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Motivated by Cuntz-Krieger-Toeplitz systems associated to undirected graphsand representations of groupoids, we obtain a generalisation of the Sz-Nagy'sDilation Theorem for operator valued partially positive semidefinite maps on$*$-semigroupoids with unit, with varying degrees of aggregation, firstly by$*$-representations with unbounded operators and then we characterise theexistence of the corresponding $*$-representations by bounded operators. Bylinearisation of these constructions, we obtain similar results for operatorvalued partially positive semidefinite maps on $*$-algebroids with unit andthen, for the special case of $B^*$-algebroids with unit, we obtain ageneralisation of the Stinespring's Dilation Theorem. As an application of thegeneralisation of the Stinespring's Dilation Theorem, we show that some naturalquestions on $C^*$-algebroids are equivalent.

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