A product system E over a semigroup P is a family of Hilbert spaces [E s : s # P] together with multiplications E s _E t Γ E st . We view E as a unitary-valued cocycle on P, and consider twisted crossed products A < ;, E P involving E and an action ; of P by endomorphisms of a C*-algebra A. When P is quasi-lattice ordered in the sense of Nica, we isolate a class of covariant representations of E, and consider a twisted crossed product B P < {, E P which is universal for covariant representations of E when E has finite-dimensional fibres, and in general is slightly larger. In particular, when P=N and dim E 1 = , our algebra B N < {, E N is a new infinite analogue of the Toeplitz-Cuntz algebras TO n . Our main theorem is a characterisation of the faithful representations of B P < {, E P.
1998 Academic Press
Crossed products of C*-algebras by semigroups of endomorphisms have been profitably used to model Toeplitz algebras [2, 1, 13], and the Hecke algebras arising in the Bost Connes analysis of phase transitions in number theory [3,11,14]. There are two main ways of studying such a crossed product. First, one can try to embed it as a corner in a crossed product by an automorphic action of an enveloping group, and then apply the established theory. The algebra on which the group acts is typically a direct limit, and the success of this approach depends on being able to recognise the direct limit and the action on it [7,17,23]. Or, second, one can use the techniques developed in [2,5,13] to deal directly with the semigroup crossed product and its representation theory. Here the goal is a characterisation of the faithful representations of the crossed product, and such characterisations have given important information about a wide range of semigroup crossed products [2,3,13,14].
For ordinary crossed products A < : G (those involving an action : of G by automorphisms of A ), an important adjunct are the twisted crossed article no. FU973227