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000163 | $\sigma^\phi=(\sigma_t^\phi)_{t\in\R}$ for the modular automorphism
000164 | group of $\phi$.  The following are equivalent:
000165 | \begin{enumerate}[label=\textup{(\roman*)}]
000166 | \item $N$ is globally invariant under the modular group,
000167 |   $\sigma_t^\phi(N)=N$ for every $t\in\R$;
000168 | \item there exists a normal faithful conditional expectation
000169 |   $E_\phi:M\to N$ such that
000170 |   $\phi\circ E_\phi=\phi$ on $M_+$.
000171 | \end{enumerate}
000172 | When these conditions hold, $E_\phi$ is unique among the normal
000173 | conditional expectations preserving $\phi$, it commutes with the
000174 | modular group,
000175 | \[
000176 |  E_\phi\sigma_t^\phi=\sigma_t^{\phi|_N}E_\phi,
000177 | \tag{2.2}\label{eq:takesaki-intertwine}
000178 | \]
000179 | and the modular group of the restricted weight is the restriction:
000180 | \[
000181 |  \sigma_t^{\phi|_N}=\sigma_t^\phi|_N.
000182 | \tag{2.3}\label{eq:mod-restrict}
000183 | \]
000184 | \end{theorem}
000185 | 
000186 | This is M.~Takesaki, \emph{Conditional expectations in von Neumann
000187 | algebras}, J.\ Funct.\ Anal.\ \textbf{9} (1972), 306--321, and
000188 | \emph{Theory of Operator Algebras II}, Springer, 2003,
000189 | Theorem~IX.4.2.
000190 |