In this paper we introduce the crossed product construction for a discrete group action on an operator system. In analogy to the work of E. Katsoulis and C. Ramsey, we describe three canonical crossed products arising from such a dynamical system. We describe how these crossed product constructions behave under G-equivariant maps, tensor products, and the canonical C * -covers. We show that hyperrigidity is preserved under two of the three crossed products. Finally, using A. Kavruk's notion of an operator system that detects C * -nuclearity, we give a negative answer to a question on operator algebra crossed products posed by Katsoulis and Ramsey.

We show that hyperrigidity for a C*-correspondence (A, X) is equivalent to non-degeneracy of the left action of the Katsura ideal $$\mathcal {J}_X$$ J X on X. This extends the work of Kakariadis (Bull Lond Math Soc 45(6):1119–1130, 2013, Theorem 3.3) and Dor-On and Salomon (J Lond Math Soc 98(2):416–438, 2018, Theorem 3.5) who establish this equivalence for discrete graphs as well as the work of Katsoulis and Ramsey (The non-selfadjoint approach to the Hao–Ng isomorphism, arXiv preprint arXiv:1807.11425, 2018, Theorem 3.1), who establish one direction of this equivalence.