The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions

Authors:
Yoshikazu Katayama, Colin E. Sutherland and Masamichi Takesaki

Journal:
Electron. Res. Announc. Amer. Math. Soc. **1** (1995), 43-47

MSC (1991):
Primary 46L40

DOI:
https://doi.org/10.1090/S1079-6762-95-01006-7

MathSciNet review:
1336699

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Abstract: We announce in this article that i) to each approximately finite dimensional factor $\mathcal {R}$ of any type there corresponds canonically a group cohomological invariant, to be called the **intrinsic invariant** of $\mathcal {R}$ and denoted $\Theta (\mathcal {R})$, on which $\operatorname {Aut}(\mathcal {R})$ acts canonically; ii) when a group $G$ acts on $\mathcal {R}$ via $\alpha : G \mapsto \operatorname {Aut}(\mathcal {R})$, the pull back of Orb($\Theta (\mathcal {R})$), the orbit of $\Theta (\mathcal {R})$ under $\operatorname {Aut}(\mathcal {R})$, by $\alpha$ is a cocycle conjugacy invariant of $\alpha$; iii) if $G$ is a discrete countable amenable group, then the pair of the module, mod($\alpha$), and the above pull back is a complete invariant for the cocycle conjugacy class of $\alpha$. This result settles the open problem of the general cocycle conjugacy classification of discrete amenable group actions on an AFD factor of type $\mathrm {III}_1$, and unifies known results for other types.

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Connes, A., *Outerconjugacy classes of automorphisms of factors*, Ann. Sci. École Norm. Sup. **8** (1975), 383-419.
Connes, A., *On the classification of von Neumann algebras and their automorphisms*, Symposia Math. **XX** (1976), 435-478.
Conne, A., *Periodic automorphisms of the hyperfinite factor of type $\mathrm {II}_1$*, Acta Sci. Math. **39** (1977), 39-66.
Connes, A., *Factors of type $\mathrm {III}_1$, property $L’_\lambda$ and closure of inner automorphisms*, J. Operator Theory **14** (1985), 189-211.
Connes, A. Takesaki, M, *The flow of weights on factors of type $\mathrm {III}$*, Tohoku Math. J. **29** (1977), 473-555.
Haagerup, U., *Connes’ bicentralizer problem and uniqueness of the injective factor of type $\mathrm {III}_1$*, Acta Math. **158** (1987), 95-147.
Haagerup, U. & Størmer, E., *Pointwise inner automorphisms of von Neumann algebras with an appendix by C. Sutherland*, J. Funct. Anal. **92** (1990), 177-201.
Jones, V. F. R., *Actions of finite groups on the hyperfinite type $\mathrm {II}_1$ factor*, Mem. Amer. Math. Soc. **237** (1980).
Jones, V. F. R. & Takesaki, M., *Actions of compact abelian groups on semifinite injective factors*, Acta Math. **153** (1984), 213-258.
Katayama, Y., Sutherland, C. E. & Takesaki, M., *The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions , to appear*.
Kawahigashi, Y., Sutherland, C. E. & Takesaki, M., *The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions*, Acta Math. **169** (1992), 105-130.
Ocneanu, A., *Actions of discrete amenable groups on factors*, vol. 1138, Lecture Notes in Math., Springer, Berlin, 1985.
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Sutherland, C. E. & Takesaki, M., *Actions of discrete amenable groups on injective factors of type $\mathrm {III}$l, $\lambda \neq 1$*, Pacific J. Math. **137** (1989), 405-444.
Wong, S. Y. R., *On the dictionary between ergodic transformations, Krieger factors and ergodic flows*, Thesis, Univ. Newsouth Wales (1986), 72 + v.

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Additional Information

**Yoshikazu Katayama**

Affiliation:
address Yoshikazu Katayama, Department of Mathematics, Osaka Kyoiku University, Osaka, Japan.

Email:
F61021@sinet.adjp

**Colin E. Sutherland**

Affiliation:
address Colin E. Sutherland, Department of Mathematics, University of New South Wales, Kensington, NSW, Australia.

Email:
colins@solution.maths.unsw.edu.au

**Masamichi Takesaki**

Affiliation:
Department of Mathematics, University of California, Los Angeles, Califnornia 90024-1555.

Email:
mt@math.ucla.edu

Received by editor(s):
May 17, 1995

Additional Notes:
This research is supported in part by NSF Grant DMS92-06984 and DMS95-00882, and also supported by the Australian Research Council Grant

Article copyright:
© Copyright 1995
American Mathematical Society