Gheondea, Aurelian2025-02-272025-02-272024-09-110378-620Xhttps://hdl.handle.net/11693/116898Motivated by current investigations in dilation theory, in both operator theory and operator algebras, and the theory of groupoids, we obtain a generalisation of the Sz-Nagy’s Dilation Theorem for opera- tor valued positive semidefinite maps on ∗-semigroupoids with unit, with varying degrees of aggregation, firstly by ∗-representations with unbounded operators and then we characterise the existence of the cor- responding ∗-representations by bounded operators. By linearisation of these constructions, we obtain similar results for operator valued posi- tive semidefinite maps on ∗-algebroids with unit and then, for the special case of B∗-algebroids with unit, we obtain a generalisation of the Stine- spring’s Dilation Theorem. As an application of the generalisation of the Stinespring’s Dilation Theorem, we show that some natural questions on C∗-algebroids are equivalent.EnglishCC BY 4.0 Deed (Attribution 4.0 International)https://creativecommons.org/licenses/by/4.0/Primary 47L75Secondary 43A3547A2047L6046L99∗-semigroupoid∗-algebroidPositive semidefiniteCompletely positiveDilation∗-representationArticle10.1007/s00020-024-02777-41420-8989