Discrete product systems of Hilbert bimodules

Abstract

A Hilbert bimodule is a right Hilbert module X over a C∗-algebra A together with a left action of A as adjointable operators on X. We consider families X = {Xs: s ∈ P} of Hilbert bimodules, indexed by a semigroup P, which are en-dowed with a multiplication which implements isomorphisms Xs ⊗A Xt → Xst; such a family is a called a product system. We define a generalized Cuntz-Pimsner algebra OX, and we show that every twisted crossed product of A by P can be re-alized as OX for a suitable product system X. Assuming P is quasi-lattice ordered in the sense of Nica, we analyze a certain Toeplitz extension Tcv(X) of OX by embedding it in a crossed product BPoτ,XP which has been “twisted ” by X; our main Theorem is a characterization of the faithful representations of BPoτ,XP. Introduction.