On Certain Cuntz-pimsner Algebras

Let A be a separable unital C*-algebra and let π : A → L(H) be a faithful representation of A on a separable Hilbert space H such that π(A) ∩ K(H) = {0}. We show that OE , the Cuntz-Pimsner algebra associated to the Hilbert A-bimodule E = H⊗C A, is simple and purely infinite. If A is nuclear and belongs to the bootstrap class to which the UCT applies, then the same applies to OE . Hence by the Kirchberg-Phillips Theorem the isomorphism class of OE only depends on the K-theory of A and the class of the unit. In his seminal paper [Pm], Pimsner constructed a C*-algebra OE from a Hilbert bimodule over a C*-algebra A as a quotient of a concrete C*-algebra TE , an analogue of the Toeplitz algebra, acting on the Fock space associated to E. There has recently been much interest in these CuntzPimsner algebras (or Cuntz-Krieger-Pimsner algebras), which generalize both crossed products by Z and Cuntz-Krieger algebras, as well as the associated Toeplitz algebras. The structure of these C*-algebras is not yet fully understood, although considerable progress has been made. For example, Pimsner found a six-term exact sequence for the K-theory of OE which generalizes the Pimsner-Voiculescu exact sequence (see [Pm, Theorem 4.8]); conditions for simplicity were found in [Sc2, MS, KPW1, DPW] and for pure infiniteness in [Z]. The purpose of the present note is to analyze the structure of Cuntz-Pimsner algebras associated to a certain class of Hilbert bimodules. Let A be a separable unital C*-algebra and let π : A → L(H) be a faithful representation of A on a separable Hilbert space H such that π(A)∩K(H) = {0}. Then E = H⊗C A is a Hilbert bimodule over A in a natural way. We show that OE is separable, simple and purely infinite. If A is nuclear and in the bootstrap class, then the same holds for OE and thus by the Kirchberg-Phillips theorem the isomorphism class of OE is completely determined by the K-theory of A together with the class of the unit (since OE is KK-equivalent to A). Many examples of Cuntz-Pimsner algebras found in the literature arise from Hilbert bimodules which are finitely generated and projective; in such cases the left action must consist entirely of compact operators. Our examples do not fall in this class; in fact, the left action has trivial intersection with the compacts. And this has some interesting consequences: OE ∼= TE (see [Pm, Corollary 3.14]) and the natural embedding A →֒ OE induces a KK-equivalence (see [Pm, Corollary 4.5]). In §1 we review some basic facts concerning the construction of TE as operators on the Fock space of E and the gauge action λ : T → Aut (TE). We assume that the left action of A does not meet the compacts K(E) and identify OE with TE . The fixed point algebra FE , the analogue of the AF-core of a Cuntz-Krieger algebra, contains a canonical descending sequence of essential ideals indexed by N with trivial intersection. The crossed product OE ⋊λ T has a similar collection of essential ideals indexed by Z on which the dual group of automorphisms acts in a natural way. By Takesaki-Takai duality OE ⊗K(L (T)) ∼= (OE ⋊λ T)⋊λ̂ Z; Date: 28 August 2001. 1991 Mathematics Subject Classification. Primary 46L05; Secondary 46L55.