turkmath.org

It has been long known that for maps taking values in operator spaces the assumption of complete positivity is strong enough to obtain dilation results. Often, complete positivity makes it possible to obtain a construction which leads to a "bigger" space, and "simpler" dilated objects acting on them. In this talk, we review some of the classical results, namely, GNS construction, Naimark's dilation theorems, Stinespring's dilation theorem and its consequences, and finally state and discuss a proof of Kasparov's generalization of Stinespring's theorem.