On classification of approximately inner actions of discrete amenable groups on

subfactors

MASUDA Toshihiko ( )

Kochi University

Outline

(1) Preliminary

(2) Connes-Radon-Nikodym type cocycle and κ invariant

(3) Cocycle conjugacy invariants

(4) Classification N ⊂ M : subfactor of type II_{1}, [M : N] < ∞, extremal,

N ⊂ M ⊂ M_{1} ⊂ M_{2} ⊂ ⋯, Jones tower, e_{i} ∈ M_{i}: i-th Jones projection,

Assumption: Normalizer of N (resp. M) in M(resp. M_{1}) is trivial. ⇒ For α ∈ Aut(M, N), α ∈ Aut(M) outer ⇔ α ∈ Aut(N) outer,

α can be extended to M_{k} by α(e_{k}) = e_{k} Preliminary

Def. (Choda-Kosaki, Popa) σ ∈ Aut(M, N), σ : strongly outer or properly outer ⇔ a ∈ M_{k} σ(x)a = ax, ∀x ∈ M → a = 0.

Th. (Choda-Kosaki, Goto) σ is non-strongly outer ⇔ _{M}M_{kM} ≻ _{M}M_{σM}.

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

Def. (Loi) Φ(α) := {α|_{M′∩M_{k}}}_{k=0}^{∞} Φ(α) : The Loi invariant of α

Th. (Loi) N ⊂ M : strongly amenable subfactor of type II_{1}, ⇒ Ker Φ = \overline{Int}(M, N)

Th. (Popa) N ⊂ M : strongly amenable subfactor of type II_{1} ⇒ Cnt_{r}(M, N) = Cnt (M, N)

Th. (Popa) N ⊂ M : strongly amenable subfactor of type II_{1}, G : discrete amenable group, α : strongly outer action of G on N ⊂ M, ⇒ Φ(α): complete cocycle conjugacy invariant.

Def. and Th. (Goto)

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

χ_{a}(M, N) : (algebraic) χ group. χ_{a}(M, N) becomes an abelian group. N ⊂ M strongly amenable ⇒ χ_{a}(M, N) = χ(M, N)

= \frac{\overline{Int(M, N)} ∩ Cnt (M, N)}{Int(M, N)}

Connes-Radon-Nikodym cocycle

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

(1) ασα^{−1} = Ad u(α, σ) σ,

(2) u(α, σ_{1}σ_{2}) = u(α, σ_{1})σ_{1}(u(α, σ_{2})),

(3) u(α_{1}α_{2}, σ) = α_{1}(u(α_{2}, σ))u(α_{1}, σ),

(4) u(α, Ad u) = α(u)u^{∗}, u ∈ U(N), (4’) u(Ad u, σ) = uσ(u^{∗}), u ∈ U(N),

(5) σ(x)a = ax, ∀ x ⇒ α(a) = u(α, σ)a.

(cf. R : type III factor, α ∈ Aut(R) ⇒ ασ_{t}^{φ}α^{−1} = Ad [Dφ ∘ α^{−1} : Dφ]_{t}σ_{t}^{φ} )

How to construct u(α, σ).

Fix T ∈ Hom (_{M}M_{kM},_{M}M_{σM}) with T T^{∗} = 1. U_{α} : canonical implementing unitary of α on L^{2}(M_{k}).

u(α, σ) := T U_{α}T^{∗}(ˆ1). (ˆ1 ∈ L^{2}(M)).

α ∈ Ker Φ ⇒ u(α, σ) does not depend on the choice of k or T.

The rest of this talk, we assume the following.

(1) χ_{a}(M, N) : finite (abelian) group (2) Ker Φ = Aut(M, N)

(Hence χ_{a}(M, N) = \frac{Cnt_{r}(M, N)}{Int(M, N)} )

(3) ∃ σ : χ_{a}(M, N) → Cnt_{r}(M, N) s.t. [σ_{k}] = k in χ_{a}(M, N) and σ_{k}σ_{l} = σ_{kl}.

Remark. Jones subfactors or almost class of Wenzl subfactors satisfy these assumptions.

Algebraic κ invariant For k, l ∈ χ_{a}(M, N), u(σ_{l}, σ_{k}) ∈ T. Def. κ_{a}(k, l) := u(σ_{l}, σ_{k})^{∗}. κ_{a} is a bicharacter of χ_{a}(M, N).

Example. Principal graph A_{2n+1} case.

[figure: *]

σ ∈ Ker Φ ∩ Cnt_{r}(M, N), σ^{2} = id. Hence χ_{a}(M, N) = Z/2Z

Principal graph A_{4n−3} → κ_{a}(σ, σ) = 1 Principal graph A_{4n−1} → κ_{a}(σ, σ) = −1 Cocycle conjugacy invariants G : discrete group, α : action of G. H := {n ∈ G | α_{n} ∈ Cnt_{r}(M, N)}, H is a normal subgroup. Define ν ∈ Hom (H, χ_{a}(M, N)) by ν(n) = [α_{n}]. Then ν(g^{−1}ng) = ν(n) holds. ∃v_{n} ∈ U(N) s.t α_{n} = Ad v_{n}σ_{ν(n)}. α_{g}α_{g^{−1}ng}α_{g} = α_{n},

(LHS) = α_{g} Ad v_{g^{−1}ng}σ_{ν(n)}α_{g}^{−1} = Ad α_{g}(v_{g^{−1}ng})α_{g}σ_{ν(n)}α_{g}^{−1} = Ad (α_{g}(v_{g^{−1}ng})u(α_{g}, σ_{ν(n)}))σ_{ν(n)}

(RHS) = Ad v_{n}σ_{ν(n)},

Then ∃ λ(g, n) ∈ T s.t. α_{g}(v_{g^{−1}ng})u(α_{g}, σ_{ν(n)}) = λ(g, n)v_{n}.

From α_{m}α_{n} = α_{mn}, m, n ∈ H, ∃ μ(m, n) ∈ T s.t. v_{m}σ_{ν(m)}(v_{n}) = μ(m, n)v_{mn}.

Relation of λ(g, n) and μ(m, n).

g, h ∈ G, l, m, n ∈ H.

(1) μ(l, m)μ(lm, n) = μ(m, n)μ(l, mn)

(2) λ(gh, n) = λ(g, n)λ(h, g^{−1}ng)

(3) λ(g, mn)\overline{λ(g, n)λ(g, n)} = μ(m, n)\overline{μ(g^{−1}mg, g^{−1}ng)}

(4) λ(m, n) = μ(m, m^{−1}nm)\overline{μ(n, m)} × \overline{κ_{a}(ν(n), ν(m))}

(5) λ(g, e) = λ(e, n)

= μ(e, n) = μ(m, e) = 1 Z(G, H|κ_{a}, ν) : Set of (λ, μ) satisfying the previous relations.

(λ, μ) ∼ (λ′, μ′) ⇔ λ(g, h) = λ′(g, n)\overline{c_{g^{−1}ng}}c_{n}, μ(m, n) = μ′(m, n)\overline{c_{m}c_{n}}c_{mn} for some c : G → T with c_{e} = 1

Λ(G, H|κ_{a}, ν) := Z(G, H|κ_{a}, ν)/ ∼. [λ, μ] ∈ Λ(G, H|κ_{a}, ν) does not depend on the choice of v_{n} ∈ U(N).

(H, [λ, μ], ν) : cocycle conjugacy invariant.

Conversely, for given (H, [λ, μ], ν) ∃ α : action of G with given invariant?

Yes if N ⊂ M is strongly amenable and G is amenable.

Remark.

(1) If κ invariant is trivial, → Λ(G, H|κ_{a}, ν) = Λ(G, H).

(2) Λ(G, H|κ, ν) : Λ(G, H) space. (Λ(G, H) : usual characteristic invariant group.)

Classification

Th 1. N ⊂ M : strongly amenable type II _{1} subfactor, (Ker Φ = Aut(M, N)) G : discrete amenable group, α, β : actions of G, Assumption : κ invariant is trivial. then α, β cocycle conjugate ⇔ (H_{α}, [λ_{α}, μ_{α}], ν_{α}) = (H_{β}, [λ_{β}, μ_{β}], ν_{β}).

Th 2. P ⊂ Q : strongly amenable type II_{1} subfactor, (We never assume Ker Φ = Aut(Q, P).) G : discrete amenable group, α, β : actions of G s.t H := α^{−1}(Int(Q, P)) = α^{−1}(Cnt (Q, P)) = β^{−1}(Int(Q, P)) = β^{−1}(Cnt (Q, P)) then, α ∼ β ⇔ (Φ(α), [λ_{α}, μ_{α}]) = (Φ(β), [λ_{β}, μ_{β}])

Proof. Popa’s classification + Brauer group technique.

Outline of proof of Th. 1

Extension of automorphism Set K := χ_{a}(M, N), P ⊂ Q := N ⋊_{σ} K ⊂ M ⋊_{σ} K, w_{k} : implementing unitary.

For α ∈ Aut(M, N)(= Ker Φ), define α̃ ∈ Aut(Q, P) by

(1) α̃(a) = α(a), a ∈ M,

(2) α̃(w_{k}) = u(α, σ_{k})w_{k}.

(The above is an analogue of Haagerup-Størmer’s canonical extension.)

Prop. α, β ∈ Aut(M, N)

(1) α̃β = α̃β̃, α̃^{−1} = α̃^{−1}

(2) α̃σ̂_{k} = σ̂_{k}α̃, σ̂_{k} : dual action of K̂,

(3) Ãd u = Ad u, u ∈ U(N).

(4) If κ_{a}(l, k) ≡ 0 , σ̃_{k} = Ad w_{k}.

Proof of (4). σ̃_{k}(w_{l}) = u(σ_{k}, σ_{l})w_{l}

= \overline{κ_{a}(l, k)}w_{l} = w_{l} = Ad w_{k}(w_{l}), σ̃_{k}(a) = σ_{k}(a) = Ad w_{k}(a), a ∈ M.

Consider actions of G × K̂ on P ⊂ Q by α̃ : (g, k) → α̃_{g}σ̂_{k} and β̃ : (g, k) → β̃_{g}σ̂_{k}.

By (4), α̃^{−1}(Int(Q, P)) = α̃^{−1}(Cnt (Q, P)) = β̃^{−1}(Int(Q, P)) = β̃^{−1}(Cnt (Q, P)). We have [λα̃, μα̃] = [λβ̃, μβ̃] and Φ(α̃) = Φ(β̃).

By Th. 2, α̃ ∼ β̃. ˜α̃, ˜ β̃ : natural extension of α̃, β̃ to P ⋊ K̂ ⊂ Q ⋊ K̂ Then ˜ α̃ ∼ ˜ β̃. By Takesaki duality, α ∼ β. □

When κ invariant is not trivial.

Th. 3. Assumption : ∃ν̃ ∈ Hom (G, χ_{a}(M, N)) s.t ν_{α} = ν̃|_{H_{α}}. Then (H_{α}, [λ_{α}, μ_{α}], ν_{α}) = (H_{β}, [λ_{β}, μ_{β}], ν_{β}) ⇒ α, β stably conjugate. i.e. second dual is conjugate.

Idea. Kawahigashi-Sutherland-Takesaki’s classification of discrete abelian groups on the AFD type III_{1} factor.

By cocycle perturbation, we have α_{g}σ_{k} = σ_{k}α_{g} and β_{g}σ_{k} = σ_{k}β_{g}.

Consider α_{g}σ_{ν_{α}(g)}^{−1}, β_{g}σ_{ν_{β}(g)}^{−1} and apply Th. 2.

Example. (Kawahigashi) N ⊂ M : A_{4n−1} subfactor. α ∈ Aut(M, N), α^{p} ∈ Cnt (M, N) ν_{α}(pn) = [σ^{n}] ∈ χ_{a}(M, N). (G = Z and H = pZ).

If p is odd, define ν̃ by ν̃(n) = [σ^{n}]. Since [σ^{2}] = [id], ν̃(p) = [σ^{p}] = [σ] = ν(p).

Construction of model action.

R : AFD type II_{1} factor, α^{0} : outer action of G on R.

γ_{n} := σ_{ν(n)^{−1}} ⊗ α_{n}^{0}, n ∈ H. μ̃(m, n) := μ(m, n)κ_{a}(ν(m), ν(n)).

On (N ⊗ R) ⋊_{γ,μ̃} H ⊂ (M ⊗ R) ⋊_{γ,μ̃} H, define α as follows.

(1) α_{g}(a) = id ⊗ α_{g}^{0}(a), a ∈ M ⊗ R

(2) α_{g}(u_{g^{−1}ng}) = λ(g, n)u_{n}. This α is a desired one.