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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 31st 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * [[Roberto Longo]], _The emergence of time_ ([arxiv:1910.13926](https://arxiv.org/abs/1910.13926)) <a href="https://ncatlab.org/nlab/revision/diff/modular+theory/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+theory/13">v13</a>, <a href="https://ncatlab.org/nlab/show/modular+theory">current</a>

   added pointer to today’s

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

   diff, v13, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 28th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: *** ## nPOV definition 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](https://mathoverflow.net/questions/386464/making-sense-of-every-non-commutative-algebra-has-its-own-internal-time-evoluti/386481#386481). *** <a href="https://ncatlab.org/nlab/revision/diff/modular+theory/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+theory/16">v16</a>, <a href="https://ncatlab.org/nlab/show/modular+theory">current</a>

   Added:

   ---

   nPOV definition

   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.

   ---

   diff, v16, current

3. • CommentRowNumber3.
   • CommentAuthorTom Mainiero
   • CommentTimeAug 5th 2024
   • PermaLink
   Author: Tom Mainiero
   Format: MarkdownItexAdded link to video lecture series/Kostecki non-commutative integration reference. <a href="https://ncatlab.org/nlab/revision/diff/modular+theory/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+theory/21">v21</a>, <a href="https://ncatlab.org/nlab/show/modular+theory">current</a>

   Added link to video lecture series/Kostecki non-commutative integration reference.

   diff, v21, current