Normal *-homomorphism - Imported Knowlpedia Content

Definition

Let $M$ and $N$ be von Neumann algebras. A $*$ -homomorphism $\pi:M\to N$ is normal if it is normal as a linear map, equivalently continuous from the ultraweak topology of $M$ to that of $N$ . Thus there is a bounded preadjoint $\pi_*:N_*\to M_*$ satisfying

\pi_*(\omega)=\omega\circ\pi\qquad(\omega\in N_*).

Normality does not by itself require $\pi$ to be unital or injective. If the chosen category uses unital morphisms, preservation of the identity is an additional axiom.

Equivalent order criterion

Because every $*$ -homomorphism is positive, $\pi$ is normal exactly when it preserves suprema of bounded increasing nets of positive elements:

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

It is enough to test the corresponding monotone-continuity condition on increasing nets of projections. This criterion is order-theoretic but is equivalent to ultraweak continuity only because the map is positive Takesaki, chapter on von Neumann algebra topologies.

Examples and a non-example

The identity map, inclusions of von Neumann subalgebras, and spatial amplifications $x\mapsto x\otimes I_K$ are normal. In contrast, let $\mathcal U$ be a free ultrafilter on $\mathbb N$ . The ultrafilter character

\ell^\infty(\mathbb N)\longrightarrow\mathbb C,\qquad (x_n)\longmapsto\lim_{\mathcal U}x_n

is a unital $*$ -homomorphism but is not normal: the projections onto the first $m$ coordinates increase to $1$ , while every one has ultrafilter limit $0$ .

Stability and conventions

Composites of normal $*$ -homomorphisms are normal, and restricting the codomain to a von Neumann subalgebra containing the range does not change normality. The terms “normal,” “ultraweakly continuous,” and “weak-star continuous” agree here when each algebra carries its canonical predual.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: the chapters on von Neumann algebra topologies, preduals, and normal maps.
Shôichirô Sakai, C-Algebras and W-Algebras, Springer, 1971; Classics in Mathematics reprint, 1998. DOI record. Relevant: the abstract predual formulation and normal homomorphisms.