Partially Positive Semidefinite Maps on $\ast$-Semigroupoids and Linearisations

Abstract

Motivated by Cuntz-Krieger-Toeplitz systems associated to undirected graphs and representations of groupoids, we obtain a generalisation of the Sz-Nagy's Dilation Theorem for operator valued partially positive semidefinite maps on $\ast$-semigroupoids with unit, with varying degrees of aggregation, firstly by $\ast$-representations with unbounded operators and then we characterise the existence of the corresponding $\ast$-representations by bounded operators. By linearisation of these constructions, we obtain similar results for operator valued partially positive semidefinite maps on $\ast$-algebroids with unit and then, for the special case of $B^{\ast }$-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 $C^{\ast }$-algebroids are equivalent.

arXiv comment42 pages

References in corpus (1)

• Full Cuntz-Krieger dilations via non-commutative boundaries