Measurable operator affiliated with a von Neumann algebra

Definition

Let $M\subseteq B(H)$ be a semifinite von Neumann algebra. A closed densely defined operator affiliated with $M$ is measurable with respect to $M$ if its domain is strongly dense: there are projections $p_n\in M$ such that $p_n\uparrow1$ strongly, $p_nH\subseteq\operatorname{dom}(T)$ , and each complementary projection $1-p_n$ is finite in $M$ . This condition uses Murray–von Neumann finiteness, not a numerical trace. It permits unbounded operators while ensuring that $T$ is bounded on successively larger corners whose omitted parts are finite.

Spectral characterization

Write $E^{|T|}$ for the spectral measure of the positive operator $|T|$ . Measurability is equivalent to the existence of $s\geq0$ for which the spectral tail $E^{|T|}((s,\infty))$ is a finite projection. Once one such tail is finite, every higher tail is finite and the projections $E^{|T|}([0,n])$ supply a strongly dense domain. This formulation makes the definition independent of a chosen approximating sequence Nelson, pp. 103–106.

Algebra and examples

The measurable operators form a unital involutive algebra $S(M)$ when sums, products, and adjoints are taken with their natural closed extensions. This is the noncommutative analogue of the algebra of almost-everywhere finite measurable functions Segal, pp. 401–457.

If $M$ is finite, every closed densely defined affiliated operator is measurable. At the other extreme, for $M=B(H)$ on an infinite-dimensional Hilbert space, finite projections have finite-dimensional range and $S(M)=B(H)$ ; the definition admits no genuinely unbounded operators there Nelson, pp. 103–106.

Conventions and scope

References

Irving E. Segal, “A Non-Commutative Extension of Abstract Integration,” Annals of Mathematics 57 (1953), 401–457. DOI record. Relevant: the strongly dense domain and measurable-operator algebra.
Edward Nelson, “Notes on Non-Commutative Integration,” Journal of Functional Analysis 15 (1974), 103–116. DOI record. Relevant: pp. 103–106 on affiliated measurable operators and the measure topology.