space $X$ and a cocycle provided by a Čech 1-cocycle with coefficients in the sheaf of germs of continuous $\mathbb{T}$-valued functions. We prove that the $C^*$-algebra of the resulting twist is continuous trace and we compute its Dixmier-Douady invariant.

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arXiv2017

Graded C-algebras, graded K-theory, and twisted P-graph C-algebras

We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish graded versions of Pimsner's six-term sequences for graded Hilbert bimodules whose left action is injective and by compacts, and a graded Pimsner-Voiculescu sequence. We introduce the notion of a twisted P-graph C*-algebra and establish connections with graded C*-algebras. Specifically, we show how a functor from a P-graph into the group of order two determines a grading of the associated C*-algebra. We apply our graded version of Pimsner's exact sequence to compute the graded K-theory of a graph C*-algebra carrying such a grading.

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arXiv2017

Twisted Topological Graph Algebras Are Twisted Groupoid C*-algebras

The second author showed how Katsura's construction of the C*-algebra of a topological graph E may be twisted by a Hermitian line bundle L over the edge space E. The correspondence defining the algebra is obtained as the completion of the compactly supported continuous sections of L. We prove that the resulting C*-algebra is isomorphic to a twisted groupoid C*-algebra where the underlying groupoid is the Renault-Deaconu groupoid of the topological graph with Yeend's boundary path space as its unit space.

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arXiv2016

Simplicity of twisted C*-algebras of higher-rank graphs and crossed products by quasifree actions

We characterise simplicity of twisted C*-algebras of row-finite k-graphs with no sources. We show that each 2-cocycle on a cofinal k-graph determines a canonical second-cohomology class for the periodicity group of the graph. The groupoid of the k-graph then acts on the cartesian product of the infinite-path space of the graph with the dual group of the centre of any bicharacter representing this second-cohomology class. The twisted k-graph algebra is simple if and only if this action is minimal. We apply this result to characterise simplicity for many twisted crossed products of k-graph algebras by quasifree actions of free abelian groups.

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arXiv2016

Topological spaces associated to higher-rank graphs

We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the quotient is again a higher-rank graph, and show that identifying isomorphic co-hereditary subgraphs in a disjoint union of two rank-$k$ graphs gives rise to pullbacks of the associated $C^*$-algebras. We describe a combinatorial version of the connected-sum operation and apply it to the rank-2-graph realisations of the four basic surfaces to deduce that every compact 2-manifold is the topological realisation of a rank-2 graph. We also show how to construct $k$-spheres and wedges of $k$-spheres as topological realisations of rank-$k$ graphs.

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arXiv2016

A Stabilization Theorem for Fell Bundles over groupoids

We study the $C^*$-algebras associated to upper-semicontinuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer--Raeburn "Stabilization Trick," we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced $C^*$-algebras of any saturated upper-semicontinuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the $C^*$-algebra of a continuous Fell-bundle by applying Renault's results about the ideals of the $C^*$-algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle $C^*$-algebra of a bundle over $G$ in terms of an action, described by the first and last named authors, of $G$ on the primitive-ideal space of the $C^*$-algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted $k$-graph algebras, where the components of our results become more concrete.

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arXiv2014

Crossed products and twisted $k$-graph algebras

An automorphism $β$ of a $k$-graph $Λ$ induces a crossed product $C^* ( Λ) \rtimes_β\mathbb{Z}$ which is isomorphic to a $(k+1)$-graph algebra $C^* ( Λ\times_β\mathbb{Z})$. In this paper we show how this process interacts with $k$-graph $C^*$-algebras which have been twisted by an element of their second cohomology group. This analysis is done using a long exact sequence in cohomology associated to this data. We conclude with some examples

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arXiv2014

Groupoid Actions on Fractafolds

We define a bundle over a totally disconnected set such that each fiber is homeomorphic to a fractal blowup. We prove that there is a natural action of a Renault-Deaconu groupoid on our fractafold bundle and that the resulting action groupoid is a Renault-Deaconu groupoid itself. We also show that when the bundle is locally compact the associated $C^*$-algebra is primitive and has a densely defined lower-semicontinuous trace.

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arXiv2014

Quantum Heisenberg Manifolds as Twisted Groupoid $C^*$-Algebras

The quantum Heisenberg manifolds are noncommutive manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds and have been studied by various authors. Rieffel constructed the quantum Heisenberg manifolds as the generalized fixed-point algebras of certain crossed product $C^*$-algebras, and they also can be realized as crossed products of $C(\mathbb{T}^2)$ by Hilbert $C^*$-bimodules in the sense of Abadie et al. In this paper, we describe how the quantum Heisenberg manifolds can also be realized as twisted groupoid $C^*$-algebras.

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arXiv2013

On twisted higher-rank graph C*-algebras

We define the categorical cohomology of a k-graph Λ and show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This leads to an alternative characterisation of the twisted k-graph C*-algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted k-graph C*-algebra is isomorphic to a twisted groupoid C*-algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted k-graph C*-algebras are nuclear and belong to the bootstrap class.

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arXiv2012

On the K-theory of twisted higher-rank-graph C*-algebras

We investigate the K-theory of twisted higher-rank-graph algebras by adapting parts of Elliott's computation of the K-theory of the rotation algebras. We show that each 2-cocycle on a higher-rank graph taking values in an abelian group determines a continuous bundle of twisted higher-rank graph algebras over the dual group. We use this to show that for a circle-valued 2-cocycle on a higher-rank graph obtained by exponentiating a real-valued cocycle, the K-theory of the twisted higher-rank graph algebra coincides with that of the untwisted one.

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arXiv2012

Hausdorff Measures and KMS States

Given a compact metric space $X$ and a local homeomorphism $T:X\to X$ satisfying a local scaling property, we show that the Hausdorff measure on $X$ gives rise to a KMS state on the $C^{*}$-algebra naturally associated to the pair $(X,T)$ such that the inverse temperature coincides with the Hausdorff dimension. We prove that the KMS state is unique under some mild hypothesis. We use our results to describe KMS states on Cuntz algebras, graph algebras, and $C^{*}$-algebras on fractafolds.

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