Abstract. We give new characterizations to ensure that a free product of groups with amalgamation has a simple reduced group C * -algebra, and provide a concrete example of an amalgam with trivial kernel, such that its reduced group C * -algebra has a unique tracial state, but is not simple.Moreover, we show that there is a radical class of groups for which the reduced group C * -algebra of any group is simple precisely when the group has a trivial radical corresponding to this class.

We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form:Some a are R-related to some b; Some a are R-related to all b; All a are R-related to some b;All a are R-related to all b.Such primitives formalize sentences from natural language like 'All students read some textbooks'. Here a, b denote arbitrary sets (of objects), and R denotes an arbitrary binary relation between objects. The language of the logic contains only variables denoting sets, determining the class of set terms, and variables denoting binary relations between objects, determining the class of relational terms. Both classes of terms are closed under the standard Boolean operations. The set of relational terms is also closed under taking the converse of a relation. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem.

We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K0 group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the latter result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag–Solitar groups BS(m, n).

Abstract. Toeplitz C * -algebras of right-angled Artin groups were studied by Crisp and Laca. They are a special case of the Toeplitz C * -algebras T (G, P ) associated with quasi-lattice ordered groups (G, P ) introduced by Nica. Crisp and Laca proved that the so-called "boundary quotients" C * Q (Γ) of C * (Γ) are simple and purely infinite. For a certain class of finite graphs Γ we show that C * Q (Γ) can be represented as a full corner of a crossed product of an appropriate C * -subalgebra of C * Q (Γ) built by using C * (Γ ), where Γ is a subgraph of Γ with one less vertex, by the group Z. Using induction on the number of the vertices of Γ we show that C * Q (Γ) are nuclear and moreover belong to the small bootstrap class. We also use the Pimsner-Voiculescu exact sequence to find their K-theory. Finally we use the Kirchberg-Phillips classification theorem to show that those C * -algebras are isomorphic to tensor products of O n with 1 ≤ n ≤ ∞.