On the universal property of Pimsner–Toeplitz -algebras and their continuous analogues

We consider -algebras generated by a single -correspondence (Pimsner–Toeplitz algebras) and by a product systems of -correspondences. We give a new proof of a theorem of Pimsner, which states that any representation of the generating -correspondence gives rise to a representation of the Pimsner–Toeplitz algebra. Our proof does not make use of the conditional expectation onto the subalgebra fixed under the dual action of the circle group. We then prove the analogous statement for the case of product systems, generalizing a theorem of Arveson from the case of product systems of Hilbert spaces.