Pointwise inner automorphisms of von Neumann algebras. With an appendix by Colin Sutherland.

*(English)*Zbl 0751.46037In 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.

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.

Reviewer: M.E.Walter (Boulder)

##### MSC:

46L40 | Automorphisms of selfadjoint operator algebras |

46L35 | Classifications of \(C^*\)-algebras |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

##### Keywords:

von Neumann algebra; modular automorphism group; faithful normal strictly semifinite lacunary weights; pointwise inner automorphisms; factors of type \(III_ \lambda\); separable predual; infinite multiplicity; modular automorphism; cohomology group; natural Borel structure
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\textit{U. Haagerup} and \textit{E. Størmer}, J. Funct. Anal. 92, No. 1, 177--201 (1990; Zbl 0751.46037)

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