Ideal Structure of C * -algebras Associated with C * -correspondences

We study the ideal structure of C∗-algebras arising from C∗-correspondences. We prove that gauge-invariant ideals of our C∗-algebras are parameterized by certain pairs of ideals of original C∗-algebras. We show that our C∗-algebras have a nice property which should be possessed by generalization of crossed products. Applications to crossed products by Hilbert C∗-bimodules and relative Cuntz-Pimsner algebras are also discussed. 0. Introduction For a C∗-algebra A, a C∗-correspondence over A is a (right) Hilbert A-module with a left action of A. Since endomorphisms (or families of endomorphisms) of A define C∗-correspondences over A, we can regard C∗-correspondences as (multivalued) generalizations of automorphisms or endomorphisms. This point of view has same philosophy as the idea that topological correspondences defined in [K2] are generalizations of continuous maps (see [K2, Section 1]). A crossed product by an automorphism is a C∗-algebra which has an original C∗-algebra as a C∗-subalgebra, and reflects many aspects of the automorphism. For example, the set of ideals of the crossed product which are invariant under the dual action of the one-dimensional torus T corresponds bijectively to the set of ideals of the original C∗-algebra which are invariant under the automorphism. As C∗-correspondences are generalizations of endomorphisms, a natural problem is to define “crossed products” by C∗-correspondences. There are plenty of evidence that the construction of the C∗-algebra OX from a C ∗-correspondence X in [K4] is the right one. One piece of evidence is that this generalizes many constructions which were or were not considered as generalizations of crossed products (see [K4]). We are going to explain another piece of evidence. For a C∗-correspondence X, we can naturally define a notion of representations of X (Definition 2.7). Thus one C∗-algebra which is naturally associated with a C∗-correspondence X is a C∗-algebra TX having a universal property with respect to representations of X (Definition 3.1). This C∗algebra TX is nothing but an (augmented) Cuntz-Toeplitz algebra defined in [Pi]. When a C∗-correspondence X is defined by an automorphism, the C∗-algebra TX is isomorphic to the Toeplitz extension of the crossed product by the automorphism defined in [PV]. This C∗-algebra is too large to reflect the informations of X. In order to get “crossed products”, we have to go to a quotient of TX . There are two 1991 Mathematics Subject Classification. Primary 46L05.