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Pointwise inner automorphisms of von Neumann algebras. With an appendix by Colin Sutherland. (English) Zbl 0751.46037
In this paper a complete description of the pointwise inner automorphisms for factors of type $$III_ \lambda$$, $$0\leq\lambda<1$$ is given (assuming a separable predual). Recall that an automorphism $$\alpha$$ of a von Neumann algebra $$M$$ is pointwise inner if for each normal state $$\varphi$$ there is a unitary $$u=u(\varphi)$$ in $$M$$ such that $$\varphi\circ\alpha=u\varphi u^*$$. The proofs rely on the existence of faithful normal strictly semifinite lacunary (“zero is isolated”) weights of infinite multiplicity. (Thus the $$III_ 1$$ case remains completely open.)
The authors’ main result is the following
Theorem. Let $$M$$ be a factor of type $$III_ \lambda$$, $$0\leq\lambda<1$$, with separable predual. Let $$\omega$$ be a dominant weight and $$\alpha\in\text{Aut}(M)$$. Then $$\alpha$$ is pointwise inner if and only if there are $$v\in U(M)$$, the unitary group of $$M$$, and an extended modular automorphism $$\bar\sigma_ c^ \omega$$ (in the sense of Connes and Takesaki’s “Flow of weights” paper) such that $$\alpha=\text{Adv}\circ\bar\sigma_ c^ \omega$$. (Note the $$c$$ is a cocycle in the flow of weights.)
In the nonseparable case the authors show some factors of type $$II_ 1$$ with pointwise inner automorphisms which are outer.
In an appendix by Colin Sutherland it is shown that in the $$III_ 0$$ case the cohomology group $$H^ 1(Z,U(C_ \varphi))$$, and hence $$H^ 1(F^ M)$$, is nonsmooth in its natural Borel structure, hence is a very big space. Note that in this paper a new proof of the isomorphism between $$H^ 1(Z,U(C_ \varphi))$$ and $$H^ 1(F^ M)$$ is given. This isomorphism was proven in the “Flow of weights” paper by Connes and Takesaki.

##### MSC:
 46L40 Automorphisms of selfadjoint operator algebras 46L35 Classifications of $$C^*$$-algebras 46L80 $$K$$-theory and operator algebras (including cyclic theory)
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##### References:
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