We define two notions of Morita equivalence for graded C*-algebras (graded Morita equivalence and homogeneous Morita equivalence) and provide Brown-Green-Rieffel Stabilization Type Theorems for both notions of graded equivalence. We apply our results to finite regular graphs by establishing an explicit connection between graded C*-algebras and coactions. Lastly, we incorporate Cartan subalgebras with totally disconnected spectra and obtain Brown-Green-Rieffel Stabilization Type Theorems for these cases.

We develop new techniques for the construction and classification of representations of row-finite and locally convex higher-rank graph C*-algebras O. This class includes Cuntz–Krieger algebras associated to row-finite directed graphs. Our approach relies on the representation theory of a certain non-self-adjoint algebra and a lifting process of representations. We introduce a novel dimension vector for representations of O yielding a countable partition of the spectrum. Given a Cuntz–Krieger algebra and a finite dimension vector, we construct a smooth manifold parametrising the corresponding spectral component. Our techniques are both explicit and functorial.

We compute the nuclear dimension of extensions of C*-algebras involving commutative unital quotients and stable Kirchberg ideals. We identify the finite directed graphs whose C*-algebras are covered by this theorem.

We show that the C*-algebra of a row-finite source-free k-graph is Rieffel-Morita equivalent to a crossed product of an approximately finite-dimensional (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 et al. ['Groups acting simply transitively on the vertices of a building of type A(2) I', Geom. Dedicata 47 (1993), 143-166], Cartwright et al. 'Groups acting simply transitively on the vertices of a building of type A(2) II', Geom. Dedicata 47 (1993), 167-226] and Robertson and Steger ['Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras', J. reine angew. Math. 513 (1999), 115-144] from the theory of A(2)-groups, which do embed.

We study the K-theory of the Cuntz-Nica-Pimsner C*-algebra of a rank-two product system that is an extension determined by an invariant ideal of the coefficient algebra. We use a construction of Deaconu and Fletcher that describes the Cuntz-Nica-Pimsner C*-algebra of the product system in terms of two iterations of Pimsner's original construction of a C*-algebra from a right-Hilbert bimodule. We apply our results to the product system built from two commuting surjective local homeomorphisms of a totally disconnected space, where the Cuntz-Nica-Pimsner C*-algebra is isomorphic to the C*-algebra of the associated rank-two Deaconu–Renault groupoid. We then apply a theorem of Spielberg about stable finiteness of an extension to obtain sufficient conditions for stable finiteness of the C*-algebra of the Deaconu-Renault groupoid.

We establish conditions under which an inclusion of finitely aligned left-cancellative small categories induces inclusions of twisted C*-algebras. We also present an example of an inclusion of finitely aligned left-cancellative monoids that does not induce a homomorphism even between (untwisted) Toeplitz algebras. We prove that the twisted C*-algebras of a jointly faithful self-similar action of a countable discrete amenable groupoid on a row-finite k-graph with no sources, with respect to homotopic cocycles, have isomorphic K-theory.

We show that every strongly ℤ-graded C*-algebra (equivalently, every C*-algebra carrying a strongly continuous 𝕋-action with full spectral subspaces) is a Cuntz–Pimsner algebra, and describe subalgebras and subspaces that can be used as the coefficient algebra and module in the construction. We deduce that for surjective graded homomorphisms ϕ of C*-algebras A graded by torsion-free abelian groups H, if the restriction ϕ_0 of ϕ to the zero-graded component A_0 of A induces isomorphisms in K-theory, so does ϕ itself. When H is free abelian, we show how to pick out smaller subalgebras of A_0 on which it suffices to check that ϕ induces isomorphisms in K-theory.

We determine the primitive ideal space and hence the ideal lattice of a large class of separable groupoid C*-algebras that includes all 2-graph C*-algebras. A key ingredient is the notion of harmonious families of bisections in etale groupoids associated to finite families of commuting local homeomorphisms. Our results unify and recover all known results on ideal structure for crossed products of commutative C*-algebras by free abelian groups, for graph C*-algebras, and for Katsura's topological graph C*-algebras.

We consider étale Hausdorff groupoids in which the interior of the isotropy is abelian. We prove that the norms of the images under regular representations, of elements of the reduced groupoid C^*-algebra whose supports are contained in the interior of the isotropy vary upper semicontinuously. This corrects an error in [T.M. Carlsen, E. Ruiz, A. Sims and M. Tomforde, "Reconstruction of groupoids and C*-rigidity of dynamical systems," Adv. Math 390 (2021), 107923].

We present an example of a twist over a minimal Hausdorff etale groupoid such that the restriction of the twist to the interior of the isotropy is not topologically trivial; that is, the restricted twist is not induced by a continuous 2-cocycle.

We prove a sandwiching lemma for inner-exact locally compact Hausdorff etale groupoids. Our lemma says that every ideal of the reduced C-& lowast;-algebra of such a groupoid is sandwiched between the ideals associated to two uniquely defined open invariant subsets of the unit space. We obtain a bijection between ideals of the reduced C-& lowast;-algebra, and triples consisting of two nested open invariant sets and an ideal in the C-& lowast;-algebra of the subquotient they determine that has trivial intersection with the diagonal subalgebra and full support. We then introduce a generalisation to groupoids of Ara and Lolk's relative strong topological freeness condition for partial actions, and prove that the reduced C-& lowast;-algebras of inner-exact locally compact Hausdorff etale groupoids satisfying this condition admit an obstruction ideal in Ara and Lolk's sense.

We show how to construct a family of groups with simple commutator subgroups from aperiodic 1-vertex, finitely aligned higher rank graphs (which are, in fact, a class of cancellative monoids). Inverse semigroups form the intermediary between these cancellative monoids and the family of groups we are interested in. These groups can naturally be viewed as higher-dimensional generalizations of the classical Thompson groups since the finite direct products of free monoids are examples of the appropriate 1-vertex higher rank graphs.

We consider Deaconu-Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces.We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2-cocycle.When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids.We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy.We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal.We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions.

We study the categorical homology of Zappa-Sz\'ep products of small categories, which include all self-similar actions. We prove that the categorical homology coincides with the homology of a double complex, and so can be computed via a spectral sequence involving homology groups of the constituent categories. We give explicit formulae for the isomorphisms involved, and compute the homology of a class of examples that generalise odometers. We define the C*-algebras of self-similar groupoid actions on k-graphs twisted by 2-cocycles arising from this homology theory, and prove some fundamental results about their structure.

We extend Nekrashevych’s K K KK -duality for C ∗ C^* -algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid ( G , E ) (G,E) acting faithfully on a finite directed graph E E , we associate two C ∗ C^* -algebras, O ( G , E ) \mathcal {O}(G,E) and O ^ ( G , E ) \widehat {\mathcal {O}}(G,E) , to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in K K KK -theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author.

We show that the Hilbert bimodule associated with a compact topological graph can be recovered from the $C^*$ -algebraic triple consisting of the Toeplitz algebra of the graph, its gauge action and the commutative subalgebra of functions on the vertex space of the graph. We discuss connections with work of Davidson–Katsoulis and of Davidson–Roydor on local conjugacy of topological graphs and isomorphism of their tensor algebras. In particular, we give a direct proof that a compact topological graph can be recovered up to local conjugacy from its Hilbert bimodule, and present an example of nonisomorphic locally conjugate compact topological graphs with isomorphic Hilbert bimodules. We also give an elementary proof that for compact topological graphs with totally disconnected vertex space the notions of local conjugacy, Hilbert bimodule isomorphism, isomorphism of $C^*$ -algebraic triples, and isomorphism all coincide.

We describe how to recover a Lie structure on a twist over a Hausdorff étale groupoid from functional-analytic data in the spirit of Connes' reconstruction theorem for manifolds. We first characterise when a smooth structure on the unit space of a Hausdorff étale groupoid can be extended to a Lie-groupoid structure on the whole groupoid. We introduce Lie twists over Hausdorff Lie groupoids, building on Kumjian's notion of a twist over a topological groupoid. We establish necessary and sufficient conditions on a family of sections of a twist over a Lie groupoid under which the twist can be made into a Lie twist so that all the specified sections are smooth. We use these results in the setting of twists over étale groupoids to describe conditions on a Cartan pair of C*-algebras and a family of normalisers of the subalgebra, under which Renault's Weyl twist for the pair can be made into a Lie twist for which the given normalisers correspond to smooth sections.

We study stable finiteness of extensions of 2-graph C*-algebras determined by saturated hereditary sets of vertices. We use two iterations of the Pimsner-Voiculescu sequence to calculate the map in K-theory induced by the inclusion of a hereditary subgraph into the larger 2-graph it lives in. We then apply a theorem of Spielberg about stable finiteness of extensions to provide a sufficient condition for the C*-algebra of the larger 2-graph to be stably finite. We illustrate our results with examples.

We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of twists for which the local bisection hypothesis holds automatically.

We show that every nuclear O∞-stable ⁎-homomorphism with a separable exact domain has nuclear dimension at most 1. In particular separable, nuclear, O∞-stable C⁎-algebras have nuclear dimension 1. We also characterise when O∞-stable C⁎-algebras have finite decomposition rank in terms of quasidiagonality and primitive-ideal structure, and determine when full O2-stable ⁎-homomorphisms have nuclear dimension 0.