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 ×_AX_t → 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_cv(X) of O_X by embedding it in a crossed product B_P⋊_τ,XP which has been “twisted” by X; our main Theorem is a characterization of the faithful representations of B_P⋊_τ,XP.

Authors