Flow of weights - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra with separable predual and choose a normal semifinite faithful weight $\varphi$ . Form the continuous core

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

as the continuous core, hence a von Neumann crossed product by the modular automorphism group. Let $\theta:\mathbb R\curvearrowright c_\varphi(M)$ be its dual action. The center $Z(c_\varphi(M))$ is $\theta$ -invariant. The flow of weights of $M$ is the restricted action

\theta|_{Z(c_\varphi(M))}:\mathbb R\curvearrowright Z(c_\varphi(M)).

Changing $\varphi$ produces a conjugate flow, so its conjugacy class is an isomorphism invariant of $M$ .

Factor types

When $M$ is a factor, its flow of weights is ergodic. For a type $\mathrm{III}_1$ factor the flow is the one-point flow. For type $\mathrm{III}_\lambda$ , $0<\lambda<1$ , it is periodic; in the logarithmic translation normalization its least positive period is $\lvert\log\lambda\rvert$ . Type $\mathrm{III}_0$ gives a properly ergodic, nonperiodic flow. These correspondences connect the flow with the Connes type III classification Connes–Takesaki, §§4–5.

Why the center of the core appears

The continuous core is semifinite even when $M$ is type III. Its center retains the residual scaling information carried by modular automorphisms, while the dual action records how that information changes with time. Restricting to the center converts modular data that initially depends on a weight into a flow intrinsic to $M$ .

Classification scope

The flow is much finer than the single parameter $\lambda$ in the type label. In the separable approximately finite-dimensional type $\mathrm{III}_0$ case, its conjugacy class is a complete isomorphism invariant Takesaki, Chapter XII, §5. It is not a complete invariant for arbitrary type III factors.

Conventions and scope

References

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 canonical flow and its relation to type III invariants.
Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter XII, §4 on the flow of weights and §5 on approximately finite-dimensional type III₀ factors.