Crossed products of operator systems

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.