Modular automorphism group - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra and let $\varphi$ be a normal semifinite [[operator-algebras/faithful-weight|faithful weight]]. The modular automorphism group of $\varphi$ is the unique ultraweakly continuous one-parameter group

\sigma^\varphi:\mathbb R\longrightarrow\operatorname{Aut}(M)

obtained from the modular operator in the weight's GNS construction or standard representation:

\pi_\varphi(\sigma_t^\varphi(x)) =\Delta_\varphi^{it}\pi_\varphi(x)\Delta_\varphi^{-it}.

The Tomita–Takesaki theorem ensures that the right-hand side again belongs to $\pi_\varphi(M)$ . Thus the definition is independent, up to canonical equivalence, of the chosen realization.

Characterization by the KMS condition

The weight $\varphi$ is invariant under its modular group. On the $\sigma^\varphi$ -analytic elements, the group is characterized by the boundary relation

\varphi(xy)=\varphi\!\left(y\,\sigma_{-i}^\varphi(x)\right)

whenever the terms are in the weight's finite domains. This is the Kubo–Martin–Schwinger condition at inverse temperature $1$ , and it characterizes the modular dynamics under the standard continuity and faithfulness hypotheses Takesaki, vol. II, Chapter VIII, §1.

Traces and inner modular flows

If $\varphi$ is a faithful normal semifinite trace, then $\sigma_t^\varphi=\operatorname{id}_M$ for every $t$ . For a faithful normal state on $M_n(\mathbb C)$ with density matrix $\rho$ ,

\sigma_t^\varphi(x)=\rho^{it}x\rho^{-it},

so the modular group is inner and is trivial exactly when $\rho$ is scalar. In general, the outer class of modular flow carries essential information about type III von Neumann algebras.

Dependence on the weight

Different n.s.f. weights generally produce different automorphism groups. Connes's Radon–Nikodym cocycle relates them by a time-dependent inner perturbation, so their images in the outer automorphism group agree in the appropriate sense. This weight-independent outer flow underlies the flow of weights and type III classification Takesaki, vol. II, Chapter VIII.

Conventions and scope

References

Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter VIII, §1 on modular automorphism groups and the Radon–Nikodym theorem for weights.