The reduced Hao-Ng isomorphism problem for product systems

Abstract

We prove that the reduced Hao-Ng isomorphism problem has an affirmative answer for every generalised gauge action of a locally compact Hausdorff group on a product system over a unital subsemigroup of a discrete group, without any assumptions on the product system or the subsemigroup. We show that the Fock C*-algebra of the induced product system is canonically isomorphic to the reduced crossed product of the Fock C*-algebra, by identifying the Fock space of the induced product system with the reduced crossed product of the Fock space. This also yields the corresponding completely isometric isomorphism for the tensor algebras. Using the representations of the fixed point algebra that detect Sehnem's strong covariance relations, we show that the canonical representation of the induced product system in the reduced crossed product of the reduced strong covariant C*-algebra is strong covariant.

arXiv comment25 pages. v2: major revision, title changed. Non-degeneracy of the product system is no longer assumed. We use an identification of Fock spaces to obtain the canonical isomorphisms for the (reduced) operator algebras