Isometric dilations for representations of product systems

We discuss representations of product systems (of W*-correspondences) over the semigroup Zⁿ₊ and show that, under certain pureness and Szegö positivity conditions, a completely contractive representation can be dilated to an isometric representation. For n=1,2 this is known to hold in general (without assuming the conditions), but for n ≥ 3, it does not hold in general (as is known for the special case of isometric dilations of a tuple of commuting contractions). Restricting to the case of tuples of commuting contractions, our result reduces to a result of Barik, Das, Haria, and Sarkar (Isometric dilations and von Neumann inequality for a class of tuples in the polydisc. Trans. Amer. Math. Soc. 372 (2019), 1429-1450). Our dilation is explicitly constructed, and we present some applications.