TFPT scientific source excerpt
Original repository path: documents/articles--2026-08-29--holomorphic_kms_extension_en/sections/002-subsection-the-frozen-car-covariance.tex
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000181 | \emph{The modular-theory chain.}
000182 | We spell out the Takesaki step because it is frequently reversed
000183 | incorrectly.  In the GNS representation of the faithful KMS state,
000184 | the modular group and the physical dynamics are related, with the
000185 | present KMS convention, by
000186 | \[
000187 |  \sigma_s^{\Phi_\beta^{\B}}(\pi_\Phi(x))
000188 |  =\pi_\Phi(\tau_{-\beta s}(x)).
000189 | \tag{4.6}\label{eq:kms-modular}
000190 | \]
000191 | Rotations preserve the subnet, hence
000192 | \[
000193 |  \sigma_s^{\Phi_\beta^{\B}}
000194 |    (\pi_\Phi(\A(I)))=\pi_\Phi(\A(I)).
000195 | \]
000196 | Takesaki's theorem now gives the unique
000197 | $\Phi_\beta^{\B}$-preserving normal faithful expectation.  By
000198 | \eqref{eq:E-preserves}, the group average is such an expectation, so
000199 | it is the Takesaki expectation.  Moreover,
000200 | \[
000201 |  \sigma_s^{\omega_\beta}
000202 |  =\sigma_s^{\Phi_\beta^{\B}}|_{\A(I)}
000203 |  =\tau_{-\beta s}|_{\A(I)},\qquad
000204 |  E_I\sigma_s^{\Phi_\beta^{\B}}
000205 |  =\sigma_s^{\omega_\beta}E_I.
000206 | \tag{4.7}\label{eq:modular-chain}
000207 | \]
000208 | Notice the logical order: existence of the Gibbs extension gives a
000209 | state on $\B$; modular invariance then identifies its
000210 | state-preserving expectation.  Takesaki's criterion alone would not
000211 | manufacture a KMS extension from an arbitrary state on $\A$.