The Toeplitz algebra of a Hilbert bimodule

Suppose a C ∗ -algebra A acts by adjointable operators on a Hilbert A -module X. Pimsner constructed a C ∗ -algebra 𝒪 X which includes, for particular choices of X , crossed products of A by Z , the Cuntz algebras 𝒪 n , and the CuntzKrieger algebras 𝒪 B. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the ToeplitzCuntz-Krieger algebras of directed graphs, which includes Cuntz’s uniqueness theorem for 𝒪 ∞.