Added:

The modern approach to defining the modular automorphism group is through the theory of noncommutative L_p-spaces. This was pioneered by Haagerup in 1979 \cite{Haagerup} and Yamagami in 1992 \cite{Yamagami}.

In this approach, given a von Neumann algebra $M$, a faithful semifinite normal weight $\mu$ on $M$, and an imaginary number $t$, the modular automorphism associated to $M$, $\mu$, and $t$ is

$\sigma_\mu^t\colon M\to M,\qquad m\mapsto \mu^t m \mu^{-t}.$This approach makes it easy to deduce various properties of the modular automorphism group.

For more details, see a MathOverflow answer.

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added pointer to today’s

- Roberto Longo,
*The emergence of time*(arxiv:1910.13926)