Tau-measurable operator - Imported Knowlpedia Content

Definition

Let $M\subseteq B(H)$ carry a faithful normal semifinite trace $\tau$ . A closed densely defined operator $T$ affiliated with $M$ is $\tau$ -measurable if its domain is $\tau$ -dense: for every $\varepsilon>0$ , there is a projection $e\in M$ such that

eH\subseteq\operatorname{dom}(T) \quad\text{and}\quad \tau(1-e)<\varepsilon.

Thus the domain may omit a subspace, but the omitted projection can be made arbitrarily small according to the chosen trace. The definition depends on $\tau$ , not only on the underlying von Neumann algebra.

Distribution-function characterization

The distribution function of $T$ is

d_T(s)=\tau\!\left(E^{|T|}((s,\infty))\right),\qquad s\geq0.

The operator $T$ is $\tau$ -measurable exactly when $d_T(s)\to0$ as $s\to\infty$ . Indeed, the low-spectrum projections $E^{|T|}([0,s])$ provide bounded restrictions of $T$ , while their complements have trace $d_T(s)$ . This criterion is the operator-algebraic analogue of a measurable function being finite almost everywhere Nelson, pp. 103–106.

Measure topology and algebra

The $\tau$ -measurable operators form an involutive algebra $S(M,\tau)$ under closed sums and products. Its measure topology has basic zero-neighborhoods consisting of operators $T$ for which there is a projection $e\in M$ with

\lVert Te\rVert\leq\varepsilon \quad\text{and}\quad \tau(1-e)\leq\delta.

This algebra is complete for the measure topology. Generalized singular numbers turn convergence in measure into scalar estimates Fack–Kosaki, §§1–3.

Examples and distinctions

For $M=L^\infty(X,\mu)$ with the integration trace, $S(M,\tau)$ identifies with the almost-everywhere finite measurable functions, acting by multiplication. For $B(H)$ with the usual operator trace, every nonzero projection has trace at least one; taking $\varepsilon<1$ forces $e=1$ , so the $\tau$ -measurable operators are precisely the bounded operators.

References

Edward Nelson, “Notes on Non-Commutative Integration,” Journal of Functional Analysis 15 (1974), 103–116. DOI record. Relevant: pp. 103–116 on $\tau$ -measurability and the measure topology.
Thierry Fack and Hideki Kosaki, “Generalized s-Numbers of $\tau$ -Measurable Operators,” Pacific Journal of Mathematics 123 (1986), 269–300. DOI record. Relevant: §§1–3 on measure topology, distribution functions, and generalized singular numbers.