The celebrated theorem of Ando says that a pair of commuting contraction on a Hilbert space always dilates to an isometric dilation. Building on this, Solel proved that every completely contractive covariant representation of a product system \(\mathbb E\) over \(\mathbb N^2_0\) dilates to an isometric covariant representation of \(\mathbb E\) over \(\mathbb N^2_0.\) This result generalizes Ando’s theorem in the special case where the \(C^{*}\) -algebra \(\mathcal {M}=\mathbb {C}\) and \(\mathbb {E}(\textbf{n})=\mathbb {C}\) for all \(\textbf{n}\in \mathbb N^2_0.\) This naturally leads to the question: What class of isometric covariant representations of a product system \(\mathbb E\) over \(\mathbb N^2_0\) are sufficient to serve as dilations for completely contractive covariant representations, as established in Solel’s dilation framework? The central objective of this article is to investigate such isometric covariant representations that serve as dilations of completely contractive covariant representations.
