Higher-rank Graphs Research Articles

Real rank and topological dimension of higher rank graph algebras

We study dimension theory for the C � -algebras of row-finite k-graphs with no sources. We establish that strong aperiodicity—the higher-rank analogue of condi- tion (K)—for a k-graph is necessary and sufficient for the associatedC � -algebra to have topological dimension zero. We prove that a purely infinite 2-graph algebra has real-rank zero if and only if it has topological dimension zero and satisfies a homological condition that can be characterised in terms of the adjacency matrices of the 2-graph. We also show that a k-graph C � -algebra with topological dimension zero is purely infinite if and only if all the vertex projections are properly infinite. We show by example that there are strongly purely infinite 2-graphs algebras, both with and without topological dimension zero, that fail to have real-rank zero.

The cycline subalgebra of a Kumjian-Pask algebra

Let $\Lambda$ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalgebra $\mathcal{M}$ inside the Kumjian-Pask algebra ${\rm KP}_R(\Lambda)$. We also prove a generalized Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras which says that a representation of ${\rm KP}_R(\Lambda)$ is injective if and only if it is injective on $\mathcal{M}$.

Cohn Path Algebras of Higher-Rank Graphs

In this article, we introduce Cohn path algebras of higher-rank graphs. We prove that for a higher-rank graph Λ, there exists a higher-rank graph T Λ such that the Cohn path algebra of Λ is isomorphic to the Kumjian-Pask algebra of T Λ. We then use this isomorphism and properties of Kumjian-Pask algebras to study Cohn path algebras. This includes proving a uniqueness theorem for Cohn path algebras.

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.

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.

Amenability of groupoids arising from partial semigroup actions and topological higher rank graphs

We consider the amenability of groupoids $G$ equipped with a group valued cocycle $c : G → Q$ with amenable kernel $ c^{ −1} (e)$. We prove a general result which implies, in particular, that $G$ is amenable whenever $Q$ is amenable and if there is countable set $D ⊂ G$ such that $c(G^ u)D = Q$ for all $u ∈ G ^{(0)}$. We show that our result is applicable to groupoids arising from partial semigroup actions. We explore these actions in detail and show that these groupoids include those arising from directed graphs, higher rank graphs and even topological higher rank graphs. We believe our methods yield a nice alternative groupoid approach to these important constructions.

Topological aperiodicity for product systems over semigroups of Ore type

We prove a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type. To this end, we introduce Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product system of 'regular' C*-correspondences. Under a certain aperiodicity condition on the latter, we obtain the uniqueness theorem and a simplicity criterion for the algebras in question. These results generalize the corresponding ones for crossed products by discrete groups, due to Archbold and Spielberg, and for Exel's crossed products, due to Exel and Vershik. They also give interesting conditions for topological higher rank graphs and $P$-graphs, and apply to the new Cuntz C*-algebra $\mathcal{Q}_\mathbb{N}$ arising from the "$ax+b$"-semigroup over $\mathbb{N}$.

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

This paper continues our investigation into the question of when a homotopy $\omega = \{\omega_t\}_{t \in [0,1]}$ 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: $K_*(C^*(\mathcal{G}, \omega_0)) \cong K_*(C^*(\mathcal{G}, \omega_1)).$ In particular, we build on work by Kumjian, Pask, and Sims to show that if $\mathcal{G} = \mathcal{G}_\Lambda$ is the infinite path groupoid associated to a row-finite higher-rank graph $\Lambda$ with no sources, and $\{c_t\}_{t \in [0,1]}$ is a homotopy of 2-cocycles on $\Lambda$, then $K_*(C^*(\mathcal{G}_\Lambda, \sigma_{c_0})) \cong K_*(C^*(\mathcal{G}_\Lambda, \sigma_{c_1})),$ where $\sigma_{c_t}$ denotes the 2-cocycle on $\mathcal{G}_\Lambda$ associated to the 2-cocycle $c_t$ on $\Lambda$. 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.

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 L2(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, if Λ is aperiodic, we obtain a faithful representation of C⁎(Λ) on L2(Λ∞,M), where M is the Perron–Frobenius probability measure on the infinite path space Λ∞ 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 L2(X,μ), where X is a fractal subspace of [0,1] by embedding Λ∞ into [0,1] as a fractal subset X of [0,1]. Moreover, when the Radon–Nikodym derivatives of a Λ-semibranching function system are constant, we show that we can associate to it a KMS state for C⁎(Λ). Finally, we construct a wavelet system for L2(Λ∞,M) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.

Topological realizations and fundamental groups of higher-rank graphs

AbstractWe investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graphΛ, this functor determines a category equivalence between the category of coverings ofΛand the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions fork-graphs: projective limits and crossed products by finitely generated free abelian groups.

The Socle and Semisimplicity of a Kumjian–Pask Algebra

The Kumjian–Pask algebra KP(Λ) is a graded algebra associated to a higher-rank graph Λ and is a generalization of the Leavitt path algebra of a directed graph. We analyze the minimal left ideals of KP(Λ), and identify its socle as a graded ideal by describing its generators in terms of a subset of vertices of the graph. We characterize when KP(Λ) is semisimple, and obtain a complete structure theorem for a semisimple Kumjian–Pask algebra. As a consequence of this structure theorem, every semisimple Kumjian–Pask algebra can be obtained as a Leavitt path algebra of a directed graph.

On twisted higher-rank graph 𝐶*-algebras

We define the categorical cohomology of a k k -graph Λ \Lambda 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 k -graph C ∗ C^* -algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted k k -graph C ∗ C^* -algebra is isomorphic to a twisted groupoid C ∗ C^* -algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted k k -graph C ∗ C^* -algebras are nuclear and belong to the bootstrap class.

Von Neumann algebras of strongly connected higher-rank graphs

We investigate the factor types of the extremal KMS states for the preferred dynamics on the Toeplitz algebra and the Cuntz--Krieger algebra of a strongly connected finite $k$-graph. For inverse temperatures above 1, all of the extremal KMS states are of type I$_\infty$. At inverse temperature 1, there is a dichotomy: if the $k$-graph is a simple $k$-dimensional cycle, we obtain a finite type I factor; otherwise we obtain a type III factor, whose Connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the graph.

Spatial realisations of KMS states on the [formula omitted]-algebras of higher-rank graphs

Several authors have recently been studying the equilibrium or KMS states on the Toeplitz algebras of finite higher-rank graphs. For graphs of rank one (that is, for ordinary directed graphs), there is a natural dynamics obtained by lifting the gauge action of the circle to an action of the real line. The algebras of higher-rank graphs carry a gauge action of a higher-dimensional torus, and there are many potential dynamics arising from different embeddings of the real line in the torus. Previous results show that there is nonetheless a “preferred dynamics” for which the system exhibits a particularly satisfactory phase transition, and that the unique KMS state at the critical inverse temperature can then be implemented by integrating vector states against a measure on the infinite path space of the graph. Here we obtain a similar description of the KMS state at the critical inverse temperature for other dynamics. Our spatial implementation is given by integrating against a measure on a space of paths which are infinite in some directions but finite in others. Our results are sharpest for the algebras of rank-two graphs.

KMS states on the [formula omitted]-algebra of a higher-rank graph and periodicity in the path space

We study the KMS states of the C⁎-algebra of a strongly connected finite k-graph. We find that there is only one 1-parameter subgroup of the gauge action that can admit a KMS state. The extreme KMS states for this preferred dynamics are parameterised by the characters of an abelian group that captures the periodicity in the infinite-path space of the graph. We deduce that there is a unique KMS state if and only if the k-graph C⁎-algebra is simple, giving a complete answer to a question of Yang. When the k-graph C⁎-algebra is not simple, our results reveal a phase change of an unexpected nature in its Toeplitz extension.

Ionescu's theorem for higher rank graphs

We will define new constructions similar to the graph systems of correspondences described by Deaconu et al. We will use these to prove a version of Ionescu's theorem for higher rank graphs. Afterwards we will examine the properties of these constructions further and make contact with Yeend's topological k-graphs and the tensor groupoid valued product systems of Fowler and Sims.

Centers of algebras associated to higher-rank graphs

The Kumjian–Pask algebras are path algebras associated to higher-rank graphs, and generalize the Leavitt path algebras. We study the center of a simple Kumjian–Pask algebra and characterize commutative Kumjian–Pask algebras

Groupoids and $C^*$-algebras for categories of paths

In this paper we describe a new method of defining C*-algebras from oriented combinatorial data, thereby generalizing the constructions of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the C*-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems.

JOINS AND COVERS IN INVERSE SEMIGROUPS AND TIGHT -ALGEBRAS

AbstractWe show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the ${C}^{\ast } $-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight ${C}^{\ast } $-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph $\Lambda $, the tight ${C}^{\ast } $-algebra of the inverse semigroup associated to $\Lambda $ is the same as the ${C}^{\ast } $-algebra of $\Lambda $.

Twisted $C^*$-algebras associated to finitely aligned higher-rank graphs

We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish versions of the usual uniqueness theorems and the classification of gauge-invariant ideals. We show that all twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs are nuclear and satisfy the UCT, and that for twists that lift to real-valued cocycles, the K -theory of a twisted relative Cuntz-Krieger algebra is independent of the twist. In the final section, we identify a sufficient condition for simplicity of twisted Cuntz-Krieger algebras associated to higher-rank graphs which are not aperiodic. Our results indicate that this question is significantly more complicated than in the untwisted setting.