  Furber and Jacobs have shown in their study of quantum computation that the
category of commutative C*-algebras and PU-maps (positive linear maps which
preserve the unit) is isomorphic to the Kleisli category of a comonad on the
category of commutative C*-algebras with MIU-maps (linear maps which preserve
multiplication, involution and unit). [Furber and Jacobs, 2013]
  In this paper, we prove a non-commutative variant of this result: the
category of C*-algebras and PU-maps is isomorphic to the Kleisli category of a
comonad on the subcategory of MIU-maps.
  A variation on this result has been used to construct a model of Selinger and
Valiron's quantum lambda calculus using von Neumann algebras. [Cho and
Westerbaan, 2016]
