Operator affiliated with a von Neumann algebra

Let $M\subseteq B(H)$ be a von Neumann algebra. A densely defined operator $T$ on $H$ that is closed is affiliated with $M$ , written $T\eta M$ , if every unitary $u$ in the commutant $M'$ preserves $\operatorname{Dom}(T)$ and

Tu\xi=uT\xi\qquad(\xi\in\operatorname{Dom}(T)).

Equivalently, $uTu^*=T$ as unbounded operators for every unitary $u\in M'$ . Affiliation therefore extends the relation $T\in M$ from bounded operators to closed unbounded operators while retaining invariance under all symmetries commuting with $M$ .

Spectral characterization

If $T$ is self-adjoint, then $T\eta M$ exactly when every spectral projection of $T$ belongs to $M$ . More generally, for the polar decomposition $T=v|T|$ , affiliation is equivalent to $v\in M$ together with affiliation of $|T|$ . These criteria turn an unbounded commutation condition into bounded operator-algebra data.

Examples and non-examples

Every bounded $T\in M$ is affiliated with $M$ . For $M=L^\infty(X,\mu)$ acting by multiplication on $L^2(X,\mu)$ , a measurable function defines an affiliated multiplication operator on its maximal $L^2$ -domain. In contrast, a closed operator whose spectral projections do not lie in $M$ is not affiliated, even if some bounded functions of that operator happen to belong to $M$ .

Role in noncommutative integration

Affiliated operators supply the unbounded observables and measurable operators used in noncommutative integration. Additional conditions such as $\tau$ -measurability control the size of the domain relative to a trace; affiliation alone imposes no integrability, boundedness, or finite-trace condition. This distinction is essential for constructions such as $\tau$ -compact operators.

References

Edward Nelson, “Notes on Non-Commutative Integration,” Journal of Functional Analysis 15 (1974), 103–116. DOI record. Relevant: §§1–2 on affiliated and measurable operators.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on unbounded operators affiliated with von Neumann algebras.