# Journal of Operator Theory

Volume 88, Issue 2,  Fall 2022  pp. 247-274.

Zappa-Szep actions of groups on product systems

Authors:  Boyu Li (1), Dilian Yang (2)
Author institution: (1) Department of Mathematics and Statistics, University of Victoria, Victoria, B.C. V8W 3R4, Canada
(2) Department of Mathematics and Statistics, University of Windsor, Windsor, ON. N9B 3P4, Canada

Summary:  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 $G$. We propose a notion of a Zappa-Sz\'ep action of $G$ on $X$, and 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, one can also construct another 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\'{e}p product systems and their associated C*-algebras.

DOI: http://dx.doi.org/10.7900/jot.2021jan26.2336
Keywords:  Zappa-Szep action, product system, Nica-Toeplitz representation,right LCM semigroup