The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions

Kawahigashi, Yasuyuki; Sutherland, Colin E.; Takesaki, Masamichi

doi:10.1007/bf02392758

Cited by 82 publications

(81 citation statements)

References 16 publications

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“…This is based on the Connes technique given in [9, page 466]. (Also see [25,Theorem 20] and a remark at the end of [22].) Note that the case p a = 0 is covered by the relative version of the Connes classification theorem [7, Theorem 1] as in [27].…”

Section: The Second Classification Theoremmentioning

confidence: 99%

“…We then study Hecke algebra subfactors of Wenzl constructed in [37]. Finally, we discuss analogy between our classification here and the classification of automorphisms of injective type III factors in [9], [25], [35].…”

Section: Examples and Remarksmentioning

confidence: 99%

“…Because we are interested in approximately inner automorphisms, take the (unique) injective type III 1 factor M, where any automorphism is approximately inner by [9, [25,Theorem 1]. Then the classification of automorphisms of this M is given as follows.…”

Section: Example 63mentioning

confidence: 99%

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Classification of approximately inner automorphisms of subfactors

Kawahigashi¹

1997

Mathematische Annalen

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For classification of approximately inner automorphisms of subfactors, we introduce a new invariant, a higher obstruction. From an algebraic viewpoint, this can be regarded as a generalization of the Connes obstruction, and from an analytic viewpoint, this can be regarded as a generalization of the Jones invariant κ. We have two classification theorems for approximately inner automorphisms of strongly amenable subfactors with known invariants and this new one. In particular, our theorems give a complete classification of automorphisms, up to outer conjugacy, of AFD subfactors of type II 1 with index less than four except for one special case for A 4n−1 and E 6 .

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Section: The Second Classification Theoremmentioning

confidence: 99%

Section: Examples and Remarksmentioning

confidence: 99%

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Classification of approximately inner automorphisms of subfactors

Kawahigashi¹

1997

Mathematische Annalen

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“…Special cases of the result in the III 1 case have been established in [11]. The general case will appear in [10].…”

Section: Theorem 1 Let R Be An Approximately Finite Dimensional Sepamentioning

confidence: 99%

“…The celebrated work of Connes, [1,3], surveyed in [2], Ocneanu's analysis, [12], and the previous work of Kawahigashi, and the second and third authors, [11], reveal the beatiful structure of the automorphism group of an approximately finite dimensional, or AFD, factor R. In this note, we announce that it is possible to describe the structure of Aut(R) independently of the type of R. This structure of Aut(R) enables us to define a cohomological invariant which we call the intrinsic invariant of R. If a group G acts on R via α, then the pull back of the orbit of the intrinsic invariant of R under the natural action of the group Aut(R) gives a cocyle conjugacy invariant which is complete if G is a countable discrete amenable group. This completes the cocylce conjugacy classification of discrete amenable group actions on an AFD factor R including the type III 1 case.…”

Section: Introductionmentioning

confidence: 99%

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Katayama

Sutherland

Takesaki

1995

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We announce in this article that i) to each approximately finite dimensional factor R of any type there corresponds canonically a group cohomological invariant, to be called the intrinsic invariant of R and denoted Θ(R), on which Aut(R) acts canonically; ii) when a group G acts on R via α : G → Aut(R), the pull back of Orb(Θ(R)), the orbit of Θ(R) under Aut(R),by α is a cocycle conjugacy invariant of α; iii) if G is a discrete countable amenable group, then the pair of the module, mod(α), and the above pull back is a complete invariant for the cocycle conjugacy class of α. This result settles the open problem of the general cocycle conjugacy classification of discrete amenable group actions on an AFD factor of type III 1 , and unifies known results for other types.

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The Non-commutative Flow of Weights on a Von Neumann Algebra

Falcone¹,

Takesaki²

2001

Journal of Functional Analysis

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this work is respectfully dedicated to the memory of y. misonouThe flow of weights of Connes and Takesaki is a canonical functor from the category of separable factors to the category of ergodic flows. The non-commutative flow of weights is another canonical functor from the category of separable factors to the category of covariant systems of semi-finite von Neumann algebras equipped with trace scaling one parameter automorphism groups with conjugations as morphisms. The constructions of these two functors are very similar. The flow of weights functor is obtained by looking at all semi-finite normal weights on a factor with the Murray von Neumann equivalence relation. The non-commutative flow of weights functor is obtained by relating an arbitrary pair of faithful semi-finite normal weights by the Connes cocycle. Not only does this construction put a period to the search for a canonical construction of the core of a factor of type III, but it also allows us to put the characteristic square of a factor obtained by Katayama, Sutherland, and Takesaki in a new perspective. The power of this new approach is seen in an ultimate solution to a long standing question of extending the extended modular automorphism of a dominant weight to an arbitrary weight, which has been left open ever since the introduction of extended modular automorphisms by Connes and Takesaki over 20 years ago. The construction of the functor ties together the theory of L p -spaces of Haagerup, Kosaki, Hilsum, Terp, and Izumi to the structure theory of a factor of type III. In fact, the non-commutative flow of weights is obtained by the analytic continuation of L p -spaces to a pure imaginary value of p.

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The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions

Cited by 82 publications

References 16 publications

Classification of approximately inner automorphisms of subfactors

Classification of approximately inner automorphisms of subfactors

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The Non-commutative Flow of Weights on a Von Neumann Algebra

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