The extension of unital completely positive semigroups on operator systems to semigroups on C*-algebras

The theory of markovian dynamics of open quantum systems relies on the notion of one-parameter completely positive semigroup on C$^*$-algebras. Generally, the theory of operator semigroups on Banach spaces is an important field of study with an abundance of applications [1].
The development of functional analysis and quantum mechanics lead to the idea that the completely positive maps are naturally defined on operator systems [2]. Therefore we may regard the completely positive unital semigroups on operator systems to be the natural framework for studying quantum markovian dynamics.
The category of operator systems and completely positive maps has a number of good extension properties, and every operator system has an injective envelope, which is a C$^*$-algebra [3]. Using these properties, we show that every unital completely positive semigroup on operator system can be uniquely extended to a completely positive semigroup on its injective envelope.
The work is supported by the Russian Science Foundation under the grant no. 19-11- 00086 and performed in the Steklov Mathematical Institute of the Russian Academy of Sciences.