Dilation theorems for VH-spaces
Uğurcan, Barış Evren
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In the Appendix of the book Le¸cons d’analyse fonctionnelle by F. Riesz and B. Sz.-Nagy, B. Sz.-Nagy  proved an important theorem on operator valued positive definite maps on ∗-semigroups, which today can be considered as one of the pioneering results of dilation theory. In the same year W.F. Stinespring  proved another celebrated theorem about dilation of operator valued completely positive linear maps on C ∗ -algebras. Then F.H. Szafraniec  showed that these theorems are actually equivalent. Due to reasons coming from multivariate stochastic processes R.M. Loynes , considered a generalization of B. Sz.-Nagy’s Theorem for vector Hilbert spaces (that he called VH-spaces). These VH-spaces have “inner products” that are vector valued, into the so-called “admissible spaces”. This work is aimed at providing a detailed proof of R.M. Loynes Theorem that generalizes B. Sz.-Nagy, a detailed proof of the equivalence of Stinespring’s Theorem in the Arveson formulation  for B∗ -algebras with B. Sz.-Nagy’s Theorem following the lines in  together with some ideas from , and to get VHvariants of Stinespring’s Theorem for C ∗ -algebras and B∗ -algebras. Relations between these theorems are also considered.