Browsing by Subject "Operator algebras."

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    Complete positivity in operator algebras
    (2006) Kavruk, Ali Şamil
    In this thesis we survey positive and completely positive maps defined on operator systems. In Chapter 3 we study the properties of positive maps as well as construction of positive maps under certain conditions. In Chapter 4 we focus on completely positive maps. We give some conditions on domain and range under which positivity implies complete positivity. The last chapter consists of Stinespring’s dilation theorem and its applications to various areas.
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    Dilation theorems for VH-spaces
    (2009) Uğurcan, Barış Evren
    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.