Group Actions on Product Systems

We introduce the concept of a crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We generalize a theorem of Hao and Ng about the crossed product of the Cuntz-Pimsner algebra of a $C^{\ast}$-correspondence by a group action to the context of product systems. We present examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.