The Extension of Unital Completely Positive Semigroups on Operator Systems to Semigroups on $C^*$-algebras

TL;DR

This paper demonstrates that continuous unital completely positive semigroups on matricial operator systems can be extended to semigroups on finite-dimensional $C^*$-algebras, with uniqueness in the invertible case, advancing the mathematical framework for open quantum systems.

Contribution

It establishes the extension of unital completely positive semigroups from operator systems to finite-dimensional $C^*$-algebras, including a uniqueness result for invertible semigroups.

Findings

Any continuous unital CP semigroup on matricial systems can be extended to a finite-dimensional $C^*$-algebra.

The extension is unique if the semigroup is invertible.

The extension uses the injective envelope of the operator system.

Abstract

The study of open quantum systems relies on the notion of unital completely positive semigroups on $C^*$-algebras representing physical systems. The natural generalisation would be to consider the unital completely positive semigroups on operator systems. We show that any continuous unital completely positive semigroup on matricial system can be extended to a semigroup on a finite-dimensional $C^*$-algebra, which is an injective envelope of the matricial system. In case the semigroup is invertible, this extension is unique.