Semifinite weight - Knowlpedia

Let $M$ be a von Neumann algebra and let $\varphi:M_+\to[0,+\infty]$ be a weight. The weight $\varphi$ is semifinite if its finite part is order-dense in $M_+$ : for every nonzero $x\in M_+$ , there exists a nonzero $y\in M_+$ such that

0\leq y\leq x\qquad\text{and}\qquad\varphi(y)<\infty.

Equivalently, the left ideal $\mathfrak n_\varphi=\{a\in M:\varphi(a^*a)<\infty\}$ is dense in $M$ for the ultraweak topology. Semifiniteness is a domain condition; it does not require faithfulness or normality and does not assert that $\varphi$ is finite on every positive element.

Approximation by finite elements

If $\varphi$ is also normal, each $x\in M_+$ can be recovered as the supremum of finite-weight positive elements below $x$ . Normality then gives

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

The order approximation and the recovery of the value play different roles: semifiniteness supplies enough finite elements, while normality makes $\varphi$ preserve their increasing supremum.

Examples and non-examples

The canonical operator trace on $B(H)$ is semifinite because every nonzero positive operator dominates a nonzero finite-rank positive operator of finite trace. Every finite positive functional is automatically a semifinite weight, whether or not it is faithful, because every positive element already has finite value. The weight taking $+\infty$ on every nonzero positive element is not semifinite. These examples also show that semifiniteness and faithfulness are independent conditions.

Role in integration

For a faithful normal semifinite weight, the finite left ideal $\mathfrak n_\varphi$ is large enough to build the weight-GNS Hilbert space, normality controls limits, and faithfulness removes its null ideal. These three independent hypotheses are the standard starting point for modular theory. A semifinite von Neumann algebra is instead an algebra admitting a faithful normal semifinite trace; it should not be confused with a particular semifinite weight on an arbitrary algebra.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter VII, §1 on semifinite weights and their finite ideals.
Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: the opening chapters on faithful normal semifinite weights and modular theory.