DISCRETE PRODUCT SYSTEMS AND TWISTED CROSSED-PRODUCTS BY SEMIGROUPS

A product system E over a semigroup P is a family of Hilbert spaces {E -s:s is an element of P} together with multiplications E(s)xE(t)-->E-s t. We view E as a unitary-valued cocycle on P; and consider twisted cr ossed products Ax(beta,E) P involving E and an action beta of P by end omorphisms of a C-algebra A. When P is quasi-lattice ordered in the s ense of Nica, we isolate a class of covariant representations of E, an d consider a twisted crossed product B-p x(tau,E) P which is universal for covariant representations of E when E has finite-dimensional fibr es, and in general is slightly larger. In par ticular, when P=N and di m E-1 = infinity, our algebra B-N x(tau,E) N is a new infinite analogu e of the Toeplitz-Cuntz algebras FOn. Our main theorem is a characteri sation of the faithful representations of B-P x(tau,E) P. (C) 1998 Aca demic Press.