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 endowed with a multiplication which
implements isomorphisms Xs⊗AXt→ 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 realized 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 BP⋊τ,XP which has been “twisted” by X;
our main Theorem is a characterization of the faithful representations of
BP⋊τ,XP.
