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nLab > Latest Changes: modular theory

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeOct 31st 2019
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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
- CommentRowNumber2.
- CommentAuthorDmitri Pavlov
- CommentTimeJul 28th 2021
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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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ , a faithful semifinite normal weight $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>$ on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ , and an imaginary number $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>$ , the modular automorphism associated to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>$ , and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>$ is
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>σ</mi> <mi>μ</mi> <mi>t</mi></msubsup><mo lspace="0.11111em">:</mo><mi>M</mi><mo>→</mo><mi>M</mi><mo>,</mo><mspace width="2em"/><mi>m</mi><mo>↦</mo><msup><mi>μ</mi> <mi>t</mi></msup><mi>m</mi><msup><mi>μ</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>t</mi></mrow></msup><mo>.</mo></mrow><annotation encoding="application/x-tex">\sigma_\mu^t\colon M\to M,\qquad m\mapsto \mu^t m \mu^{-t}.</annotation></semantics></math>$
This approach makes it easy to deduce various properties of the modular automorphism group.

For more details, see a MathOverflow answer.

diff, v16, current
- CommentRowNumber3.
- CommentAuthorTom Mainiero
- CommentTimeAug 5th 2024
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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

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nLab > Latest Changes: modular theory

nPOV definition