We construct reduced and full semigroup C*-algebras for left cancellative semigroups. Our new construction covers particular cases already considered by A. Nica and also Toeplitz algebras attached to rings of integers in number fields due to J. Cuntz.Moreover, we show how (left) amenability of semigroups can be expressed in terms of these semigroup C*-algebras in analogy to the group case.2000 Mathematics Subject Classification. Primary 46L05; Secondary 20Mxx, 43A07.

We study C*-algebras associated with subsemigroups of groups. For a large class of such semigroups including positive cones in quasi-lattice ordered groups and left Ore semigroups, we describe the corresponding semigroup C*algebras as C*-algebras of inverse semigroups, groupoid C*-algebras and full corners in associated group crossed products. These descriptions allow us to characterize nuclearity of semigroup C*-algebras in terms of faithfulness of left regular representations and amenability of group actions. Moreover, we also determine when boundary quotients of semigroup C*-algebras are UCT Kirchberg algebras. This leads to a unified approach to Cuntz algebras and ring C*-algebras.2000 Mathematics Subject Classification. Primary 46L05; Secondary 20Mxx, 43A07.Research supported by the ERC through AdG 267079. 2 XIN LI connection between nuclearity and faithfulness of the left regular representation remained mysterious. Recently, new examples of C*-algebras arising from number theory ([Cun], [Cu-Li1],[Li1], [Cu-Li2]) have motivated the author to generalize A. Nica's work. For semigroups associated with number theoretic rings, the restriction to positive cones of quasi-lattice ordered groups corresponds to only considering principal ideal domains -a restriction which, especially for rings from algebraic number theory, would exclude all the interesting examples. Making explicit use of the ideal structure of semigroups, the author was able to extend A. Nica's construction to arbitrary left cancellative semigroups in [Li2]. The same construction was introduced independently in [C-D-L] for particular examples of number theoretic interest. In general, it turns out that the full semigroup C*-algebras still have good properties. For instance, it is shown in [Li2] and also [Nor] that they are well-suited for studying amenability of semigroups. However, amenability is a strong assumption which interesting examples fail to have. One of the most striking examples is probably the n-fold free product N * n 0 of the natural numbers. This example is due to A. Nica, and he observed that it is closely related to the Cuntz algebra O n .A closely related topic is the theory of semigroup crossed products (by endomorphisms). One of the most important ideas in the analysis of semigroup crossed products is the idea of dilation. It already goes back to J. Cuntz in his work on the Cuntz algebras. This dilation theory has then been fully developed, in the probably most general setting, by M. Laca in [La2]. He shows that one can use inductive limit procedures to dilate isometries to unitaries and endomorphisms to automorphisms so that in the end, semigroup crossed products can be embedded as full corners into group crossed products. This means that questions about semigroup crossed products translate into questions about group crossed products which have already been intensively studied. However, this dilation theory as described here only works for left Ore semigroups, and the question remains what to do for semigroups like the free produ...

Abstract. We show that a separable, nuclear C*-algebra satisfies the UCT if it has a Cartan subalgebra. Furthermore, we prove that the UCT is closed under crossed products by group actions which respect Cartan subalgebras. This observation allows us to deduce, among other things, that a crossed product O 2 ⋊ α Z p satisfies the UCT if there is some automorphism γ of O 2 with the property that γ(D 2 ) ⊆ O 2 ⋊ α Z p is regular, where D 2 denotes the canonical masa of O 2 . We prove that this condition is automatic if γ(D 2 ) ⊆ O 2 ⋊ α Z p is not a masa or α(γ(D 2 )) is inner conjugate to γ(D 2 ). Finally, we relate the UCT problem for separable, nuclear, M 2 ∞ -absorbing C*-algebras to Cartan subalgebras and order two automorphisms of O 2 .

Abstract. We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterise continuous orbit equivalence in terms of isomorphisms of C*-crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and FeldmanMoore in the measurable setting. Secondly, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this is not always the case by constructing topological dynamical systems (actions of free abelian groups, and also non-abelian free groups) which are continuously orbit equivalent but not conjugate. Furthermore, we prove positive rigidity results. For instance, it turns out that general topological Bernoulli actions are rigid when compared with actions of nilpotent groups, and that topological Bernoulli actions of duality groups are rigid when compared with actions of solvable groups. The same is true for certain subshifts of full shifts over finite alphabets.

Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies the Toeplitz condition of [24] and that the Baum-Connes conjecture holds for G. We prove a formula describing the Ktheory of the reduced crossed product A ⋊α,r P by any automorphic action of P . This formula is obtained as a consequence of a result on the K-theory of crossed products for special actions of G on totally disconnected spaces. We apply our result to various examples including left Ore semigroups and quasilattice ordered semigroups. We also use the results to show that for certain semigroups P , including the ax + b-semigroup R ⋊ R × for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras C * λ (P ) and C * ρ (P ) coincide, although the structure of these algebras can be very different. Preliminaries on totally disconnected spacesRecall that a locally compact Hausdorff space Ω is totally disconnected if and only if its topology has a basis of compact open subsets. The corresponding algebras C 0 (Ω) of continuous functions which vanish at infinity are precisely the commutative AF-Algebras. In what follows, if V ⊆ Ω, then 1 V : Ω → C denotes the characteristic function of V . Definition 2.1. Let Ω be a totally disconnected locally compact Hausdorff space and let V be a family of compact open subsets in Ω. Moreover, let U c (Ω) denote the set of all compact open subsets of Ω. Then we say that V is a generating family of the compact open sets of Ω if U c (Ω) coincides with the smallest family U of compact open sets in Ω which contains V and which is closed under finite intersections, finite unions, and under taking differences U W with U, W ∈ U . Lemma 2.2. Suppose that V is a family of compact open sets in the totally disconnected space Ω. Then the following are equivalent (1) The set {1 V : V ∈ V} generates C 0 (Ω) as a C*-algebra. (2) The set V generates U c (Ω) in the sense of Definition 2.1. Moreover, if V is closed under taking finite intersections, then (1) and (2) are equivalent to(3) span{1 V : V ∈ V} is a dense subalgebra of C 0 (Ω) containing span{1 U : U ∈ U c (Ω)}.