Continuous unital dilations of completely positive semigroups

摘要

Since it began in the 1970s, the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an endomorphism semigroup. Several authors have proved the existence of dilations, but have usually only achieved unital dilations of specific examples, while general theorems have tended to produce non-unital dilations. A unique approach due to Jean-Luc Sauvageot leads to a unital dilation in a general setting, but leaves unclear the continuity of the dilation semigroup. The major purpose of this paper, therefore, is to further develop Sauvageot's theory in order to prove the existence of continuous unital dilations.