Galois groups of quantum group actions and regularity of fixed-point algebras

Yamanouchi, Takehiko

doi:10.1090/s0002-9947-03-03282-3

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Minimal Integrable Actions On Type III Factors1

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2006

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Cited by 2 publications

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“…We now have two integrable, minimal actions α andβ of K on M, both of which have the same fixed-point algebra N. From [38,Theorem 3.6], we find that Aut(M/Mβ ), the group of automorphisms of M leaving M α pointwise invariant, is exactly {β k : k ∈ K} = {α k : k ∈ K}. Hence, there exists a (topological) group automorphism of K such that T. Yamanouchî…”

Section: Minimal Integrable Actions On Type III Factorsmentioning

confidence: 99%

One-cocycles on smooth flows of weights and extended modular coactions

Yamanouchi

2006

Ergod. Th. Dynam. Sys.

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Section: Minimal Integrable Actions On Type III Factorsmentioning

confidence: 99%

One-cocycles on smooth flows of weights and extended modular coactions

Yamanouchi

2006

Ergod. Th. Dynam. Sys.

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A characterization of coactions whose fixed-point algebras contain special maximal abelian $\ast$-subalgebras

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Yamanouchi

2006

Ergod. Th. Dynam. Sys.

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Abstract. It is shown that, for the von Neumann algebra A obtained from a principal measured groupoid R with the diagonal subalgebra D of A, there exists a natural 'bijective' correspondence between coactions on A that fix D pointwise and Borel 1-cocycles on R.As an application of this result, we classify a certain type of coactions on approximately finite-dimensional type II factors up to cocycle conjugacy. By using our characterization of coactions mentioned above, we are also able to generalize to some extent those results of Zimmer concerning 1-cocycles on ergodic equivalence relations into compact groups. IntroductionFor each action α of a locally compact quantum group G = (M, ) on a von Neumann algebra A, we obtain an inclusion of A and the fixed-point algebra A α . The Galois theory for α means the natural correspondence between the intermediate subalgebras of (A α ⊆ A) and the left coideals of G. It is known that there exists a bijective Galois correspondence when G is a compact Kac algebra and the action α on a factor A is minimal, i.e. the relative commutant (A α ) ∩ A is trivial [17]. In particular, if α is a minimal action of an ordinary compact group K, then there exists a bijective correspondence between the set of intermediate subalgebras of (A α ⊆ A) and the set of closed subgroups of K (we note that the minimality of α implies that all the intermediate subalgebras of (A α ⊆ A) are factors). Motivated by this work, many results concerning minimal actions have been obtained [6,29,36]. However, there seems to be little hope that the arguments made in those results go beyond the case of minimal actions.

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Galois groups of quantum group actions and regularity of fixed-point algebras

Cited by 2 publications

References 17 publications

One-cocycles on smooth flows of weights and extended modular coactions

One-cocycles on smooth flows of weights and extended modular coactions

A characterization of coactions whose fixed-point algebras contain special maximal abelian $\ast$-subalgebras

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