C*-algebras associated to product systems of Hilbert bimodules

Abstract

Let (G,P) be a quasi-lattice ordered group and let X be a compactly aligned product system over P of Hilbert bimodules. Under mild hypotheses we associate to X a C*-algebra which we call the Cuntz-Nica-Pimsner algebra of X. Our construction generalises a number of others: a sub-class of Fowler's Cuntz-Pimsner algebras for product systems of Hilbert bimodules; Katsura's formulation of Cuntz-Pimsner algebras of Hilbert bimodules; the C*-algebras of finitely aligned higher-rank graphs; and Crisp and Laca's boundary quotients of Toeplitz algebras. We show that for a large class of product systems X, the universal representation of X in its Cuntz-Nica-Pimsner algebra is isometric.

arXiv comment24 pages. v2: material has been rearranged so that the algebra NO_X is defined only under hypotheses which ensure that the universal representation is injective. The substance of the results is unchanged. v3: minor revisions; this version to appear in J. Operator Theory

References in corpus (3)

• Higher rank graph C*-algebras
• Product systems of graphs and the Toeplitz algebras of higher-rank graphs
• A construction of $C^{\ast }$-algebras from $C^{\ast }$-correspondences

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• Topological aperiodicity for product systems over semigroups of Ore type
• On C*-algebras of irreversible algebraic dynamical systems
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• Topological freeness for $\ast$-commuting covering maps
• Self-Similar $k$-Graph C*-Algebras
• C*-envelopes of semicrossed products by lattice ordered abelian semigroups
• Iterating the Cuntz-Nica-Pimsner construction for compactly aligned product systems
• C*-envelopes of tensor algebras of product systems
• Product-system models for twisted $C^{\ast }$-algebras of topological higher-rank graphs
• The Cuntz Algebra Q_N and C*-Algebras of Product Systems
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• Homotopy of product systems and K-theory of Cuntz-Nica-Pimsner algebras
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• KMS states on $C^{\ast }$-algebras associated to a family of $\ast$-commuting local homeomorphisms
• Boundary Quotient C*-algebras of Products of Odometers
• Co-universal C*-algebras associated to generalised graphs
• Ionescu's theorem for higher rank graphs
• C^*-algebras from k group representations
• Two families of Exel-Larsen crossed products