On Two Equivalent Dilation Theorems in VH-Spaces

We prove that a generalized version, essentially obtained by R.M. Loynes, of the B. Sz.-Nagy??s Dilation Theorem for ${\mathcal{B}^*(\mathcal{H})}$ -valued (here ${\mathcal{H}}$ is a VH-space in the sense of Loynes) positive semidefinite maps on *-semigroups is equivalent with a generalized version of the W.F. Stinespring??s Dilation Theorem for ${\mathcal{B}^*(\mathcal{H})}$ -valued completely positive linear maps on B ^*-algebras. This equivalence result is a generalization of a theorem of F.H. Szafraniec, originally proved for the case of operator valued maps (that is, when ${\mathcal{H}}$ is a Hilbert space).