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:

 [1] Araki, H., Some properties of modular conjugation operator of von Neumann algebras and a noncommutative Radon-Nikodym theorem with a chain rule, Pacific J. Math., 50, 309-354 (1974) · Zbl 0287.46074 [2] Connes, A., Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. (4), 6, 133-252 (1973) · Zbl 0274.46050 [3] Connes, A., Almost periodic states and factors of type III1, J. Funct. Anal., 16, 415-445 (1974) · Zbl 0302.46050 [4] Connes, A.; Takesaki, M., The flow of weights on factors of type III, Tôhoku Math. J., 29, 473-575 (1977) · Zbl 0408.46047 [5] Doplicher, S.; Kastler, D.; Størmer, E., Invariant states and asymptotic abeliannes, J. Funct. Anal., 3, 419-434 (1969) · Zbl 0174.44604 [6] Haagerup, U., On the dual weights for crossed products of von Neumann algebras, II, Math. Scand., 43, 119-140 (1978) · Zbl 0405.46053 [8] McDuff, D., Central sequences and the hyperfinite factor, (Proc. London Math. Soc., 21 (1970)), 443-461 · Zbl 0204.14902 [9] Popa, S., Singular maximal abelian ∗-subalgebras in continuous von Neumann algebras, J. Funct. Anal., 50, 151-166 (1983) · Zbl 0526.46059 [10] Powers, R. T.; Størmer, E., Free states of the canonical anticommutation relations, Comm. Math. Phys., 16, 1-33 (1970) · Zbl 0186.28301 [11] Stratila, S., Modular Theory in Operator Algebras (1981), Abacus: Abacus Kent, UK · Zbl 0504.46043 [12] Sutherland, C., Notes on Orbit Equivalence; Krieger’s Theorem, (Lecture Notes Ser. No. 23 (1976), Dept. of Math., Univ. of Oslo) [14] Takesaki, M., Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math., 131, 249-310 (1973) · Zbl 0268.46058
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