Spectral characterization of affiliation - Imported Knowlpedia Content

Statement

Let $M\subseteq B(H)$ be a von Neumann algebra and let $T$ be a closed densely defined operator on $H$ , with polar decomposition $T=v|T|$ . Then $T$ is affiliated with $M$ if and only if the partial isometry $v$ belongs to $M$ and every spectral projection

E^{|T|}(\Delta)=\mathbf 1_\Delta(|T|)

belongs to $M$ for every Borel set $\Delta\subseteq[0,\infty)$ . Equivalently, $v\in M$ and $(|T|+i)^{-1}\in M$ . For self-adjoint $T$ , affiliation is equivalent to all spectral projections of $T$ lying in $M$ .

Why the criteria agree

The defining commutation relation for affiliation says that $T$ commutes, as an unbounded operator, with every unitary in $M'$ . Uniqueness of polar decomposition then forces both $v$ and the spectral measure of $|T|$ to commute with $M'$ ; the bicommutant theorem places them in $M$ . Conversely, these bounded data reconstruct $T$ by spectral integration and make it commute with $M'$ . See Takesaki, Chapter V, §5.

Self-adjoint and positive cases

For positive self-adjoint $T$ , the polar partial isometry is the support projection of $T$ , so membership of the spectral projections alone is sufficient. For self-adjoint $T$ , the bounded resolvent $(T-i)^{-1}$ generates the same von Neumann algebra as the spectral projections, giving a concise resolvent criterion for affiliation.

Scope and cautions

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V, §5 on closed operators affiliated with von Neumann algebras and their polar and spectral data.
Edward Nelson, “Notes on Non-Commutative Integration,” Journal of Functional Analysis 15 (1974), 103–116. DOI record. Relevant: §§1–2 on affiliated operators and spectral truncations.