We study the exactness of the reduced crossed product of a semigroup dynamical system and the reduced $C^{*}$-algebra of a product system. We show that for a semigroup dynamical system $(A, P,\alpha )$, under reasonable hypotheses (e.g., P is abelian and finitely generated), the reduced crossed product $A \rtimes _{\mathrm {red}} P$ 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.

We give necessary and sufficient conditions for nuclearity of Cuntz–Nica–Pimsner algebras for a variety of quasi-lattice ordered groups. First we deal with the free abelian lattice case. We use this as a stepping-stone to tackle product systems over quasi-lattices that are controlled by the free abelian lattice and satisfy a minimality property. Our setting accommodates examples like the Baumslag–Solitar lattice for $n=m>0$ and the right-angled Artin groups. More generally, the class of quasi-lattices for which our results apply is closed under taking semi-direct and graph products. In the process we accomplish more. Our arguments tackle Nica–Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the coefficient algebra. This is the case for CNP-relative quotients in-between the Toeplitz–Nica–Pimsner algebra and the Cuntz–Nica–Pimsner algebra. We complete this study with the relevant results on exactness.

We prove a necessary and sufficient condition for embeddability of an operator system into ${\mathcal{O}}_{2}$ . Using Kirchberg’s theorems on a tensor product of ${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{\infty }$ , we establish results on their operator system counterparts ${\mathcal{S}}_{2}$ and ${\mathcal{S}}_{\infty }$ . Applications of the results, including some examples describing $C^{\ast }$ -envelopes of operator systems, are also discussed.

Let $G$ be a finitely generated group acting on a compact Hausdorff space ${\mathcal{X}}$. We give a fixed point characterisation for the action being amenable. As a corollary, we obtain a fixed point characterisation for the exactness of $G$.

In this paper, we give two characterizations of the exactness of operator spaces.

Given two unital continuous C*-bundles, A and B, over the same compact Hausdorff base space X, we study the continuity properties of their different amalgamated free products over C(X).