Some remarks on reduced C ∗ C^* upper C Superscript asterisk-algebras of semigroup dynamical systems and product systems

We study the exactness of the reduced crossed product of a semigroup dynamical system and the reduced C ∗ C* upper C Superscript asterisk -algebra of a product system. We show that for a semigroup dynamical system ( A , P , α ) (A, P,α ) left parenthesis upper A comma upper P comma alpha right parenthesis , under reasonable hypotheses (e.g., P is abelian and finitely generated), the reduced crossed product A ⋊ red P A red P upper A right normal factor semidirect product Subscript red Baseline upper P is exact if and only if A is exact. This strengthens our earlier result (Hossain and Sundar, 2026, J. Noncommut. Geom. ), where it was assumed that the action of P on A is by injective endomorphisms. We also compare the groupoid crossed product described in Hossain and Sundar (2026) and the Fell bundle constructed in Rennie et al. (2017, Math. Proc. Cambridge Philos. Soc. , 163, 561–580) for a product system, and show that they are equivalent as Fell bundles.