Spatial derivative - Imported Knowlpedia Content

Definition

Let $M\subseteq B(H)$ be a von Neumann algebra, let $\phi$ be a normal semifinite weight on $M$ , and let $\Psi$ be a faithful normal semifinite weight on the commutant $M'$ . Put $\mathfrak n_\Psi=\{y\in M':\Psi(y^*y)<\infty\}$ . A vector $\xi$ is $\Psi$ -bounded when $y\mapsto y\xi$ is bounded for the GNS norm on $\mathfrak n_\Psi$ ; then define $R^\Psi(\xi)\Lambda_\Psi(y)=y\xi$ for $y\in\mathfrak n_\Psi$ , where $\Lambda_\Psi$ is the GNS map, and $q(\xi)=\phi(R^\Psi(\xi)R^\Psi(\xi)^*)$ ; put $q(\xi)=+\infty$ otherwise. The spatial derivative $d\phi/d\Psi$ is the unique positive self-adjoint operator on $H$ whose closed quadratic form is the closure of $q$ :

q(\xi)=\left\|\left(d\phi/d\Psi\right)^{1/2}\xi\right\|^2.

Why the commutant appears

The map $R^\Psi(\xi)$ intertwines the GNS representation of $M'$ with its given action on $H$ , so $R^\Psi(\xi)R^\Psi(\xi)^*\in M$ and $\phi$ can evaluate it. The weight $\Psi$ supplies a reference measure on the commuting algebra rather than on $M$ itself. This makes $d\phi/d\Psi$ available in any faithful concrete representation, which explains the adjective “spatial” Hiai, §13, Definition 13.4.

Relation to modular theory

The operator $d\phi/d\Psi$ is generally unbounded and need not belong to $M$ . Its imaginary powers encode relative modular data and transform covariantly under changes of the reference weight. In the standard form of $M$ , with the commutant weight obtained from a second weight by modular conjugation, the spatial derivative becomes the corresponding relative modular operator. Connes introduced this construction as a noncommutative Radon–Nikodym derivative Connes, pp. 19–143.

Commutative model and scope

In the commutative standard representation, the construction reduces to multiplication by an ordinary Radon–Nikodym density. In finite dimensions, with $M=B(K)$ acting in standard form, spatial derivatives are represented by the familiar left-right density-operator expression for relative modular operators.

References

Alain Connes, “Sur la théorie non commutative de l'intégration,” in Algèbres d'Opérateurs, Lecture Notes in Mathematics 725, Springer, 1979, 19–143. DOI record. Relevant: spatial Radon–Nikodym derivatives and noncommutative integration.
Fumio Hiai, Concise Lectures on Selected Topics of von Neumann Algebras, 2020. arXiv record. Relevant: §13, especially Definition 13.4 and formula (13.2), for the quadratic-form construction.