UNITAL DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS

Semigroups of completely positive maps arise naturally both in noncommu-tative stochastic processes and in the dynamics of open quantum systems. Since its inception in the 1970’s, the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an en-domorphism semigroup. In quantum dynamics, this amounts to embedding a given open system inside some closed system, while in noncommutative probability, it cor-responds to the construction of a Markov process from its transition probabilities. In addition to the existence of dilations, one is interested in what properties of the original semigroup (unitality, various kinds of continuity) are preserved. Several authors have proved the existence of dilations, but in general, the dilation achieved has been non-unital; that is, the unit of the original algebra is embedded as a proper projection in the dilation algebra. A unique approach due to Jean-Luc Sauvageot overcomes this problem, but leaves unclear the continuity of the dilation semigroup. The major purpose of this thesis, therefore, is to further develop Sauvageot’s theory in order to prove the existence of continuous unital dilations. Thi