Interpolation and non-dilatable families of ${\mathcal{C}}_0$-semigroups

We generalise a technique of Bhat and Skeide (2015) to interpolate commuting families $\{S_i{\}}_{i\in \mathcal{I}}$ of contractions on a Hilbert space $\mathcal{H}$, to commuting families $\{T_i{\}}_{i\in \mathcal{I}}$ of contractive ${\mathcal{C}}_0$-semigroups on $L^2({\prod }_{i\in \mathcal{I}}\mathbb{T})\otimes \mathcal{H}$. As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott’s construction (1970), we then demonstrate for $d\in \mathbb{N}$ with $d\ge 3$ the existence of commuting families $\{T_i{\}}_{i=1}^d$ of contractive ${\mathcal{C}}_0$-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt. the topology of uniform wot-convergence on compact subsets of ${\mathbb{R}}_{\ge 0}^d$ of non-unitarily dilatable and non-unitarily approximable $d$-parameter contractive ${\mathcal{C}}_0$-semigroups on separable infinite-dimensional Hilbert spaces for each $d\ge 3$. Similar results are also developed for $d$-tuples of commuting contractions. And by building on the counter-examples of Varopoulos–Kaijser (1973–74), a 0–1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, $\mathit{\text{viz.}}$ that `typical’ pairs of commuting operators can be simultaneously embedded into commuting pairs of ${\mathcal{C}}_0$-semigroups, which extends results of Eisner (2009–10).