The Toeplitz Algebra of a Hilbert Bimodule

Abstract

Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C* -algebra script O signX which includes, for particular choices of X, crossed products of A by ℤ, the Cuntz algebras script O signn, and the Cuntz-Krieger algebras script O signB. 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 script O sign∞.