Topological aperiodicity for product systems over semigroups of Ore type

Abstract

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 $\mathbb{N}$.

arXiv comment37 pages

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Cited by in corpus (9)

• On ${\mathrm{C}}^{\ast }$-algebras associated to product systems
• The boundary quotient for algebraic dynamical systems
• Nica-Toeplitz algebras associated with product systems over right LCM semigroups
• Categorial Independence and Lévy Processes
• Pure infiniteness and ideal structure of $C^{\ast }$-algebras associated to Fell bundles
• Topological freeness for $C^{\ast }$-correspondences
• Erratum to "Full and reduced C*-coactions". Math. Proc. Camb. Phil. Soc. 116 (1994), 435--450
• On Cohomology for Product Systems
• Product systems and their representations: an approach using Fock spaces and Fell bundles