On group actions on subfactors

MASUDA Toshihiko ( )

Kochi University N ⊂ M : subfactor of type II_{1} with [M : N] < ∞. We want to classify automorphisms of N ⊂ M, (or group actions on N ⊂ M.)

(1) Preliminaries.

(2) Kawahigashi’s classification results.

(3) Classification of group actions.

(1) Preliminaries.

N ⊂ M ⊂ M_{1} ⊂ M_{2} ⊂ ⋯: Jones tower. e_{k} ∈ M_{k} : Jones projection.

α ∈ Aut(M, N) → α can be extended to an automorphism of M_{k} by α(e_{k}) = e_{k}.

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

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

Cnt_{r} (M, N) : set of all non-strongly outer automorphisms.

Examples. N ⊂ M : subfactor with principal graph A_{2n+1} . → ∃ σ ∈ Ker Φ∩Cnt_{r} (M, N) s.t. σ^{2} = id.

Th.(Popa) N ⊂ M : strongly amenable subfactor of type II_{1}. → {M′ ∩ M_{k} ⊂ N′ ∩ M_{k}} : complete invariant.

Th. (Loi) N ⊂ M : strongly amenable subfactor of type II_{1}.

(1) Ker Φ = Int(M, N).

(2) α, β ∈ Int(M, N) and α^{n}, β^{n} ∉ Cnt (M, N) for ∀ n ≠ 0 → α, β : outer conjugate.

Th. (Popa) N ⊂ M : strongly amenable subfactor of type II_{1}.

(1) Cnt (M, N) = Cnt_{r} (M, N).

(2) G : discrete amenable group, α : strongly outer action of G. → Φ(α) : complete cocycle conjugacy invariant.

(2) Kawahigashi’s classification results. Idea : Analogy between type III factors and type II_{1}subfactors. Def. and Th. (Kawahigashi)

χ(M, N) := \frac{Int(M, N) ∩ Cnt (M, N)}{Int(M, N)} .

→ χ(M, N) : abelian group. N ⊂ M : A_{2n+1} subfactor → χ(M, N) = Z/2Z.

Th. (Kawahigashi) G : finite abelian group, → ∃ N ⊂ M AFD subfactor with finite depth and trivial normalizer s.t χ(M, N) = G.

In the rest, we assume the normalizer of N ⊂ M and M ⊂ M_{1} are trivial, and χ_{a}(M, N) is finite.

Def. and Th. (Goto)

χ_{a}(M, N) := \frac{Ker Φ ∩ Cnt_{r} (M, N)}{Int(M, N)} .

→ χ_{a}(M, N) : abelian group.

N ⊂ M strongly amenable, → χ(M, N) = χ_{a}(M, N).

Th. (Kawahigashi) N ⊂ M: subfactor of type II_{1}, For α ∈ Ker Φ, p(α) : strong outer period of α (i.e. the smallest positive number s.t. α^{p} ∈ Cnt_{r} (M, N).) Set σ := α^{p(α)} ∈ Cnt_{r} (M, N), and take a ∈ M_{k} s.t σ(x)a = ax. → ∃ γ_{h}(α) ∈ T s.t. α(a) = γ_{h}(α)a.

Def. (Kawahigashi) For α ∈ Ker Φ,

(1) γ_{h}(α) : higher obstruction for α.

(2) [σ] ∈ χ_{a}(M, N) : ν-invariant for α.

ν induces a homomorphism from p(α)Z to χ_{a}(M, N) by ν(p(α)n) = [σ^{n}].

Factors of type III  Subfactors of type II_{1}
Flow of weights  Standard invariant
mod (α)  Φ(α)
Modular automorphisms  χ_{a}(M, N)
(Modular) obstruction  Higher obstruction
Modular invariant  ν-invariant
Connes cocycle  ???

Difference between subfactors and type III factors.

σ ∈ Cnt_{r} (M, N) ∩ Ker Φ. (Strong outer period = 1.) Then γ_{h}(σ) may fail to be 1. → obstruction for flatness.

γ_{h}(σ) = 1 → classification goes well. γ_{h}(σ) ≠ 1 → classification is more difficult.

Th. (Kawahigashi) N ⊂ M : strongly amenable type II_{1} subfactor, For α, β ∈ Ker Φ, assume

(1) p(α) = p(β)(=: p),

(2) (γ_{h}(α), ν(α)) = (γ_{h}(β), ν(β)),

(3) For σ := α^{p}, γ_{h}(σ) = 1. Then α and β are outer conjugate.

Remark. γ_{h}(σ) = 1 → γ_{h}(α)^{p} = 1.

Th. (Kawahigashi) N ⊂ M : strongly amenable type II_{1} subfactor, For α, β ∈ Ker Φ, assume

(1) p(α) = p(β)(=: p),

(2) γ_{h}(α) = γ_{h}(β) and ν(α) = ν(β),

(3) For q:=(outer period of σ = α^{p}), (p, q) = 1. Then α and β are outer conjugate. (Never assume γ_{h}(σ) = 1.)

Condition (4) → ∃ ν̃ : Z → χ_{a}(M, N) s.t. ν̃|_{p(α)Z} = ν.

Idea of proof. Reduce to the case p(α) = (outer period of α).

The case [M : N] < 4. (Except A_{3} and D_{4} cases.)

Graph  A_{4n−3} A_{4n−1} , E_{6} D_{2n} E_{8}
χ group  Z_{2}  Z_{2}  {1} {1}
Loi part  {1}  {1}  Z_{2} {1}
Flatness obst.  1  -1  1  1
Classification  ◯  partial  ◯ ◯

D_{2n} case. (Φ(α), p(α), γ(α)) : complete invariant.

E_{8} case. (p(α), γ(α)) : complete invariant.

(In both cases, p(α)=(outer period).)

Th. (Loi) N ⊂ M strongly amenble subfactor of type II_{1}, For α ∈ Ker Φ, (p(α), γ_{h}(α), ν(α)) : complete outer conjugacy invariant.

Idea : Set

N(α) := \{( \begin{matrix}x & 0 & 0 \\ 0 & α(x) & 0 \\ 0 & 0 & α(x) \end{matrix} )| x ∈ N\} .

Classify the commuting square

\begin{aligned}N ⊗ M_{3}(C) & ⊂ & M ⊗ M_{3}(C) \\ ∪ & & ∪ \\ N(α) & ⊂ & M(α) \end{aligned} .

(3) Group actions.

What is the counterpart of Connes cocycles in subfactor theory?

Th. (M) For α ∈ Ker Φ, σ ∈ Cnt_{r} (M, N), ∃ u(α, σ) ∈ U(N) s.t.

(1) ασα^{−1} = Adu(α, σ)σ,

(2) u(α, σ) σ-cocycle,

(3) u(α, σ)^{∗} α-cocycle,

(4) ∀x ∈ M, σ(x)a = ax → α(a) = u(α, σ)a.

u(α, σ) ⇔ (Dφ ∘ α^{−1} : Dφ)_{t}.

(4) ⇔ α(φ)^{it} = (Dφ ∘ α^{−1} : Dφ)_{t} φ^{it}.

Algebraic κ-invariant.

Assumption. ∃ σ : lifting of χ_{a}(M, N) to Aut(M, N) s.t. σ_{kl} = σ_{k}σ_{l}. (i.e. σ : action of χ_{a}(M, N) such that [σ_{k}] = k.)

For k, l ∈ χ_{a}(M, N), κ_{a}(k, l) := u(σ_{l}, σ_{k})^{∗} ∈ T.

κ_{a}(k, k) = γ_{h}(σ_{k}) (flatness obstruction.)

κ_{a}(k, l) may be fail to be trivial. (Difference between subfactor case and type III factor case.)

Cocycle conjugacy invariants α : action of G on N ⊂ M with α_{g} ∈ Ker Φ. H := {h ∈ G | α_{h} ∈ Cnt_{r} (M, N)}. (Normal subgroup of G.) ν-invariant ν(h) := [α_{h}] ∈ χ_{a}(M, N) for h ∈ H.

Characteristic invariant. Fix v_{h} ∈ U(N) s.t. α_{h} = Adv_{h}σ_{ν(h)}. Then ∃ μ(k, l) and λ(g, h) s.t. v_{k}σ_{ν(k)}(v_{l}) = μ(k, l)v_{kl}, and α_{g}(v_{g^{−1}hg})u(α_{g}, σ_{ν(h)}) = λ(g, h)v_{h}

(Derived from α_{g}α_{g^{−1}hg}α_{g^{−1}} = α_{h} and α_{k}α_{l} = α_{kl}.)

[λ, μ] : characteristic invariant.

G = Z → λ(1, p(α)) = γ_{h}(α). (μ is trivial.)

For k, l ∈ H, λ(k, l) = μ(k, k^{−1}lk)μ(l, k)κ_{a}(ν(l), ν(k)) .

If κ_{a} is trivial, then [λ, μ] is a usual characteristic invariant.

Th. (M) N ⊂ M : strongly amenable subfactor of type II_{1} with Ker Φ = Aut(M, N). G : discrete amenable group. α : (approximately inner) action of G. Assume κ-invariant is trivial. → (H, [λ, μ], ν) : complete cocycle conjugacy invariant.

Subfactors with principal graph A_{4n−3} , (n ≥ 2), or SU(3)_{3k} (k ≥ 2) subfactors satisfy above assumption.

Th. (M) N ⊂ M : strongly amenable subfactors of type II_{1} with Ker Φ = Aut(M, N). G : discrete amenable group. α : action of G. Assume ∃ ν̃ : G → χ_{a}(M, N) homomorphism s.t. ν̃|_{H} = ν. → (H, [λ, μ], ν) : complete stable conjugacy invariant.

Subfactors with principal graph A_{4n−1} (n ≥ 2) have the nontrivial κ-invariant.

Proof of theorems. Reduce to the following theorem.

Th. (M) Q ⊂ P : strongly amenable subfactor of type II_{1}. (Never assume Ker Φ = Aut(P, Q)) G : discrete amenable group. α : action of G with {g ∈ G | α_{g} ∈ Int(M, N)} = {g ∈ G | α_{g} ∈ Cnt_{r} (M, N)}. → (Φ(α), H, [λ, μ]) : complete cocycle conjugacy invariant.