Positive semidefinite maps on ∗ and linearisations

Abstract

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 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.