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.

We investigate the concept of entropy in probabilistic theories more general than quantum mechanics, with particular reference to the notion of information causality recently proposed by Pawlowski et al. (arXiv:0905.2992). We consider two entropic quantities, which we term measurement and mixing entropy. In the context of classical and quantum theory, these coincide, being given by the Shannon and von Neumann entropies respectively; in general, however, they are very different. In particular, while measurement entropy is easily seen to be concave, mixing entropy need not be. In fact, as we show, mixing entropy is not concave whenever the state space is a non-simplicial polytope. Thus, the condition that measurement and mixing entropies coincide is a strong constraint on possible theories. We call theories with this property monoentropic.Measurement entropy is subadditive, but not in general strongly subadditive. Equivalently, if we define the mutual information between two systems A and B by the usual formula I(A :where H denotes the measurement entropy and AB is a non-signaling composite of A and B, then it can happen that I(A : BC) < I (A : B). This is relevant to information causality in the sense of Pawlowski et al.: we show that any monoentropic non-signaling theory in which measurement entropy is strongly subadditive, and also satisfies a version of the Holevo bound, is informationally causal, and on the other hand we observe that Popescu-Rohrlich boxes, which violate information causality, also violate strong subadditivity. We also explore the interplay between measurement and mixing entropy and various natural conditions on theories that arise in quantum axiomatics.

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).

Abstract. Let G and H be Hausdorff ample groupoids and let R be a commutative unital ring. We show that if G and H are equivalent in the sense of Muhly-RenaultWilliams, then the associated Steinberg algebras of locally constant R-valued functions with compact support are Morita equivalent. We deduce that collapsing a "collapsible subgraph" of a directed graph in the sense of Crisp and Gow does not change the Morita-equivalence class of the associated Leavitt path R-algebra, and therefore a number of graphical constructions which yield Morita equivalent C

We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the C * -algebra associated to non-Hausdorffétale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that the C * -algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in Z 2 is not simple.2010 Mathematics Subject Classification. 16S99, 16S10, 22A22, 46L05, 46L55.