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

Abstract

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.

On the normalizing groupoids and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations–subrelations

“…In connection with this, we have already succeeded to prove in [2] that the 1-cocycles on an equivalence relation bijectively correspond to the coactions on the corresponding von Neumann algebra which fix the Cartan subalgebra. Because of this result, we will focus on coactions on von Neumann algebras.…”

“…From [7, Theorem 2.2], it follows that there exist a discrete group Q and a Borel 1-cocycle ν : R → Q such that Ker(ν) = S and r * (ν) = Q. By [2,Theorem 4.2], we obtain the coaction α ν of Q on A associated to ν so that A α ν = B. So α ν is a minimal coaction.…”

“…First, condition (1) implies that α is faithful. Condition (2) implies that the subspace of A generated by {A α (k)} k∈K is σ -strongly* dense in A. Since Γ is discrete,…”

“…In [2], we showed that, roughly, the Borel 1-cocycles on R are in bijective correspondence with the coactions on W * (R, ω) leaving the Cartan subalgebra D pointwise fixed. In connection with the subject in Section 3, one natural question arises as to what kind of Borel 1-cocycles correspond to almost periodic coactions of the type described above.…”

“…So, if B ⊆ A is discrete, then the unique faithful normal conditional expectation E B from A onto B is regular in the sense of [3]. From [3,Theorem 11.16], it follows that there exist a locally compact group K and an outer coaction [2,Theorem 5.8] that there exists a Borel 1-cocycle c : R → K such that α = α c . In particular, we have S = Ker(c).…”