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 = {X_s :s\in P} of Hilbert bimodules, indexed by a semigroup P, which are endowed with a multiplication which implements isomorphisms X_s\otimes_A X_t \to X_{st}; such a family is a called a product system. We define a generalized Cuntz- Pimsner algebra O_X, and we show that every twisted crossed product of A by P can be realized as O_X for a suitable product system X. Assuming P is quasi- lattice ordered in the sense of Nica, we analyze a certain Toeplitz extension T_{cov}(X) of O_X by embedding it in a crossed product B_P \times_{\tau,X} P which has been ``twisted'' by X; our main Theorem is a characterization of the faithful representations of B_P \times_{\tau,X} P.