Zappa-Sz\'{e}p actions of groups on product systems

Abstract

Let $G$ be a group and $X$ be a product system over a semigroup $P$. Suppose $G$ has a left action on $P$ and $P$ has a right action on $G$, so that one can form a Zappa-Sz\'ep product $P\bowtie G$. We define a Zappa-Sz\'ep action of $G$ on $X$ to be a collection of functions on $X$ that are compatible with both actions from $P\bowtie G$ in a certain sense. Given a Zappa-Sz\'ep action of $G$ on $X$, we construct a new product system $X\bowtie G$ over $P\bowtie G$, called the Zappa-Sz\'ep product of $X$ by $G$. We then associate to $X\bowtie G$ several universal C*-algebras and prove their respective Hao-Ng type isomorphisms. A special case of interest is when a Zappa-Sz\'{e}p action is homogeneous. This case naturally generalizes group actions on product systems in the literature. For this case, besides the Zappa-Sz\'ep product system $X\bowtie G$, one can also construct a new type of Zappa-Sz\'{e}p product $X \widetilde\bowtie G$ over $P$. Some essential differences arise between these two types of Zappa-Sz\'ep product systems and their associated C*-algebras.