Completely Positive Maps of the Cuntz Algebras

We construct a covariant functor from the category whose objects are the complex, infinite dimensional, separable Hilbert spaces and whose morphisms are the contractions into the category whose objects are the unitalC*-algebras and whose morphisms are the completely positive, identity-preserving maps. TheC*-algebras involved are isomorphic to the Cuntz algebra O∞and our functor extends the canonical action of the group of unitaries on the generating Hilbert space. Our result cannot apply to On,n<∞.