# 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

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