On automorphism groups on strongly amenable subfactors of

type III_{1}

MASUDA Toshihiko ( ) Kochi University Setting N ⊂ M : strongly amenable subfactor of type III_{1}. Ñ ⊂ M̃ : common continuous decomposition of N ⊂ M.

N ⊂ M ⊂ M_{1} ⊂ M_{2} ⊂ ⋯: Jones tower.

Important assumption. {M′ ∩ M_{k} ⊂ N′ ∩ M_{k}}_{k=1}^{∞} = {M̃′ ∩ M̃_{k} ⊂ Ñ′ ∩ M̃_{k}}_{k=1}^{∞} .

Th. (Popa) N ⊂ M ≅ N^{st} ⊗ R_{∞} ⊂ M^{st} ⊗ R_{∞}.

Main Th. (1) \overline{Int}(M, N) = Ker Φ.

(2) Cnt_{sf} (M, N) = Cnt (M, N).

Def. (Loi.) α ∈ Aut(M, N). Φ(α) := {α|_{M′∩M_{k}}}_{k=1}^{∞} .

Def. (Choda-Kosaki, Popa.) α : non-strongly outer ⇔ ∃k > 0, and 0 ≠ ∃a ∈ M_{k} s.t. α(x)a = ax for ∀x ∈ M.

Def. (Winsløw) α : non-strongly free ⇔ ∃k > 0, and 0 ≠ ∃a ∈ M̃_{k} s.t. α̃(x)a = ax for ∀x ∈ M̃. (α̃ : canonical extension of α.)

Cnt_{so}(M, N) : the set of non-strongly outer automorphisms. Cnt_{sf} (M, N) : the set of non-strongly free automorphisms.

Th. (Kosaki) θ ∈ Cnt_{sf} (M, N) → θ = α ∘ σ_{t}^{φ} for some α ∈ Cnt_{so}(M, N).

Def. α ∈ Aut(M, N). p_{so}(α) : non-strongly outer period of α. p_{sf} (α) : non-strongly free period of α.

Set σ := α^{p_{so} (α)}, and I_{k} := {a ∈ M_{k} | σ(x)a = ax, ∀x ∈ M}. Fix the minimal k_{0} s.t. I_{k_{0}} ≠ {0}.

Def. (Kawahigashi) γ_{h}(α) : unitary operator on I_{k_{0}} arising from α.

Def. (Kawahigashi) ν(α) := [σ] ∈ Aut(M, N)/Int(M, N).

Th. (Loi) θ ∈ Aut(M, N).

M(θ) := {diag (x, θ(x), θ(x)) | x ∈ M} ⊂ M_{3}(M).

Define C(θ) as

C(θ) = \begin{aligned}N(θ) & ⊂ & M(θ) \\ ∩ & & ∩ \\ M_{3}(N) & ⊂ & M_{3}(M). \end{aligned}

Then the standard invariant of C(θ) is determined by the standard invariant of N ⊂ M and {Φ(α), p_{so}(α), γ_{h}(α), ν(α)}.

Th. θ ∈ Aut(M, N). Assume p_{so}(θ) = p_{sf} (θ). Then θ ∼ θ_{0} ⊗ id_{R_{∞}} via an isomorphism N ⊂ M ≅ (N^{st} ⊂ M^{st}) ⊗ R_{∞}.

Pf. By assumption, N(θ) ⊂ M_{3}(M) is a splitting inclusion by Popa’s Theorem. → C(θ) ≅ C(θ_{0} ⊗ id_{R_{∞}}). This implies θ ∼ θ_{0} ⊗ id_{R_{∞}}. □

Proof of Main Th.

(1) Fix θ ∈ Ker Φ. σ : an aperiodic automorphism of R_{0}. Then p_{so}(θ ⊗ σ) = p_{sf} (θ ⊗ σ) = 0, → θ ⊗ σ ≅ θ_{0} ⊗ id_{R_{∞}} and θ_{0} ∈ Ker Φ.

Since Ker Φ = \overline{Int}(M^{st}, N^{st}), θ ⊗ σ ∈ \overline{Int}(M ⊗ R_{0}, N ⊗ R_{0}). → θ ⊗ id_{R_{0}} ∈ \overline{Int}(M ⊗ R_{0}, N ⊗ R_{0}) and since θ ∼ θ ⊗ id_{R_{0}}, θ ∈ \overline{Int}(M, N).

(2) Fix θ ∈ Aut(M, N)\Cnt_{sf} (M, N). If p_{so}(θ) = p_{sf} (θ), θ ∼ θ_{0} ⊗ id_{R_{∞}}. Since Cnt (M^{st}, N^{st}) = Cnt_{so}(M^{st}, N^{st}), θ ∉ Cnt (M, N).

If p_{so}(θ) ≠ p_{sf} (θ)(=: p), θ^{p} = α ∘ σ_{t}^{φ} for some α ∈ Cnt_{so}(M, N). φ : dominant weight, and may assume φ ∘ θ = φ.

Consider θ_{1} := θ ∘ σ_{−t/p}^{φ}, → p_{so}(θ_{1}) = p_{sf} (θ_{1}) = p and θ_{1} ∉ Cnt (M, N). Since σ_{t}^{φ} ∈ Cnt (M, N), θ ∉ Cnt (M, N). □