Normal unital completely positive map - Imported Knowlpedia Content

Definition

Let $M$ and $N$ be unital von Neumann algebras. A linear map $\Phi:M\to N$ is a normal unital completely positive map, or normal UCP map, if it is a normal completely positive map and is also a unital completely positive map; explicitly,

\Phi(1_M)=1_N.

Equivalently, $\Phi$ is ultraweakly continuous, all its matrix amplifications are positive, and it preserves the identity. These three requirements are independent pieces of the definition. In particular, a normal UCP map is not required to be multiplicative or invertible.

Basic properties

A normal UCP map is contractive and satisfies the Kadison–Schwarz inequality

\Phi(x)^*\Phi(x)\leq\Phi(x^*x).

Compositions of normal UCP maps are again normal UCP. These properties follow from unital complete positivity, while normality ensures compatibility with increasing limits and preduals Paulsen, Chapters 2–4.

Examples

Normal unital $*$ -homomorphisms and normal conditional expectations are normal UCP. On $M_n(\mathbb C)$ , the map

\Phi(x)=t x+(1-t)\operatorname{tr}_n(x)1 \qquad(0\leq t\leq1)

is normal UCP, where $\operatorname{tr}_n$ is the normalized trace; for generic $0<t<1$ , it is not multiplicative. A normal state is exactly a normal UCP map $M\to\mathbb C$ .

Predual interpretation

The preadjoint of a normal UCP map acts in the opposite direction on normal functionals. In quantum-information terminology, the normal UCP map is the Heisenberg-picture evolution; its preadjoint preserves the normalization of normal states. This duality uses normality essentially—without it, a preadjoint on the canonical preduals need not exist.

References

Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. DOI record. Relevant: Chapters 2–4 on UCP maps, Schwarz inequalities, and dilations.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapters III–IV on normal maps and completely positive maps.