Normal linear map - Imported Knowlpedia Content

Definition

Let $M$ and $N$ be von Neumann algebras with preduals $M_*$ and $N_*$ . A bounded linear map $\Phi:M\to N$ is normal when it is continuous from the ultraweak topology on $M$ to the ultraweak topology on $N$ . Equivalently, there is a unique bounded linear preadjoint $\Phi_*:N_*\to M_*$ such that

\langle \Phi_*(\omega),x\rangle=\langle\omega,\Phi(x)\rangle

for every $x\in M$ and $\omega\in N_*$ . Thus normality records weak-star continuity, not norm continuity alone.

Order characterization

If $\Phi$ is additionally a positive linear map, normality is equivalent to preservation of bounded increasing suprema:

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

for every bounded increasing net $(x_i)$ in $M_+$ . Equivalently, this condition may be tested on increasing nets of projections. These order criteria, with positivity explicit, are standard in Takesaki, chapter III, section 3.

Stability and use

Composites of normal linear maps are normal. Normal unital positive maps preserve increasing limits of observables, while normal $*$ -homomorphisms are the morphisms most compatible with the weak-star structure of von Neumann algebras. The preadjoint reverses composition: $(\Psi\circ\Phi)_*=\Phi_*\circ\Psi_*$ .

Conventions and scope

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: chapter III, sections 2–3 on preduals, ultraweak continuity, and normal maps.