Normal completely positive map - Imported Knowlpedia Content

Definition

Let $M$ and $N$ be von Neumann algebras. A linear map $\Phi:M\to N$ is a normal completely positive map if it is both completely positive and normal. Equivalently, every matrix amplification of $\Phi$ is positive and $\Phi$ is ultraweakly continuous. Since $\Phi$ is positive, normality may also be tested on increasing bounded nets:

x_i\uparrow x\quad\Longrightarrow\quad \Phi(x_i)\uparrow\Phi(x).

Neither unitality nor multiplicativity is included in this definition.

Preduals and normality

Normality is equivalent to the existence of a bounded preadjoint $\Phi_*:N_*\to M_*$ satisfying

\omega(\Phi(x))=\Phi_*(\omega)(x) \qquad(\omega\in N_*,\ x\in M).

This makes normal CP maps compatible with the intrinsic preduals of von Neumann algebras. The order and predual characterizations of normality are developed in Takesaki, Chapter III, §§2–3.

Examples and closure properties

Every normal $*$ -homomorphism is normal and completely positive. For a bounded operator $V:K\to H$ , the map $B(H)\to B(K)$ , $x\mapsto V^*xV$ , is normal CP. Compositions and finite sums of normal CP maps remain normal CP. Normal conditional expectations are important idempotent examples.

Distinctions

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapters III–IV on normal maps and complete positivity.
Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. DOI record. Relevant: Chapter 4 on completely positive maps and their representations.