Connes cocycle derivative - Imported Knowlpedia Content

Definition

Let $\varphi$ and $\psi$ be normal semifinite faithful weights on a von Neumann algebra $M$ . Their Connes cocycle derivative is the strongly continuous family of unitaries

u_t=(D\psi:D\varphi)_t\in M,\qquad t\in\mathbb R,

supplied by the noncommutative Radon–Nikodym theorem. In a faithful representation, choose an n.s.f. weight $\chi$ on $M'$ . In terms of spatial derivatives,

u_t=(d\psi/d\chi)^{it}(d\varphi/d\chi)^{-it};

it lies in $M$ , independently of $\chi$ . It satisfies

u_{s+t}=u_s\,\sigma_s^\varphi(u_t) \quad\text{and}\quad \sigma_t^\psi(x)=u_t\sigma_t^\varphi(x)u_t^*

for $s,t\in\mathbb R$ , $x\in M$ . It is a $\sigma^\varphi$ -cocycle changing the modular flow of $\varphi$ to that of $\psi$ .

Radon–Nikodym role

The cocycle derivative is the noncommutative replacement for a ratio of densities. If $M$ is semifinite with faithful normal semifinite trace $\tau$ , and $\varphi(x)=\tau(h_\varphi x)$ , $\psi(x)=\tau(h_\psi x)$ for positive invertible affiliated densities, then

(D\psi:D\varphi)_t=h_\psi^{it}h_\varphi^{-it}.

The formula remains meaningful even when the two densities do not commute: the resulting family is generally a cocycle rather than a one-parameter group Takesaki, vol. II, Chapter VIII, §3.

Chain and inversion rules

For a third normal semifinite faithful weight $\theta$ , cocycle derivatives obey the chain rule

(D\psi:D\varphi)_t =(D\psi:D\theta)_t(D\theta:D\varphi)_t.

They also satisfy

(D\psi:D\varphi)_t^*=(D\varphi:D\psi)_t.

These identities make changes of reference weight coherent and show that no preferred weight is required to compare modular dynamics. Connes used this Radon–Nikodym theory in the structure and classification of type III factors Connes, §1.

Examples and scope

When $\psi=\varphi$ , the cocycle is constantly $1$ . If $\psi=\lambda\varphi$ for a scalar $\lambda>0$ , then $(D\psi:D\varphi)_t=\lambda^{it}1$ ; the modular automorphism groups are equal even though the cocycle is not constant unless $\lambda=1$ . Faithfulness makes each cocycle value unitary. For nonfaithful weights the general theory uses partial isometries and support projections, so the unitary formulation in the core should not be applied unchanged. Moreover, the cocycle and implementation identities alone do not distinguish the derivative from every other implementing cocycle; the Radon–Nikodym normalization in the core is essential.

References

Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter VIII, §3 on the Connes cocycle derivative.
Alain Connes, “Une classification des facteurs de type III,” Annales scientifiques de l’École Normale Supérieure 6 (1973), 133–252. DOI record. Relevant: §1 on the cocycle Radon–Nikodym derivative and modular automorphisms.