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.


46L40 Automorphisms of selfadjoint operator algebras
46L35 Classifications of \(C^*\)-algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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