Normal representation - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra and $H$ a Hilbert space. A normal representation of $M$ on $H$ is a $C^*$ -algebra representation

\pi:M\longrightarrow\mathcal B(H)

that is a normal $*$ -homomorphism. Equivalently, $\pi$ is continuous from the ultraweak topology of $M$ to the ultraweak operator topology of $\mathcal B(H)$ . Normality is separate from faithfulness and unitality. Under the convention that representations are nondegenerate, one additionally requires $\overline{\pi(M)H}=H$ , which for unital $M$ is equivalent to $\pi(1)=I_H$ .

Coefficient-functional criterion

The representation is normal exactly when every coefficient functional

x\longmapsto\langle\pi(x)\xi,\eta\rangle,\qquad \xi,\eta\in H,

belongs to the predual $M_*$ . Equivalently, for every bounded increasing net $(x_i)$ of positive elements, $\pi(\sup_i x_i)=\sup_i\pi(x_i)$ . These criteria connect the concrete operator representation with the canonical weak-star structure of $M$ Takesaki, treatment of normal representations.

Examples and a non-example

The defining inclusion of a concrete von Neumann algebra $M\subseteq\mathcal B(H)$ is normal, as is any spatial amplification $x\mapsto x\otimes I_K$ . Direct sums of normal representations are normal. The representation of $\ell^\infty(\mathbb N)$ on $\mathbb C$ given by a free-ultrafilter character is not normal: it fails to preserve the supremum of the increasing sequence of finite-coordinate projections.

Decomposition and scope

A degenerate normal representation splits as its nondegenerate restriction on $\overline{\pi(M)H}$ plus the zero representation on the orthogonal complement. Normal representations are therefore often defined to be unital without losing the nonzero essential part, but the convention should be stated.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: the chapters on normal representations, preduals, and ultraweak continuity.
Shôichirô Sakai, C-Algebras and W-Algebras, Springer, 1971; Classics in Mathematics reprint, 1998. DOI record. Relevant: representations of abstract $W^*$ -algebras and normality.