Normal positive map - Imported Knowlpedia Content

Definition

Let $M$ and $N$ be von Neumann algebras. A normal positive map is a positive linear map $\Phi:M\to N$ that is normal, meaning ultraweakly continuous. Equivalently, for every bounded increasing net $(x_i)$ in $M_+$ ,

\Phi\!\left(\sup_i x_i\right)=\sup_i\Phi(x_i).

No unitality, faithfulness, or complete positivity is implicit. Since von Neumann algebras are unital, positivity already makes $\Phi$ bounded; the extra adjective “normal” specifies its compatibility with the preduals and with monotone suprema.

Preadjoint and order criteria

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

\Phi_*(\omega)=\omega\circ\Phi.

For positive maps, it is enough to test monotone preservation on increasing nets of projections. These formulations let one pass between weak-star continuity, normal functionals, and order convergence Takesaki, chapters on normal maps and positive maps.

Examples and a non-example

Every normal $*$ -homomorphism is a normal positive map. If $\xi\in H$ , the vector functional

\mathcal B(H)\longrightarrow\mathbb C,\qquad x\longmapsto\langle x\xi,\xi\rangle

is normal and positive. Normal conditional expectations provide operator-valued examples. By contrast, a free-ultrafilter state on $\ell^\infty(\mathbb N)$ is positive and unital but not normal, because it sends every finite-coordinate projection to $0$ although those projections increase to $1$ .

Stability and distinctions

Compositions and positive scalar multiples of normal positive maps remain normal and positive. Matrix amplification introduces a separate condition: a normal positive map need not be completely positive, while a normal completely positive map is normal at every matrix level.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: the chapters on preduals, normal maps, positive maps, and monotone convergence.
Shôichirô Sakai, C-Algebras and W-Algebras, Springer, 1971; Classics in Mathematics reprint, 1998. DOI record. Relevant: order and weak-star formulations of normal positive maps.