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We investigate structural properties of the reduced cross-sectional algebra $C^*_r(\mathcal{B})$ of a Fell bundle $\mathcal{B}$ over a discrete group $G$. Conditions allowing one to determine the ideal structure of $C^*_r(\mathcal{B})$ are studied. Notions of aperiodicity, paradoxicality and $\mathcal{B}$-infiniteness for the Fell bundle $\mathcal{B}$ are introduced and investigated by themselves and in relation to the partial dynamical system dual to $\mathcal{B}$. Several criteria of pure infiniteness of $C^*_r(\mathcal{B})$ are given. It is shown that they generalize and unify corresponding results obtained in the context of crossed products, by the following duos: Laca, Spielberg; Jolissaint, Robertson; Sierakowski, R{\o}rdam; Giordano, Sierakowski and Ortega, Pardo. For exact, separable Fell bundles satisfying the residual intersection property primitive ideal space of $C^*_r(\mathcal{B})$ is determined. The results of the paper are shown to be optimal when applied to graph $C^*$-algebras. Applications to a class of Exel-Larsen crossed products are presented., Comment: This is the version accepted to Journal of Mathematical Analysis and Applications. Pure infiniteness criteria have been generalized. In particular, now they unify the corresponding results of Jolissaint, Robertson and Sierakowski, R{\o}rdam

We prove a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type. To this end, we introduce Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product system of 'regular' C*-correspondences. Under a certain aperiodicity condition on the latter, we obtain the uniqueness theorem and a simplicity criterion for the algebras in question. These results generalize the corresponding ones for crossed products by discrete groups, due to Archbold and Spielberg, and for Exel's crossed products, due to Exel and Vershik. They also give interesting conditions for topological higher rank graphs and $P$-graphs, and apply to the new Cuntz C*-algebra $\mathcal{Q}_\mathbb{N}$ arising from the "$ax+b$"-semigroup over $\mathbb{N}$., Comment: 37 pages

We consider two Z/2Z-actions on the Podles generic quantum spheres. They yield, as noncommutative quotient spaces, the Klimek-Lesniewski q-disc and the quantum real projective space, respectively. The C*-algebras of all these quantum spaces are described as graph C*-algebras. The K-groups of the thus presented C*-algebras are then easily determined from the general theory of graph C*-algebras. For the quantum real projective space, we also recall the classification of the classes of irreducible *-representations of its algebra and give a linear basis for this algebra., Comment: 8 pages, latex2e

The irreducible *-representations of the polynomial algebra O(S^3_{pq}) of the quantum 3-sphere introduced by Calow and Matthes are classified. The K-groups of its universal C*-algebra are shown to coincide with their classical counterparts. The U(1)-action on O(S^3_{pq}) corresponding for p=1=q to the classical Hopf fibration is proven to be Galois (free). The thus obtained locally trivial Hopf-Galois extension is shown to be relatively projective (admitting a strong connection) and non-cleft. The latter is proven by determining an appropriate Chern-Connes pairing., Comment: latex 2e, 23 pages in A4