Normal semifinite faithful weight

Let $M$ be a von Neumann algebra. A weight $\varphi:M_+\to[0,\infty]$ is a normal semifinite faithful weight, or n.s.f. weight, if all three conditions hold: it is normal, so it preserves suprema of increasing nets in $M_+$ ; semifinite, so its finite part is order-dense in $M_+$ ; and faithful, so $\varphi(x)=0$ for $x\in M_+$ implies $x=0$ . A weight may take the value $\infty$ , so an n.s.f. weight need not be a bounded functional or a state.

The three conditions

Normality requires $\varphi(\sup_i x_i)=\sup_i\varphi(x_i)$ for every increasing net in $M_+$ . Semifiniteness means

\varphi(x)=\sup\{\varphi(y):0\leq y\leq x,\ \varphi(y)<\infty\} \qquad(x\in M_+).

Faithfulness excludes nonzero positive elements of weight zero. Each condition is independent and must be checked separately; the conjunction is the standard one used in modular theory.

Examples

The ordinary trace on $B(H)$ , allowed to take the value $\infty$ , is normal, semifinite, and faithful. On a commutative von Neumann algebra $L^\infty(X,\mu)$ , integration against a faithful semifinite measure gives an n.s.f. weight. A faithful normal state is an n.s.f. weight whose value at the identity is one, hence is finite everywhere; the general weight notion allows substantially more algebras.

Role and terminology

N.s.f. weights supply the Hilbert-space construction and modular objects used in Tomita–Takesaki theory when no faithful normal state is available. Some authors order the adjectives as “faithful normal semifinite” and abbreviate this as f.n.s.; this denotes the same three properties, not a different kind of weight. The order of the adjectives carries no mathematical content.

References

Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter VII, §1 on normal, semifinite, faithful weights, and Chapter VIII on modular automorphism groups.