Positive Semidefinite Maps on $$*$$ -Semigroupoids and Linearisations

Motivated 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 operator 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 corresponding \(*\)-representations by bounded operators. By linearisation of these constructions, we obtain similar results for operator valued positive semidefinite maps on \(*\)-algebroids with unit and then, for the special case of \(B^*\)-algebroids with unit, we obtain a generalisation of the Stinespring’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.