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Is a unital, injective, ultraweakly continuous $*$-endomorphism $f:M\rightarrow M$ of a $III_1$ factor $M$ inner? I.e. is there a unitary $U\in M:$ $f(-)=U(-)U^*$?

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No. Take any automorphism in the Tomita-Takesaki modular flow of M. It is well known that an automorphism in the modular flow is inner if and only if M is semifinite.

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