We prove that the full C * -algebra of a second-countable, Hausdorff,étale, amenable groupoid is simple if and only if the groupoid is both topologically principal and minimal. We also show that if G has totally disconnected unit space, then the complex * -algebra of its inverse semigroup of compact open bisections, as introduced by Steinberg, is simple if and only if G is both effective and minimal.

Abstract. We provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the Cuntz-Krieger algebra. IntroductionA higher-rank graph is a countable category Λ endowed with a degree functor d : Λ → N k satisfying the unique factorization property: For all λ ∈ Λ and m, n ∈ N k with d(λ) = m + n, there are unique elements µ, ν ∈ Λ such that d(µ) = m, d(ν) = n and λ = µν. The rank of Λ is k and for this reason, (Λ, d) is also called a k-graph. A 1-graph is simply the finite-path category generated freely by an ordinary directed graph.1 In [2], Kumjian and Pask introduced the notion of a higher-rank graph in order to capture the essential features of the C * -algebras that Robertson and Steger associated to buildings [14,15,16,17] and to provide links between these and higher order shift dynamical systems. (See [3] also.)The C * -algebras associated to higher-rank graphs are generalizations of ordinary graph C * -algebras in that they are generated by families of partial isometries {s λ } λ∈Λ that satisfy certain relations that have received a lot of attention in recent years. Kumjian and Pask defined and studied the C * -algebra of (Λ, d), C * (Λ), in terms of a certain type of groupoid that encodes the graph. They were motivated by, and generalized, the theory in [5] which has been the source of considerable inspiration in our subject. However, just as in the setting of ordinary graphs, where the groupoid techniques of [5] require hypotheses that rule out many interesting examples, the work of Kumjian and Pask requires hypotheses that place significant limitations on the nature of the k-graphs that may be analyzed. Our first objective in this paper, then, is to overcome the limitations that Kumjian and Pask place on their k-graphs and to show how to build a groupoid that gives the C * -algebra of an arbitrary k-graph subject only to the condition that it is "finitely aligned" (see Definition 3.4). This condition seems to lie at the natural "boundary" of the subject. That is, with or without the use of groupoids, little can be said about k-graphs that are not finitely aligned.Date: March 10, 2018. 1991 Mathematics Subject Classification. Primary 46L05; Secondary 22A22; 20M18. Key words and phrases. graph algebra; Cuntz-Krieger algebra; higher-rank graph; groupoid; inverse semigroup.The research of the first two authors was supported in part by a grant from the National Science Foundation, DMS-0070405.1 For the purpose of motivation, we discuss the theory of 1-graphs at some length in the next section. For a very readable account of graph C * -algebras, including the rudiments of the C * -algebras of k-graphs, we recommend the CBMS lectures by Iain Raeburn [9]. We were motivated in part by the important contribution of Paterson [8] in which he successfully circumvented the limitations of [5] that involve finiteness hypotheses on the graphs under consideration by first introducing an inverse semi...

Let G be a locally compact, Hausdorff groupoid in which s is a local homeomorphism and G (0) is totally disconnected. Assume there is a continuous cocycle c from G into a discrete group Γ. We show that the collection A(G) of locally-constant, compactly supported functions on G is a dense * -subalgebra of C c (G) and that it is universal for algebraic representations of the collection of compact open bisections of G. We also show that if G is the groupoid associated to a row-finite graph or k-graph with no sources, then A(G) is isomorphic to the associated Leavitt path algebra or Kumjian-Pask algebra. We prove versions of the Cuntz-Krieger and graded uniqueness theorems for A(G).

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Undergraduate mathematics tutoring centres are prevalent in many countries; however, there is limited research-based evidence on effective organizational structures for these centres. In this study, we consider two research questions. First, how can the quantitative and qualitative data from 10 mathematics tutoring centres be organized for research purposes? Second, what hypotheses do expert mathematics tutoring centre leaders generate about characteristics of effective centres given data from a sample of ten centres? We collected quantitative data from over 26,000 students taking mathematics courses at ten institutions. Data collected included college entrance exam scores, high school grade point average, number of student visits to the centre per eligible student and course letter grade. We used exploratory data analysis to look for relationships between visits to the tutoring centre, student grades and other variables. Qualitative centre characteristics that were considered include: specialist–generalist tutoring system, tutoring capacity, physical layout, relationships between tutors and mathematics instructors and extent of tutor training. We used the Delphi process to generate testable hypotheses from the data, such as the following: (1) The more courses a tutor is responsible for tutoring the more likely it is that the tutor will struggle to answer student questions, when the difficulty level of the courses is roughly the same. (2) Centres with more specialized tutor models have more visits per student than centres with generalized tutor models. The preceding two hypotheses, along with the other generated hypotheses, have been identified by the experts participating in this study as plausible based on professional experience, exploratory data analysis and inferences based on prior research on tutoring. This study has not rigorously shown the validity of these hypotheses; rather it lays the groundwork for future investigations to determine what combination of features characterize an effective tutoring centre.