In this paper, we consider the Toeplitz algebra associated to actions of Ore semigroups on C * -algebras. In particular, we consider injective and surjective actions of such semigroups. We use the theory of groupoid dynamical systems to represent the Toeplitz algebra as a groupoid crossed product. We also discuss the K-theory of the Toeplitz algebra in some examples. For instance, we show that for the semigroup of positive matrices, the K-theory of the associated Toeplitz algebra vanishes.

In this paper, we consider actions of locally compact Ore semigroups on compact topological spaces. Under mild assumptions on the semigroup and the action, we construct a semi-direct product groupoid with a Haar system. We also show that it is Morita-equivalent to a transformation groupoid. We apply this construction to the Wiener-Hopf C * -algebras. 1

Abstract. The odd-dimensional quantum sphere S 2`C1 q is a homogeneous space for the quantum group SU q .`C 1/. A generic equivariant spectral triple for S 2`C1 q on its L 2 -space was constructed by Chakraborty and Pal in [4]. We prove regularity for that spectral triple here. We also compute its dimension spectrum and show that it is simple. We give a detailed construction of its smooth function algebra and some related algebras that help proving regularity and in the computation of the dimension spectrum. Following the idea of Connes for SU q .2/, we first study another spectral triple for S 2`C1 q equivariant under torus group action and constructed by Chakraborty and Pal in [3]. We then derive the results for the SU q .`C 1/-equivariant triple in the case q D 0 from those for the torus equivariant triple. For the case q ¤ 0, we deduce regularity and dimension spectrum from the case q D 0. (2010). 58B34, 46L87, 19K33. Mathematics Subject Classification

In this paper, we study E-semigroups over convex cones. We prove a structure theorem for E-semigroups which leave the algebra of compact operators invariant. Then we study in detail the CCR flows, E0-semigroups constructed from isometric representations, by describing their units and gauge groups. We exhibit an uncountable family of 2-parameter CCR flows, containing mutually non-cocycle-conjugate E0-semigroups.

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