Finite normal trace on a von Neumann algebra - Imported Knowlpedia Content

Definition

Let $M$ be a unital von Neumann algebra. A finite normal trace on $M$ is a tracial weight $\tau:M_+\to[0,+\infty]$ that is normal and satisfies

\tau(1)<\infty.

Finiteness at the identity implies $\tau(x)<\infty$ for every $x\in M_+$ , so $\tau$ extends uniquely to a bounded normal positive linear functional on $M$ satisfying $\tau(ab)=\tau(ba)$ . Faithfulness and normalization are not part of the definition: they must be imposed separately when the trace is meant to detect every positive element or be a state.

Equivalent bounded formulation

Equivalently, a finite normal trace is a normal positive linear functional $\tau:M\to\mathbb C$ satisfying

\tau(ab)=\tau(ba)\qquad(a,b\in M).

Its norm is $\|\tau\|=\tau(1)$ . Dividing a nonzero finite normal trace by $\tau(1)$ produces a normal tracial state, but it does not repair a failure of faithfulness. The equivalence between the positive-cone weight and bounded functional formulations is standard in Kadison–Ringrose, §7.2.

Examples and consequences

The normalized matrix trace $n^{-1}\operatorname{Tr}$ is a faithful finite normal trace on $M_n(\mathbb C)$ . Integration against a finite measure gives a finite normal trace on the commutative von Neumann algebra $L^\infty(X,\mu)$ . On $B(H)$ for infinite-dimensional $H$ , the canonical operator trace is normal, semifinite, and faithful but not finite because $\operatorname{Tr}(1)=+\infty$ .

A von Neumann algebra admitting a faithful finite normal trace is finite, but the zero functional is a finite normal trace on every von Neumann algebra. Faithfulness is therefore essential in any trace-based characterization of finite algebras Takesaki, Chapter V.

References

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 finite traces and normal traces.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on traces and finite von Neumann algebras.