Discrete Product Systems and Twisted Crossed Products by Semigroups

A product systemEover a semigroupPis a family of Hilbert spaces {Es : s∈P} together with multiplicationsEs×Et→Est. We viewEas a unitary-valued cocycle onP, and consider twisted crossed productsA⋊β, E PinvolvingEand an actionβofPby endomorphisms of aC*-algebraA. WhenPis quasi-lattice ordered in the sense of Nica, we isolate a class of covariant representations ofE, and consider a twisted crossed productBP⋊τ, E Pwhich is universal for covariant representations ofEwhenEhas finite-dimensional fibres, and in general is slightly larger. In particular, whenP=Nand dim E1=∞, our algebraBN⋊τ, E N is a new infinite analogue of the Toeplitz-Cuntz algebras TOn. Our main theorem is a characterisation of the faithful representations ofBP⋊τ, E P.