Dilation theorems for VH-spaces

Abstract

In the Appendix of the book Le¸cons d’analyse fonctionnelle by F. Riesz and B. Sz.-Nagy, B. Sz.-Nagy [15] 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 [11] proved another celebrated theorem about dilation of operator valued completely positive linear maps on C ∗ -algebras. Then F.H. Szafraniec [14] showed that these theorems are actually equivalent. Due to reasons coming from multivariate stochastic processes R.M. Loynes [7], 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 [2] for B∗ -algebras with B. Sz.-Nagy’s Theorem following the lines in [14] together with some ideas from [2], and to get VHvariants of Stinespring’s Theorem for C ∗ -algebras and B∗ -algebras. Relations between these theorems are also considered.