Faithful normal semifinite trace - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra. A faithful normal semifinite trace, or f.n.s. trace, is a tracial weight $\tau:M_+\to[0,+\infty]$ that is simultaneously normal, semifinite, and faithful: it preserves suprema of increasing positive nets; every positive element is the supremum of positive subelements having finite trace; and $\tau(x)=0$ for $x\in M_+$ implies $x=0$ . Equivalently, it is a normal semifinite [[operator-algebras/faithful-weight|faithful weight]] satisfying $\tau(x^*x)=\tau(xx^*)$ for every $x\in M$ .

Finite-trace approximation

Semifiniteness supplies enough finite-trace elements for analysis even when $\tau(1)=+\infty$ . In particular, finite-trace projections can approximate the identity in the strong operator topology, and they generate the relative compact ideal used in semifinite Fredholm theory. Normality ensures that the trace of an increasing approximation converges to the trace of its supremum; faithfulness ensures that trace zero detects the zero positive element Takesaki, Chapter V.

Examples and structural role

The canonical operator trace on $B(H)$ is an f.n.s. trace, although it is finite only when $H$ is finite-dimensional. On $L^\infty(X,\mu)$ , integration against a faithful semifinite measure gives an f.n.s. trace. A type $\mathrm{II}_\infty$ factor has an f.n.s. trace unique up to positive scalar multiplication.

A von Neumann algebra is semifinite exactly when it admits an f.n.s. trace. Type III algebras admit n.s.f. weights but no f.n.s. trace; traciality is the decisive extra property Takesaki, Chapter V.

Conventions and scope

The order of the adjectives varies: “normal faithful semifinite” and “faithful normal semifinite” describe the same conjunction. Semifiniteness does not mean $\tau(1)<\infty$ , and an f.n.s. trace should not be confused with the finite-dimensional matrix trace. Some sources build semifiniteness from density of the finite left ideal rather than order approximation; for a normal weight on a von Neumann algebra these formulations agree.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on traces and semifinite von Neumann algebras.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory, AMS, 1997. AMS record. Relevant: §7.2 on normal semifinite traces.