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 ∈ P } of Hilbert bimodules, indexed by a semigroup P , which are endowed with a multiplication which implements isomorphisms X s ⊗ A X 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,X P which has been "twisted" by X; our main Theorem is a characterization of the faithful representations of B P τ,X P .Introduction.

Let X be a Hilbert bimodule over a C *-algebra A. We analyse the structure of the associated Cuntz-Pimsner algebra X and related algebras using representation-theoretic methods. In particular, we study the ideals (I) in X induced by appropriately invariant ideals I in A, and identify the quotients X /(I) as relative Cuntz-Pimsner algebras of Muhly and Solel. We also prove a gauge-invariant uniqueness theorem for X , and investigate the relationship between X and an alternative model proposed by Doplicher, Pinzari and Zuccante.

Abstract. Suppose a C * -algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C * -algebra OX which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras On, and the Cuntz-Krieger algebras OB. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the Toeplitz-Cuntz-Krieger algebras of directed graphs, which includes Cuntz's uniqueness theorem for O∞.A Hilbert bimodule X over a C * -algebra A is a right Hilbert A-module with a left action of A by adjointable operators. The motivating example comes from an automorphism α of A: take X A = A A , and define the left action of A by a · b := α(a)b. In [23], Pimsner constructed a C * -algebra O X from a Hilbert bimodule X in such a way that the O X corresponding to an automorphism α is the crossed product A × α Z. He also produced interesting examples of bimodules which do not arise from automorphisms or endomorphisms, including bimodules over finite-dimensional commutative C * -algebras for which the corresponding O X are the Cuntz-Krieger algebras. The Cuntz algebra O n is O X when C X C is a Hilbert space of dimension n and the left action of C is by multiples of the identity.Here we use methods developed in [18,9] for analysing semigroup crossed products to study Pimsner's algebras. These methods seem to apply more directly to Pimsner's analogue of the Toeplitz-Cuntz algebras rather than his analogue O X of the Cuntz algebras. Nevertheless, our results yield new information about the Cuntz-Krieger algebras of some infinite graphs, giving a whole class of these algebras which behave like O ∞ .The uniqueness theorems for C * -algebras generated by algebraic systems of isometries say, roughly speaking, that all examples of a given system in which the isometries are non-unitary generate isomorphic C * -algebras. We can approach such a theorem by introducing a C * -algebra which is universal for systems of the given type, and then characterising its faithful representations. Here the systems consist of representations ψ of X and π of A on the same Hilbert space which convert the module actions and the inner product to operator multiplication; we call these Toeplitz representations of X. (The partial isometries and isometries appearing in more conventional systems are obtained by applying ψ to the elements of a basis for X.) In Section 1, we discuss these Toeplitz representations, show that there is a universal C * -algebra T X generated by a Toeplitz representation, and prove some general results relating these representations to the induced representations of Rieffel.

Abstract. We associate C * -algebras to infinite directed graphs that are not necessarily locally finite. By realizing these algebras as Cuntz-Krieger algebras in the sense of Exel and Laca, we are able to give criteria for their uniqueness and simplicity, generalizing results of Kumjian, Pask, Raeburn, and Renault for locally finite directed graphs.of edges, and range and source maps r, s : For finite graphs, these are precisely the Cuntz-Krieger algebras O A : given E, take A to be the edge matrix A E defined byconversely, given an I × I matrix A, form the graph E A with vertex set I and incidence matrix A, and then C * (E A ) is isomorphic to O A in a slightly nonobvious way (see [9, Theorem 3] or [13, Proposition 4.1]). This correspondence carries over to locally finite graphs (graphs in which vertices receive and emit finitely many edges), and the classical uniqueness and simplicity theorems for Cuntz-Krieger algebras have elegant extensions to these graph algebras [11,12].If a vertex v emits infinitely many edges, then the Cuntz-Krieger relation (1.1) does not make sense in an abstract C * -algebra. On the other hand, Fowler and Raeburn have recently noticed that if all vertices emit infinitely many edges, one can obtain uniqueness and simplicity theorems for graph algebras without demanding equality in (1.1) [8, Corollaries 4.2 and 4.5]. This suggests that a satisfactory theory should be possible if we merely insist that (1.1) holds when v emits finitely many edges.Exel and Laca have studied a generalization of the Cuntz-Krieger algebras which allows arbitrary infinite matrices, and have obtained uniqueness and simplicity theorems among other interesting results [7]. We shall show that, if we define C * (E) in the way suggested by [8], then we can pass from the graph algebra C * (E)

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 Ä 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. Academic PressCrossed 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