On C*-algebras associated to product systems

Abstract

We consider a class of Fell bundles over quasi-lattice ordered groups. We show that these are completely determined by the positive fibres and that their cross sectional C*-algebras are relative Cuntz–Pimsner algebras associated to simplifiable product systems of Hilbert bimodules. Conversely, we show that such product systems can be naturally extended to Fell bundles and this correspondence is part of an equivalence between bicategories. We also relate amenability for this class of Fell bundles to amenability of quasi-lattice orders by showing that Fell bundles extended from free semigroups are amenable. A similar result is proved for Baumslag–Solitar groups. Moreover, we construct a relative Cuntz–Pimsner algebra of a compactly aligned product system as a quotient of the associated Nica–Toeplitz algebra. We show that this construction yields a reflector from a bicategory of compactly aligned product systems into its sub-bicategory of simplifiable product systems of Hilbert bimodules. We use this to study Morita equivalence between relative Cuntz–Pimsner algebras. In a second part, we let P be a unital subsemigroup of a group G. We propose an approach to C*-algebras associated to product systems over P. We call the C*-algebra of a given product system E its covariance algebra and denote it by A x_E P, where A is the coefficient C*-algebra. We prove that our construction does not depend on the embedding P->G and that a representation of A x_E P is faithful on the fixed-point algebra for the canonical coaction of G if and only if it is faithful on A. We compare this with other constructions in the setting of irreversible dynamical systems, such as Cuntz–Nica–Pimsner algebras, Fowler’s Cuntz–Pimsner algebra, semigroup C*-algebras of Xin Li and Exel’s crossed products by interaction groups.