AUTOMORPHISMS OF CENTRAL EXTENSIONS OF TYPE III VON NEUMANN ALGEBRAS
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Abstract:
We consider the central extension of a Type III von Neumann algebra. For Type III von Neumann algebras, the algebra coincides with itself since its center is trivial. In this case, we show that any arbitrary automorphism of the algebra is inner, meaning the equality holds for some unitary element. Furthermore, we prove that every banded automorphism of the algebra is inner and acts as identity on the center. Our results are based on the Tomita-Takesaki modular theory and Connes' classification of Type III factors.
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