Continuous core of a von Neumann algebra - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra and choose a normal semifinite [[operator-algebras/faithful-weight|faithful weight]] $\varphi$ . The continuous core of $M$ , relative to $\varphi$ , is the von Neumann crossed product

c_\varphi(M)=M\rtimes_{\sigma^\varphi}\mathbb R

by the modular automorphism group. It carries a dual action $\theta:\mathbb R\curvearrowright c_\varphi(M)$ and a distinguished faithful normal semifinite trace $\tau_\varphi$ , determined by the dual weight, with $\tau_\varphi\circ\theta_s=e^{-s}\tau_\varphi$ under the Fourier convention used here.

Independence of the weight

Another n.s.f. weight $\psi$ gives a crossed product $c_\psi(M)$ canonically isomorphic to $c_\varphi(M)$ after incorporating the Connes cocycle derivative. The isomorphism fixes the embedded copy of $M$ and intertwines the dual actions. Consequently the continuous core is usually written $c(M)$ when only this canonical isomorphism class matters, even though its concrete presentation uses a weight.

Semifiniteness and type III algebras

The core is semifinite even when $M$ is type III and therefore has no faithful normal semifinite trace itself. This converts modular scaling into ordinary tracial integration on a larger algebra. Homogeneous measurable operators in the core form the Haagerup noncommutative $L^p$ spaces, while the dual action on its center defines the flow of weights.

Conventions and scope

Some authors use the opposite modular action or Fourier character. Then the dual action is time-reversed and the trace-scaling formula contains $e^s$ instead of $e^{-s}$ . These paired conventions produce isomorphic cores. The continuous core should not be confused with the algebraic core of a crossed product or with the core domain of a closed unbounded operator.

References

Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. Publisher DOI record. Relevant: Chapters X and XII on crossed products, dual weights, and the continuous decomposition.
Alain Connes and Masamichi Takesaki, “The Flow of Weights on Factors of Type III,” Tohoku Mathematical Journal 29 (1977), 473–575. DOI record. Relevant: §§4–5 on the core, its dual action, and the flow of weights.