Free dilations of families of $\mathcal{C}_{0}$-semigroups and applications to evolution families

Commuting families of contractions or contractive $\mathcal{C}_{0}$-semigroups on Hilbert spaces often fail to admit power dilations resp. simultaneous unitary dilations which are themselves commutative (see [33, 11, 13]). In the \emph{non-commutative} setting, Sz.-Nagy [45] and Bożejko [4] provided means to dilate arbitrary families of contractions. The present work extends this discrete-time result to families $\{T_{i}\}_{i \in I}$ of contractive $\mathcal{C}_{0}$-semigroups. We refer to these dilations as continuous-time \emph{free unitary dilations} and present three distinct approaches to obtain them: 1) An explicit derivation applicable to semigroups that arise as interpolations; 2) A full proof with an explicit construction, via the theory of co-generators à la Słociński [40, 41] as well as a version for topologised index sets via a reformulation of the Trotter--Kato theorem for co-generators, leading to applications to evolution families; and 3) A second full proof based on the abstract structure of semigroups, which admits a natural reformulation to semigroup defined over topological free products of $\mathbb{R}_{\geq 0}$ and leads to various residuality results.