University of Tokyo Seminary Notes 45

Quantum Symmetry, Differential Geometry of Finite Graphs and Classification of Subfactors

Lectures given at the University of Tokyo by Adrian Ocneanu (Pennsylvania State University)

Notes recorded by Yasuyuki Kawahigashi

Department of Mathematics, University of Tokyo Hongo, Tokyo, 113, JAPAN

Acknowledgement

The following notes are based on the lectures given by Professor Adrian Ocneanu at the University of Tokyo in July, 1990, and some parts are taken from his lectures given at the Institute of Statistical Mathematics, Hokkaido University, and Osaka Kyoiku University. The contents of II.6 and IV.4 are based on personal communications after the lectures. His stay in Japan was supported by Japan Association for Mathematical Sciences and his lectures were partially supported by the above institutions and Mathematical Society of Japan. We are very thankful to him for giving the lectures and permitting us to make these notes.

Yasuyuki Kawahigashi

Table of contents

I. Bimodules and the Galois functor - - - - - - - - - - - - - - - - - - - - - - 1 I.1. Bimodules and basis - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 I.2. Group-like structure — cells and connections - - - - - - - - - - - - - - - - - 4 I.3. Biunitarity - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 8 I.4. Indices and Perron-Frobenius eigenvectors - - - - - - - - - - - - - - - - - - - 11 II. String algebra and graph geometry - - - - - - - - - - - - - - - - - - - - 12 II.1. Strings and operations - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 12 II.2. Connections and embedding - - - - - - - - - - - - - - - - - - - - - - - - - - - 14 II.3. Bratteli diagrams and Jones projections - - - - - - - - - - - - - - - - - - - - 19 II.4. Infinite graphs - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 24 II.5. Transport and flatness - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 25 II.6. Computation of towers of relative commutants and flatness - - - - - - - - 31 II.7. Fourier transform and convolutions - - - - - - - - - - - - - - - - - - - - - - - 36 II.8. Knot invariant - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 38 III. Central sequences and asymptotic inclusions - - - - - - - - - - - - - 39 III.1. Asymptotic inclusion - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 39 III.2. Central sequences - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 41 III.3. Group case - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 42 IV. Computation of paragroups of small order - - - - - - - - - - - - - - - 45 IV.1. Main result for index< 4 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 45 IV.2. Concrete computations of connections - - - - - - - - - - - - - - - - - - - - 50 IV.3. Flatness - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 54 IV.4. Non-amenability of the E_{10} subfactor - - - - - - - - - - - - - - - - - - - - - 56 IV.5. Paragroups for index> 4 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 58 References - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 59 Index - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 60 I. Bimodules and the Galois functor

Let (X, μ) be a measure space. Passing from the commutative algebra L^{∞}(X, μ) to an algebra of bounded linear operators on a Hilbert space, we pass from a classical space to a quantum space. Groups are symmetries of classical spaces, then what are the symmetries of quantum spaces ? We will use an analogue of the Galois theory.

Let G be a finite group, α an action of G on a factor N. We assume that α is outer. In operator algebras, a basic construction consists in building a larger algebra out of an algebra N and some operators on N. Here we construct the crossed product algebra M = N ⋊_{α} G = ⟨N, (λ_{g})_{g∈G}⟩. (This is similar to a semidirect product of a group.) Then the action α is outer if and only if N′ ∩ M = C. (This can be taken as a definition of outerness.) We now consider conjugacy for N ⊂ M. This is an analogue of knot isotopy. We say N ⊂ M is conjugate to N_{1} ⊂ M_{1} if there is an isomorphism θ : M → M_{1} with θ(N) = N_{1}. That is, a position of N in M is a conjugacy class. Now we would like to find G out of the position of N ⊂ M = N ⋊ G.

I.1. Bimodules and basis

A bimodule is a Hilbert space _{A}H_{B} with normal ∗-representations of A and B^{op}, the opposite algebra of B, which commute with each other. Let N have a finite trace tr. By abuse of notation, we write _{N}M_{M} for _{N}L^{2}(M, tr)_{M}. We say T : _{A}X_{B} → _{A}Y_{B} is an intertwiner if T is in Hom(_{A}X_{B},_{A} Y_{B}), that is, T is linear and T (axb) = aT (x)B, a ∈ A, x ∈ X, b ∈ B. A bimodule _{A}X_{B} is called irreducible if the only self-intertwiners are the scalars. Here a self-intertwiner means an element in Hom(_{A}X_{B},_{A} X_{B}) = End(_{A}X_{B}) = B(X) ∩ A′ ∩ (B^{op})′. If this von Neumann algebra is finite dimensional, we can get an irreducible bimodule by cutting X by a minimal projection in this. Example. The bimodule _{N}N_{N} is the standard form of N. Then End(_{N}N_{N}) = N′ ∩ (N^{op})′ ∩ B(L^{2}(N)) = Z(N), the center of N. If N is a factor, then _{N}N_{N} is irreducible. Now let M = ⟨N, λ_{g}⟩ be the crossed product. An element x ∈ M is written in the form x = ∑_{g∈G} x_{g}λ_{g}, x_{g} ∈ N. This implies that the irreducible decomposition of _{N}M_{N} is ⨁_{g} _{N}(Nλ_{g})_{N}. Here each component is irreducible and not mutually equivalent because of outerness of the action. Thus each element g ∈ G is in one-to-one correspondence to an irreducible bimodule _{N}(Nλ_{g})_{N}. What is the composition law of the group in terms of these bimodules ? The answer is given by_{N}(Nλ_{g})_{N} ⊗_{N} _{N}(Nλ_{h})_{N} ≅ _{N}(Nλ_{gh})_{N} By an analogue of the Galois theory, we get a correspondence between the position and the dual of the finite group. We obtain natural symmetries of a quantum space.

Now suppose N ⊂ M and N′ ∩ M = C, the outerness condition. We try to find a group-like object acting on M so that M is a “crossed product”. We assume no algebraic conditions on N ⊂ M. That is, the data are N ⊂ M and a finite trace tr : M → C. If _{N}X is a left N-module, then X is known to be projective, by von Neumann, that is, _{N}X ≅ _{N}(L^{2}(N)^{⊕k}p), where p is a projection in Mat_{k}(N), k ≤ ∞. Then we write (Tr_{k} ⊗ tr)(p) = dim_{N} X, which is invariant though p is not unique. (This number is called a coupling constant.) For (N ⊂ M, tr), we set [M : N] = dim_{N} M, which is the Jones index [J].

We also see the modules are not only projective but also “almost” free. For any _{N}X, one can find a basis (λ_{i})_{i∈I} ⊂ X in the following sense. Denote the Hilbert space inner product on X by ⟨ , ⟩. We have an N-valued inner product ⟨ , ⟩_{N} defined on a dense subset X_{0} of X with tr_{N}(a⟨x_{1}, x_{2}⟩_{N}) = ⟨ax_{1}, x_{2}⟩, for x_{1}, x_{2} ∈ X, a ∈ N. (This is a Radon-Nikodym derivative with respect to the trace. Also note that an algebra-valued inner product is necessary for KK-theory.) The system (λ_{i})_{i∈I} is called a basis for _{N}X if for all x ∈ X we have x = ∑_{i∈I}⟨x, λ_{i}⟩_{N}λ_{i}. (See Herman-Ocneanu.) To find a basis, decompose _{N}X into cyclic pieces _{N}(Nv_{i}) and set

λ_{i} = ⟨v_{i}, v_{i}⟩_{N}^{−1/2}v_{i} = lim_{ε↘0}(⟨v_{i}, v_{i}⟩_{N} + ε)^{−1/2}v_{i},

if v_{i} is bounded in _{N}X. Then we get for av_{i} ∈ _{N}(Nv_{i})

⟨av_{i}, λ_{i}⟩_{N}λ_{i} = a⟨v_{i}, ⟨v_{i}, v_{i}⟩_{N}^{−1/2}v_{i}⟩_{N}⟨v_{i}, v_{i}⟩_{N}^{−1/2}v_{i}
= a⟨v_{i}, v_{i}⟩_{N}⟨v_{i}, v_{i}⟩_{N}^{−1}v_{i}
= av_{i}.

Hence for all x ∈ X, we get ∑_{i}⟨x, λ_{i}⟩_{N}λ_{i} = x. We define an orthogonal projection q ∈ Mat_{I×I}(N) by q_{ij} = ⟨λ_{i}, λ_{j}⟩_{N}. Then

dim_{N} X = (Tr_{I} ⊗ tr)(q) = ∑_{i} tr(⟨λ_{i}, λ_{i}⟩) = ∑_{i}⟨λ_{i}, λ_{i}⟩.

If λ̃_{j} is another basis of _{N}X, then λ̃_{j} = ∑⟨λ̃_{j}, λ_{i}⟩_{N}λ_{i} = ∑_{i} v_{ji}λ_{i}, where a partial isometry v = (v_{ji}) ∈ Mat_{J×I}(N) is defined by v_{ji} = ⟨λ̃_{j}, λ_{i}⟩_{N}.

For the case of N ⊂ M, a basis for _{N}M was introduced by Pimsner-Popa by different means [PP]. (This argument is also O.K. for type III cases with a little bit of modular automorphism groups.)

For N ⊂ M, construct the standard bimodule _{N}H_{M} ≡ _{N}M_{M}( = _{N}(L^{2}(M))_{M}.) A bimodule is a “quantum automorphism” in the following sense. For a bimodule AX_{B}, choose a basis (λ_{i}) for _{A}X. Then λ_{i}b = ∑_{j}⟨λ_{i}b, λ_{j}⟩_{A}λ_{j} for b ∈ B. Define a map θ = (θ_{ij}) : B → Mat_{I×I}(A) by θ_{ij}(b) = ⟨λ_{i}b, λ_{j}⟩_{A}. This is a ∗-homomorphism with the properties θ(1) = q, θ(B) ⊂ q(MatA)q.

Example. In the group case M = N ⋊ G, set _{N}X_{N} = _{N}(Nλ_{g})_{N}. Then λ_{g} is a basis for _{N}(Nλ_{g}). Because λ_{g}x = α_{g}(x)λ_{g}, x ∈ N, we get ⟨λ_{g}x, λ_{g}⟩_{N} = α_{g}(x) and θ = α_{g} ∈ Aut(N). This θ is defined uniquely up to inner perturbation. For the bimodule _{N}M_{N}, the homomorphism θ is given by ⨁_{g} α_{g} : N → L^{∞}(G) ⊗ N ⊂ Mat_{G×G}(N). For the bimodule_{N}M_{M}, the homomorphism θ : M → Mat_{G×G}(N) = B(L^{2}(G)) ⊗ N is given by θ(x) = (α_{g}(x))_{g} ∈ L^{∞}(G) ⊗ N, x ∈ N, and θ(λ_{g}) = λ_{g} ⊗1 ∈ B(L^{2}(G))⊗N. Determining θ for the bimodule _{M}M_{N} is left as an exercise.

I.2. Group-like structure — cells and connections

In the above sense, bimodules are “quantum homomorphisms”. Then what is the group-like global structure ? Recall Weyl’s method of making tensor powers of the standard one and decomposing them into pieces. We take tensor powers of H = _{N}M_{M} and H̄ = _{M}M_{N}^{∗} = _{M}M_{N}. Take a finite power ⋯ H ⊗H̄ ⊗H ⊗H̄ ⋯ and decompose it into irreducible components. We have four kinds of bimodules; N-N, N-M, M-N, M-M bimodules. We choose a representative in each equivalence class.

For an irreducible bimodule _{M}X_{M}, take H ⊗ X, which means _{N}H_{M} ⊗_{M} _{M}X_{M}, and decompose it into irreducible N-M sub-bimodules: H ⊗ X ≅ ⨁_{i} Y_{i}. We write the following graph for this.

\begin{matrix}X \\ ↓ & H⊗· \\ Y_{i} \end{matrix}

For each edge v_{i}, choose an N-M intertwiner T (v_{i}) : H ⊗ X → Y_{i}. (That is, we have partial isometries with mutually orthogonal support.) We draw a following graph.

M-M bimodules

···

H^{#} ⊗ · · ⊗H^{#}

N-M bimodules ⋯  ⋯ M-N bimodules

· ⊗ H^{#} H^{#} ⊗ · ···

N-N bimodules

where H^{#} stands for H or H̄.

The terminology “finite depth” means that only finitely many bimodules appear in repeating this procedure. We also call depth the longest distance from ∗ to a vertex on the graph.

Example(Group case). Let M = N ⋊ G, and assume G = S_{3} for simplicity. By Frobenius duality, we get Hom(H ⊗ X, Y) ≅ Hom(X, H̄ ⊗ Y). Write S_{3} = {1, a, a^{2}, b, ab, a^{2}b}, with a^{3} = b^{2} = (ab)^{2} = 1, and Ŝ_{3} = {1, ε, σ}, where ε denotes the sign representation and σ denotes the 2-dimensional representation. Then we get the following graph.

1_{•}  σ_{•}  ε_{•}  M-M
NM_{M} • •_{M}M_{N}

1^{•} a^{•} a^{2}^{•} b^{•} ab^{•} a^{2}b^{•} N-N

Note that _{M}M_{M} is obtained as H̄ ⊗ H = _{M}(M ⊗_{N} M)_{M}. For ρ ∈ Rep(G), set ξ_{ij}^{ρ} = ∑_{g} ρ_{ij}(g^{−1})λ_{g^{−1}} ⊗_{N} λ_{g} ∈ H̄ ⊗ H. If x ∈ N, then ξ_{ij}^{ρ}x = xξ_{ij}^{ρ}. We also have

ξ_{ij}^{ρ}λ_{h} = ∑_{g} ρ_{ij}(g^{−1})λ_{g^{−1}} ⊗ λ_{gh}

= ∑_{k,s} ρ_{is}(h)ρ_{sj}(k^{−1})λ_{hk^{−1}} ⊗ λ_{k}

= λ_{h} ∑_{s} ρ_{is}ξ_{sj}^{ρ}.

Thus, X_{j}^{ρ} = ⨁_{s} Mξ_{sj}^{ρ} is an M-M bimodule. One can show that H̄ ⊗ H = ⨁_{ρ,j} X_{j}^{ρ} and X_{i}^{ρ} ≅ X_{j}^{ρ}, where the intertwiner T_{ij}^{ρ} : X_{i}^{ρ} → X_{j}^{ρ} is given by T_{ij}^{ρ}(ξ_{si}^{ρ}) = ξ_{sj}^{ρ}. Each _{M}(X_{i}^{ρ})_{M} is irreducible as seen below. Notice that N′ ∩ X_{i}^{ρ} ≡ {x ∈ X_{i}^{ρ} | ax = xa for all a ∈ N} is a linear span of {ξ_{si}^{ρ}}_{s}. Any intertwiner T : X_{i}^{ρ} → X_{i}^{ρ} must map N′ ∩ X_{i}^{ρ} to N′ ∩ X_{i}^{ρ}. Now use T (ξλ_{g}) = T (ξ)λ_{g} and that ρ is irreducible to show that T ∈ C. Now what is the group structure ? We have the following diagram of N-N bimodules.

MX_{M} ⟶_{T_{2}}^{·⊗H̄} _{M}Y_{2N}
H⊗·↓^{T_{1}}  ^{S_{2}}↓^{H⊗·}
NY_{1M} ⟶_{·⊗H̄}^{S_{1}} _{N}Z_{N},

that is,

H ⊗ X ⊗ H̄ ⟶^{1_{H} ⊗T_{2}} H ⊗ Y_{2}
T_{1}⊗1H̄ ↓ ↓^{S_{2}}
Y_{1} ⊗ H̄ ⟶^{S_{1}} Z.

Because S_{2}(1 ⊗ T_{2})(T_{1} ⊗ 1)^{∗}S_{1}^{∗} is a N-N bimodule map from Z to Z, it is a scalar by the irreducibility. We write this

W ( \begin{matrix}T_{1} & ↙ & ↘ & T_{2} \\ S_{1} & ↘ & ↙ & S_{2} \end{matrix} ) ∈ C,

which depends on choices of the bimodules and the intertwiners. We call four edges, one from each graph, with common vertices, a cell. In the above example, we have 36 cells. A map W from the set of cells to C is called a connection (or a Boltzmann weight). This is unique up to a gauge coming from the choice of vertices and edges of the graphs just like a cohomological invariant. In the above case,

W ( \begin{matrix}i↙ & σ & _{↘j} \\ ↘ & g & ↙ \end{matrix} ) = σ_{ij}(g) ∈ C,

as seen below. For a subfactor R ⋊ H ⊂ R ⋊ G, we get the Mackey machine in this way.

In the following graph,

1_{•}  σ_{•}  ε_{•}
•
1^{•} a^{•} a^{2}^{•} b^{•} ab^{•} a^{2}b^{•}

we get the following connection. For the cell

M(X_{j}^{ρ})_{M} ⟵^{T} H ⊗_{N}M_{M}
↓ ↑
MM_{N} ⟶ _{N}(Nλ_{g})_{N},

the connection is given by

λ_{g} ∑_{s} σ_{is}(g)ξ_{sj}^{σ} = ξ_{ij}^{σ}λ_{g} ⟵  λ_{g}
k↓ ↑
σ_{ik}(g)λ_{g}  ⟶ σ_{ik}(g)λ_{g}, λ_{g}.

Thus W = σ_{ik}(g).

I.3. Biunitarity In the following diagram, let X, Z be fixed and Y vary.

\begin{matrix}X \\ T_{1} & ↓ \\ Y & ⟶^{S_{1}} & Z. \end{matrix}

Then S_{1}(T_{1} ⊗ 1)’s make a basis for Hom(H ⊗ X ⊗ H̄, Z). Similarly, in

\begin{aligned}X & ⟶^{S_{2}} & Y_{2} \\ S_{2} & ↓ \\ & Z, \end{aligned}

we get a basis for Hom(H ⊗ X ⊗ H̄, Z). These mean that W ( \begin{matrix}i & ↙ & ↘ & j \\ & ↘ & ↙ \end{matrix} ) is

unitary when X, Z are fixed. For example, for fixed σ, g in the group case, (σ_{ij}(g))_{ij} is unitary.

Similarly, if Y_{1}, Y_{2} are fixed in

X ⟶ Y_{2}
↑ ↑
Y_{1} ⟶ Z,

then W ( \begin{matrix}↗ & i & ↘ \\ ↘ & j & ↗ \end{matrix} ) is unitary, where we need a renormalization

W ( \begin{matrix}k & ↗ & i & ↘ & l \\ ↘ & j & ↗ \end{matrix} ) = \sqrt{\frac{μ(i)μ(j)}{μ(k)μ(l)}}\overline{W ( \begin{matrix}k & ↙ & i & ↘ & l \\ & ↘ & j & ↙ \end{matrix} )}.

In the group case, (\frac{|σ|^{1/2}}{|G|^{1/2}} σ_{ij}(g))_{(σ,i,j),g} is a unitary, which is |G|×|G|-matrix. That is,

( \begin{matrix}1_{11}(1) & 1_{11}(a) & 1_{11}(a^{2}) & ⋯ \\ σ_{11}(1) & σ_{11}(a) & σ_{11}(a^{2}) & ⋯ \\ σ_{12}(1) & σ_{12}(a) & σ_{12}(a^{2}) & ⋯ \\ ⋮ & ⋮ & ⋮ & ... \end{matrix} )

is unitary (up to renormalization). (cf. Peter-Weyl Theorem.)

The biunitarity axioms states that the both matrices

(W ( \begin{matrix}↙ & i & ↘ \\ ↘ & j & ↙ \end{matrix} )) , (W ( \begin{matrix}k & ↗ & ↘ & l \\ ↘ & ↗ \end{matrix} )) ,

are unitary for each fixed (i, j) and (k, l). (Note that this renormalization convention is slightly different from that in [O1].)

In the following we write

W (\begin{matrix}a & ⟶ & c \\ ↓ & & ↓ \\ b & ⟶ & d \end{matrix})

for one of

W (( \begin{matrix}b & ↙ & a & ↘ & c \\ & ↘ & d & ↙ \end{matrix} )) , W (( \begin{matrix}a & ↗ & b & ↘ & d \\ ↘ & c & ↗ \end{matrix} )) ,

depending the position of the vertex a. In this notation, the renormalization rule and biunitarity are expressed as follows.

W (\begin{matrix}a & ⟶ & c \\ ↓ & & ↓ \\ b & ⟶ & d \end{matrix}) = \sqrt{\frac{μ(b)μ(c)}{μ(a)μ(d)}} · \overline{W (\begin{matrix}b & ⟶ & d \\ ↓ & & ↓ \\ a & ⟶ & c \end{matrix})},

∑_{ξ,η} W (\begin{matrix}a & ⟶^{ξ} & c \\ ^{ζ}↓ & & ↓^{η} \\ b & ⟶^{ρ} & d \end{matrix}) · W (\begin{matrix}c & ⟵^{ξ} & a \\ ^{η}↓ & & ↓^{ζ′} \\ d & ⟵^{ρ′} & b \end{matrix} ) = δ_{ζ·ρ,ζ′·ρ′},

where we used the notation

W (\begin{matrix}a & ⟶ & c \\ ↓ & & ↓ \\ b & ⟶ & d \end{matrix}) = \overline{W (\begin{matrix}c & ⟵ & a \\ ↓ & & ↓ \\ d & ⟵ & b \end{matrix})}.

I.4. Indices and Perron-Frobenius eigenvectors

We have

dim(_{N}M ⊗_{N} X) = [M : N] dim_{N} X,
dim(_{N}M ⊗_{M} X) = dim_{N} X,

where we mean by “dim” the dimension as N-module. By these, we get a relation between the index and an eigenvalue. For the vector μ_{0}, which assigns the dimension to each bimodule, the adjacency matrix Δ of the graph satisfies

Δμ_{0}(x) = ∑_{\begin{matrix}v∈edges \\ s(v)=x \\ r(v)=y \end{matrix}} μ_{0}(y),

where s(v) and r(v) denote the source and the range of an edge v.

μ_{0} = 1_{•}  2_{•}  1_{•} ; 6^{•} ,  Δμ_{0} = 6_{•}  12_{•}  6_{•} ; 6^{•}

Renormalizing μ_{0} by multiplying the dim of _{M}X by [M : N]^{−1/2}, we get an eigenvector μ for Δ.

μ = 1_{•}  2_{•}  1_{•} ; \sqrt{6}^{•} ,  Δμ = \sqrt{6}μ = \sqrt{6}_{•}  2\sqrt{6}_{•}  \sqrt{6}_{•} ; 6^{•}

That is, the index is equal to the square of the eigenvalue.

II. String algebra and graph geometry

II.1. Strings and operations

Let G be an unoriented bipartite finite graph with a distinguished vertex.

even:  ∗  •  •
odd:  •  •

We will see how a graph is viewed as a manifold, which is called a granifold. The adjacency matrix Δ plays a role of the Laplacian, and the vector μ the harmonic non-negative function. An (oriented) path on G is a succession of edges: ξ = (v_{1}, v_{2},…, v_{n}), where r(v_{i}) = s(v_{i+1}). We write ṽ for the edge v with the reversed orientation. A string on G is a pair of paths, ρ = (ρ_{+}, ρ_{−}), with s(ρ_{+}) = s(ρ_{−}), r(ρ_{+}) = r(ρ_{−}), and |ρ_{+}| = |ρ_{−}|, where | · | denote the length of a path. We draw a picture as follows to denote a string.

ρ_{+} \begin{matrix}∗  •  • \\ •  • \\ ∗  • ρ_{−} \\ • \\ • \end{matrix}

Define an algebra String^{(n)}G with the linear basis of the n-strings, that is, the strings with length n. We define a multiplication and a ∗-operation by

(ρ_{+}, ρ_{−}) · (η_{+}, η_{−}) = δ_{ρ_{−},η_{+}}(ρ_{+}, η_{−}),
(ρ_{+}, ρ_{−})^{∗} = (ρ_{−}, ρ_{+}),

which makes String^{(n)}G into a finite dimensional C^{∗}-algebra.

For the moment, we work with strings ρ having s(ρ) = ∗, the given distinguished point on the graph G. Define tr : String^{(n)} → C by tr(ρ) = δ_{ρ_{+},ρ_{−}}β^{−|ρ|}μ(r(ρ)), where β is the eigenvalue for μ, Δ(μ) = βμ. Define an embedding i_{n}^{n+k} : String^{(n)}G → String^{(n+k)}G by i_{n}^{n+k}(ρ_{+}, ρ_{−}) = ∑_{|ξ|=k}(ρ_{+} · ξ, ρ_{−} · ξ), where ρ_{±} · ξ denotes the concatenation of the paths. (Remark that the identity in matrix algebra is written as ∑ e_{ii}.) This i_{n}^{n+k} is compatible with the trace tr defined above: tr(i_{n}^{n+k}(ρ)) = tr(ρ).

The above tr is the only trace with this property. Let A_{∞} = (⋃_{n} String^{(n)}, tr) and we complete this as follows. (The GNS construction with respect to tr.) Define an inner product on A_{∞} by ⟨x, y⟩ = tr(xy^{∗}) to make a pre-Hilbert space. By completion, we get a Hilbert space L^{2}(A_{∞}, tr). The algebra A_{∞} acts by multiplication on the left. Taking a weak closure, we get a von Neumann algebra, which is the (Murray-von Neumann) hyperfinite II_{1} factor R.

II.2. Connections and embedding

Now start with four graphs G_{1}, G_{2}, G_{3}, G_{4} with common vertices.

⋯ ⟶^{G_{2}} ⋯
G_{1}↓ ↓^{G_{3}}
⋯ ⟶_{G_{4}} ⋯

Assume that we have a vector μ : vertices → C, which is harmonic for each G_{i}, such that the eigenvalues β_{G_{i}} and β_{G_{i+2}} are equal. (The index i is in mod 4.) Further assume we have a biunitary connection W. We then construct string algebras.

∗ ⟶^{G_{2}} • ⟶^{G̃_{2}} • ⋯
G_{1}↓ ↓^{G_{3}} ^{G_{1}}↓
• ⟶_{G_{4}} • ⟶_{G̃_{4}} • ⋯
G̃_{1}↓ ↓G̃_{3} G̃_{1}↓
• ; ⋮ ⟶^{G_{2}} • ; ⋮ ⟶^{G̃_{2}} • ; ⋮ ⋯

Take Hilbert spaces H_{1}, H_{2} with a orthonormal basis:

•  • ⟶ •
↓ ↓
• ⟶ • ,  •

We can define a unitary matrix: H_{1} → H_{2} by W, which determines the matrix entries. We write the following picture for this unitary.

\begin{matrix}• & ⟶ & • \\ ↓ & & ↓ \\ • & ⟶ & • \end{matrix} ↙ W

We can use these unitaries for several times for identifying Hilbert spaces. The result does not depend on the order in which we use W.

∗ ⟶ •  ∗  ∗  ∗
↓  ↓  ↓  ↓
•  • ⟶ •  •  •
↓  ↓  ↓  ↓
•  •  • ⟶ •  •
↓ ^{⟶}  ↓ ^{⟶}  ↓ ^{⟶}^{···}  ↓
⋮  ⋮  ⋮  ⋮
↓  ↓  ↓  ↓
•  •  •  • ⟶ •

Thus all the Hilbert spaces of paths of the following form are unitarily equivalent using W.

∗^{(0,0)}

↓ • ↓

• ⟶ ⋯ ⟶ •

↓ ⋮ ↓

•_{(k,n)}

The corresponding string algebras are isomorphic to each other, using Ad(W). For any k, n ≥ 0, we have an algebra A_{k,n} given by strings from ∗ = (0, 0) to (k, n). The trace is compatible with this identification.

Note that A_{k,n} is embedded into A_{k,n+1} just like before. Similarly, A_{k,n} ⊂ A_{k+1,n}. We have a double complex of algebras.

⋮  ⋮
⋯ A_{k,n} ⊂ A_{k,n+1} ⋯ A_{k,∞}
∩  ∩
⋯ A_{k+1,n} ⊂ A_{k+1,n+1} ⋯ A_{k+1,∞}
⋮ ⋮
⋯ A_{∞,n} ⊂ A_{∞,n+1} ⋯

The inclusion A_{0,∞} ⊂ A_{1,∞} is our string model for N ⊂ M. Note that by definition of identification using W, the coefficient of the embedding ρ = (ρ_{+}, ρ_{−}) ∈ A_{0,n} ↦ ∑ c_{ρ,η}η, where c_{ρ,η} ∈ C, and a string η = (η_{+}, η_{−}) ∈ A_{k,n} is given in the form of vertical k-paths composed with horizontal n-paths, is given by

c_{ρ,η} = ∑_{configurations} ∏_{cells} W (cell).

Here a “configuration” means a choice of cells which fill the following k×2n-diagram.

∗ ⟶ • · · · · · · •^{ρ_{+}} ⟶ • ⟵ • · · · · · · •^{ρ_{−}} ⟵ ∗
η_{+}↓  ↓  ↓^{η_{−}}
⋮  ⋮  ⋮
η_{+}↓  ↓  ↓^{η_{−}}
• ⟶ • · · · · · · •_{η_{+}} ⟶ • ⟵ • · · · · · · •_{η_{−}} ⟵ •

This is just by the definition of the embeddings and matrix multiplication rule. We also denote this coefficient c_{ρ,η} by

W (\begin{matrix}∗ & ⟶ & ⋯ & ⟵ & ∗ \\ ↓ & & & & ↓ \\ ⋮ & & & & ⋮ \\ ↓ & & & & ↓ \\ • & ⟶ & ⋯ & ⟵ & • \end{matrix}) .

This is a trace-preserving homomorphism. Most of known subfactors are from this construction. Note that the string algebra construction works for any four graphs with biunitary connection without assuming flatness which will be explained in II.5. (This construction is equivalent to that based on commuting squares. The commuting square condition corresponds to biunitarity.) But there is a following example, which does not come from the string algebra.

Example. Choose a non-amenable finitely generated group G and consider the non-commutative Bernoulli shift α of G on the hyperfinite II_{1} factor R. This actions is ergodic on the central sequence algebra by Jones [J2]. Consider a subfactor

N = \{( \begin{matrix}x \\ & α_{g_{1}(x)} \\ & & . & . & . \\ & & & & & α_{g_{n}}(x) \end{matrix} ) | x ∈ R\} ⊂ M = R ⊗ Mat_{n+1},

where g_{1},… g_{n} are generators of G. By ergodicity on the central sequence algebra, we get N^{ω} ∩ M′ = C. But for string algebra construction, we always get the sequence of Jones projections (e_{n})_{n} (on the horizontal string algebras) in N^{ω} ∩ M′. (See II.3 below.) Thus this subfactor does not arise from the string algebra construction.

The main theorem states that the Galois functor and the string algebra construction are inverses to each other under good conditions.

Also see Roche [R] for a string algebra construction.

II.3. Bratteli diagrams and Jones projections

Now we move to local analysis of finite dimensional algebras. Let A be a finite dimensional C^{∗}-algebra. Then A is isomorphic to ⨁_{n} B(H_{n}) ≅ ⨁_{n} H_{n}^{∗}⊗H_{n}, where each H_{n} is a finite dimensional Hilbert space. The irreducible A-modules are, up to isomorphism, _{A}H_{n}’s. If there is a homomorphism φ : A → B with φ(1) = 1, _{A}H_{i} is an A-module, and _{B}K_{j} is a B-module, then K_{j} becomes an A-module using φ. Then _{A}K_{j} ≅ ⨁_{i} m_{ij}_{A}H_{i}, for some integers m_{ij}. We draw the following diagram for this.

•^{i}  •  •  _{A}H_{i}
m_{ij}
•  •_{j}  •  •  _{B}K_{j}

For each edge v, choose an intertwiner (partial isometries with mutually orthogonal ranges) T (v) : _{A}H_{i} → _{A}K_{j}, that is, with relations

T (v)^{∗}T (v) = 1_{H_{i}}, ∑_{v,r(v)=j} T (v)T (v)^{∗} = 1_{K_{j}}.

Now suppose one has an inductive system A_{0} = C ⊂ A_{1} ⊂ A_{2} ⊂ ⋯ of finite dimensional algebras. Make a graph G with vertices on level n corresponding to irreducible A_{n}-modules and edges corresponding to intertwiners as above.

•
•
• •
•  •  •
•  • • •  •

If an oriented graph G with a distinguished point is given, we let A_{n} be the n-string algebra with source ∗. We assume that G has no dead ends (i.e., every vertex is a source) and every vertex is accessible (i.e., the range of a path starting at ∗). The two functors (correspondences between the system of algebras and the graph) are inverse to each to other.

Start with (A_{n}), A_{0} = C, build its Bratteli diagram G as above, and denote for each vertex x of G, T (x) the corresponding module and for each edge v, T (v) intertwiner. Now build the string algebra (String^{(n)}G)_{n} on G. For every string ρ = (v_{1} · v_{2} ⋯ v_{n}, w_{1} · w_{2} ⋯ w_{n}), the map

T (ρ) = T (v_{n})T (v_{n−1}) ⋯ T (v_{1})T (w_{1})^{∗} ⋯ T (w_{n})^{∗}

is from T (x) to T (x), hence T (ρ) ∈ B(T (x)) ⊂ A_{n} = ⨁_{x} B(T (x)).

For the other way, start with a graph which is a Bratteli diagram; it is N-graded, on the level 0 it has one vertex ∗, and every edge v has level(r(v)) = level(s(v)) + 1, each vertex is accessible, and there are no dead ends.

Build the string algebras (String^{(n)}G). We want to show that the invariant of this inductive system is again G.

For each vertex x of G, let Path_{∗,x}^{(n)}G be the Hilbert space with orthonormal basis consisting of the paths on G from ∗ to x, n = level of x.

Now Path_{∗,x}^{(n)}G is a left module for String^{(n)}G by (ρ_{+}, ρ_{−}) · ξ = δ_{ρ_{−},ξ}ρ_{+}. This module is irreducible and

String^{(n)}G = ⨁_{level(x)=n}(Path^{(n)}G)^{∗} ⊗ Path^{(n)}G

by the correspondence (ρ_{+}, ρ_{−}) ↔ ρ_{+}^{∗} ⊗ ρ_{−}. The intertwiner T (v) corresponding to an edge x →^{v} y maps ξ ∈ Path_{x}^{(n)}G into ξ · v ∈ Path_{y}^{(n+1)}G, that is, “adding an edge”.

The systems (A_{n}), (B_{n}) are inductive systems of algebras and φ : ⋃ A_{n} → ⋃ B_{n} is a homomorphism with φ(A_{n}) ⊂ B_{n}. We obtain, besides the graphs, (G for (A_{n}) and H for (B_{n})), a graph I with edges having source in G and range in H.

∗ ∗
• • •
G  •  •  • • • H
•  •  •  •  • • •
• • •  •  •  • • • • • •

For every cell

x ⟶_{c}^{I} •
G↓^{a}  ^{b}↓^{H}
• ⟶_{I}^{d} y,

we have an intertwiner

T (c)^{∗}T (b)^{∗}T (d)T (a) : T (x) → T (x).

Since T (x) is irreducible, this is a scalar, and we write W ( \begin{matrix}↙ & ↘ \\ ↘ & ↙ \end{matrix} ) ∈ C.

If we keep x, y fixed, then {T (b)T (c)}_{b,c} and {T (d)T (a)}_{d,a} are both basis for

Hom_{A_{n}}(T (x), T (y)), so W ( \begin{matrix}↗ & x & ↘ \\ ↘ & y & ↗ \end{matrix} ) gives a unitary matrix. Now φ(ρ_{+}, ρ_{−}) =

∑ c_{ρ,η}(η_{+}, η_{−}), where (ρ_{+}, ρ_{−}) ∈ String^{(n)}G, (η_{+}, η_{−}) ∈ String^{(n)}H, and

c_{ρ,η} = ∑_{configuration} ∏_{cells} W (cell).

A configuration is for the following diagram.

• ⟶ ⋯ ⟶ •
ρ_{+}↓  ↓^{η_{+}}
⋮  ⋮
ρ_{−}↑  ↑^{η_{−}}
• ⟶ ⋯ ⟶ •

Here we have a similar convention for the cells in the lower half of the diagram to that in I.3. That is,

W (\begin{matrix}b & ⟶ & d \\ ↑ & & ↑ \\ a & ⟶ & c \end{matrix}) = \overline{W (\begin{matrix}a & ⟶ & c \\ ↓ & & ↓ \\ b & ⟶ & d \end{matrix})}

(Compare this to partition functions in statistical mechanics.) Let G be a connected finite unoriented graph with a distinguished vertex x and consider the Laplacian Δ ≡ Δ_{G}, and let μ be its unique harmonic measure Δμ = βμ on vertices of G with μ(∗) = 1. We construct Jones projections e_{n} ∈ String_{∗}^{(n+1)} by

e_{n} = ∑_{\begin{matrix}|α|=n−1 \\ |v|=|w|=1 \end{matrix}} \frac{μ(r(v))^{1/2}μ(r(w))^{1/2}}{μ(r(α))} (α · v · ṽ, α · w · w̃).

Then we get, by simple computations,

(I)  \begin{aligned}e_{i}e_{j} & = e_{j}e_{i},  |i − j| ≠ 1, \\ e_{i} & = e_{i}^{∗} = e_{i}^{2}, \\ e_{i}e_{i±1}e_{i} & = β^{−2}e_{i}. \end{aligned}

Recall that

tr(ρ) = δ_{ρ_{+},ρ_{−}}β^{−|ρ|}μ(r(ρ))

is a trace on String(G). This give a conditional expectation

E_{n−1} : String^{(n)}G → String^{(n−1)}G,

tr(ab) = tr(aE_{n−1}(b)), a ∈ String^{(n−1)}G, b ∈ String^{(n)}G.

Expressing ρ = (ρ_{+}, ρ_{−}) as ρ_{+} = η_{+} · v_{+}, ρ_{−} = η_{−} · v_{−}, |v_{+}| = |v_{−}| = 1, we get

E_{n−1}(ρ) = β^{−1}δ_{v_{+},v_{−}}μ(r(v_{+}))μ(s(v_{+}))^{−1} · (η_{+}, η_{−}).

Now we also have

(II)

e_{n} ∈ (String_{∗}^{(n−1)}G)′ ∩ String_{∗}^{(n+1)}G,
e_{n}ρe_{n} = E_{n−1}(ρ)e_{n},  for all ρ ∈ String^{(n)}G.

Theorem. If the inductive system (A_{n}) of finite dimensional algebras is endowed with a trace tr, and there are projections e_{n} ∈ A′_{n−1} ∩A_{n+1} satisfying (I), (II), then A_{n} is isomorphic to the string algebras on an unoriented graph G.

II.4. Infinite graphs

Let G, μ, β as above. Even if G is not finite but just locally finite, there exits an eigenvector μ for Δ_{G} as follows. Let G_{n} be a system of increasing graphs, containing ∗, whose union is G. Then there exists μ_{n} for each n such that

Δ_{G_{n}}μ_{n} = β_{n}μ_{n},
μ_{n}(∗) = 1.

Because {μ_{n}(x)}_{n} is bounded at each x, there exists a weak limit μ with Δ_{G}μ = (lim β_{n})μ = βμ, where β = ∥Δ_{G}∥. But unlike finite graph cases, this μ is not unique as the following example for the graph A_{∞} shows.

∗ =1  2  3  4  ⋯,  Δμ = 2μ,
∗ =1  3  8  21  ⋯,  Δμ = 3μ,

For any unoriented graph G, setting

m = sup_{x∈vertex(G)} #{e ∈ edge(G) | s(e) = x},

we get m^{1/2} ≤ ∥G∥ ≤ m. In general, if Δμ = βμ, we get β ≥ ∥Δ_{G}∥.

II.5. Transport and flatness

We consider an auxiliary graph H in addition to G, and now our four graphs are G_{1} = G, G_{2} = H, G_{3} = G, and G_{4} = H. Let W be a biunitary connection on these. Our basic observation is that strings are similar to tensors. For example, a string

ρ = (v_{1}·v_{2}·v_{3}, w_{1}·w_{2}·w_{3}) corresponds to a tensor dv_{1}⊗dv_{2}⊗dv_{3}⊗ \frac{d}{dw_{3}} ⊗ \frac{d}{dw_{2}} ⊗ \frac{d}{dw_{1}} ,

where the first three components are covariant and the other three are contravariant. The string product corresponds to the tensor product as associative algebras.

Transport along (ξ_{+}, ξ_{−}) is defined the formula

Transp_{(x,y)}^{(ξ_{+},ξ_{−})}(ρ) = ∑ c_{ρ,η}^{(ξ_{+},ξ_{−})}η,

where x = s(ρ) = s(ξ_{+}) = s(ξ_{−}), y = r(ξ_{+}) = r(ξ_{−}) = s(η), and the coefficient c_{ρ,η}^{(ξ_{+},ξ_{−})} ∈ C is determined by conjugation using W as before. This is not a homomorphism for general ξ_{+}, ξ_{−}. If we make ξ_{+}, ξ_{−} vary and consider the map

ρ ↦ ∑_{ξ_{+},η_{+},ξ_{−},η_{−}}(coefficient)(ξ_{+} · η_{+}, ξ_{−} · η_{−}),

then it is a homomorphism, but not for a fixed (ξ_{+}, ξ_{−}). The map Transp_{x,y}^{(ξ_{+},ξ_{−})} is just completely positive in general.

A field of strings on G is a map x ∈ vertex(G) ↦ f_{x} ∈ String_{x}G. They form an algebra naturally. A field f is said to be flat if for all vertices x, y of G which can be joined by H and any paths ξ_{+}, ξ_{−} of the same length from x to y on H, we have

Transp_{x,y}^{(ξ_{+},ξ_{−})}(f_{x}) = δ_{ξ_{+},ξ_{−}}f_{y}.

Flat fields form a ∗-subalgebra of all the fields. Note that the restriction of flat fields (f_{x})_{x} ↦ f_{∗} ∈ String_{∗}G is an injective homomorphism by flatness. If this restriction for flat fields on G_{1}^{(even)} with respect to the horizontal graph G_{2} · G̃_{2} is surjective in the string algebra construction, we say the connection on the four graphs is flat. Note that such a flat field of k-strings is characterized by the property that it is embedded into a vertical k-string algebra A′_{0,2n}∩A_{k,2n} with the same form of strings for all 2n ≥ depth. (Also see the proof of the theorem in II.6.) By this, flatness of a connection means that A_{k,0} ⊂ A′_{0,n} ∩ A_{k,n} for all k, n. This condition is equivalent to saying that A_{0,∞} and A_{∞,0} commute. (In this way, it is easy to see a connection arising via the Galois functor satisfies flatness.) Note that A′_{0,2n} ∩ A_{k,2n} is spanned by elements of the form ∑_{ζ}(ζ ·η_{+}, ζ ·η_{−}), where ζ is a horizontal 2n-path and η is a vertical k-string. Thus, seeing a coefficient of a vertical k-string (ξ_{+}, ξ_{−}) embedded in A_{k,2n}, flatness can be stated in the form

(∗) W ( \begin{matrix}∗ & ⟶ & ⋯^{ξ_{+}} & ⟶ & • & ⟵ & ⋯^{ξ_{−}} & ⟵ & ∗ \\ ^{ζ_{+}}↓ & & & & ↓ & & & & ↓^{ζ_{−}} \\ ⋮ & & & & ⋮ & & & & ⋮ \\ ^{ζ_{+}}↓ & & & & ↓ & & & & ↓^{ζ_{−}} \\ x & ⟶ & ⋯_{η_{+}} & ⟶ & • & ⟵ & ⋯_{η_{−}} & ⟵ & y \end{matrix} ) = δ_{ζ_{+},ζ_{−}}C_{ξ,η},

where C_{ξ,η} ∈ C depends only on ξ = (ξ_{+}, ξ_{−}), η = (η_{+}, η_{−}). This form is mentioned in Remarks on page 153 of [O1]. Example. The field id^{(n)} is given by ∑_{s(ξ)=x,|ξ|=n}(ξ, ξ) at every point x. Using the biunitarity ∑ W (i, j)\overline{W (k, j)} = δ_{i,j}, we get

Transp_{x,y}^{(ξ_{+},ξ_{−})}id_{x}^{(n)} = δ_{ξ_{+},ξ_{−}}id_{y}^{(n)},

which shows flatness, because we have

∑_{ζ} W ( \begin{matrix}x & ⟶^{ξ_{+}} & y \\ ^{ζ}↓ & & ↓^{η_{+}} \\ • & & • \\ ^{ζ}↑ & & ↑^{η_{−}} \\ x & ⟶^{ξ_{−}} & y \end{matrix} ) = δ_{η_{+},η_{−}}δ_{ξ_{+},ξ_{−}}.

(The reason is as follows. Around the left middle •, we get, by biunitarity,

∑_{v,w} W ( \begin{matrix}· & ⟶ & · \\ ^{v}↓ & & ↓ \\ • & ⟶^{w} & · \\ ^{v}↑ & & ↑ \\ · & ⟶ & · \end{matrix}) = δ_{→↓,→↑},

where v, w are edges. Repeating this procedure, we get the above formula.) Example. Let e_{n,x} be the n-th Jones projection at x and e_{n} the corresponding field. This field is flat. By the same kind of computation as above, we get flatness of e_{n}. Note that the coefficients of the Jones projections come from the renormalization rule:

W (\begin{matrix}a & ⟶^{ξ_{1}} & b \\ η̃_{1}↑ & & ↑ξ̃_{2} \\ c & ⟶_{η_{2}} & d \end{matrix} ) = \sqrt{\frac{μ(a)μ(d)}{μ(b)μ(c)}} W (\begin{matrix}a & ⟶^{ξ_{1}} & b \\ ^{η_{1}}↓ & & ↓^{ξ_{2}} \\ c & ⟶_{η_{2}} & d \end{matrix} ) ,

for four edges ξ_{1}, ξ_{2}, η_{1}, η_{2}.

Remark. Because for k ≥ depth, A_{k+1,0} is generated by A_{k,0} and the Jones the projection, which is flat, we need to check the formula (∗) for flatness only for 2n × 2k-diagrams with k, n ≤ depth. Thus flatness for a given connection on finite graphs can be checked by finite times of computations. This corresponds to a remark on page 154 of [O1].

Starting from G, H, W, we get a subfactor N = A_{0,∞} ⊂ M = A_{1,∞}. From this subfactor, we get a principal graph K via the Galois functor. We will later show that this K is given in terms of flatness. If the algebra of flat fields are spanned by e_{n}, then the graph K is A_{∞}. This situation often happens in generic cases and means that the subfactor is non-amenable in the sense that spanning condition does not hold. (See [O1].)

The smallest known index value above 4 for an irreducible subfactor is 4.026 ⋯, and this is obtained for G = E_{10}, H = P (E_{10}), where P is a certain polynomial of degree 10. (See [HSO]).

Theorem. This subfactor with index= 4.026 ⋯ has A_{∞} as a principal graph, In other words, the higher relative commutants consist of only Jones projections. Hence this subfactor is non-amenable.

The proof will be given in IV.4.

Example. Consider the following subfactor.

N = \{( \begin{matrix}x & 0 & 0 \\ 0 & α(x) & 0 \\ 0 & 0 & β(x) \end{matrix} ) | x ∈ R\} ⊂ M = \{( \begin{matrix}a & b & c \\ d & e & f \\ g & h & i \end{matrix} ) | a, b, c, d, e, f, g, h, i ∈ R\} .

Here α, β gives an outer action of F_{2}, the free group of two generators. Then N′ ∩ M = C^{3}, and the higher relative commutants are given by the Cayley graph.

•

The norm of this graph is 1 + \sqrt{2}, and the eigenvalue is 3 > 1 + \sqrt{2}. Example. Consider the following subfactor.

N = \{( \begin{matrix}x & 0 \\ 0 & α(x) \end{matrix} ) | x ∈ R\} ⊂ M = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) | a, b, c, d ∈ R\} .

We have index = 4. If α^{n} = Ad(u) ∈ Int(R) for some n > 0, then the graph obtained by the Galois functor is four times Ã_{2n−1}. (See IV.1 for this notation.) We have a one-parameter family of biunitarity connection on this graph parametrized by λ. Flatness forces λ^{2n} = 1. If α(u) = γu in the above, then γ^{n} = 1. (Connes’ obstruction.) Here we have γ = λ^{2}. If we use a λ without λ^{2n} = 1 for construction, then we get a free α and the graph A_{∞} as an invariant.

In the basic construction,

M_{0} ⊂ M_{1} ⊂ End(_{M_{0}}M_{1}) ≅ M_{1} ⊗_{M_{0}} M_{1},

we get End(M_{1}) ≅ B(L^{2}(M_{1})) and End(_{M_{0}}M_{1}) ≅ M′_{0}∩B(L^{2}(M_{1})). (cf. Sauvageot.) In the above isomorphism, we get e_{0} ↔ 1 ⊗_{m_{0}} 1 and 1 ↔ ∑ λ_{i}^{∗} ⊗_{M_{0}} λ_{i}, where λ_{i}’s make a basis for _{M_{0}}M_{1}. We then have

M′_{0} ∩ M_{2} = M′_{0} ∩ End(_{M_{0}}M_{1}) = End(_{M_{0}}M_{1}_{M_{0}}).

In this way, we get a correspondence between the towers of relative commutants and endomorphisms of bimodules.

We have chosen a basis in End(_{M_{0}}M_{k}_{M_{0}}). This basis is String_{∗}^{(2k)}:

M′_{0} ∩ M_{n} ≅ String_{∗}^{(n)}(the principal graph),
M′_{1} ∩ M_{n+1} ≅ String_{∗}^{(n)}(the dual graph).

II.6. Computation of towers of relative commutants and flatness Consider a subfactor A_{0,∞} = M_{0} ⊂ A_{1,∞} = M_{1} constructed as the string algebra

on four graphs. Using the Jones projection on the vertical algebra, we can easily show that the inclusion

A_{0,∞} = M_{0} ⊂ A_{1,∞} = M_{1} ⊂ A_{2,∞} = M_{2}

is standard, that is, the third algebra is obtained as the basic construction from the first two. Similarly,

A_{0,∞} = M_{0} ⊂ A_{1,∞} = M_{1} ⊂ A_{2,∞} = M_{2} ⊂ A_{3,∞} = M_{3} ⋯

is the Jones tower. The problem is what M′_{0} ∩ M_{k} = A′_{0,∞} ∩ A_{k,∞} is. We show the following theorem.

Theorem. In the string algebra construction from finite graphs G_{1}, G_{2}, G_{3}, G_{4}, the relative commutant A′_{0,∞} ∩ A_{k,∞} is given by fields of flat k-strings on G_{1}^{(even)} [resp. G_{3}^{(odd)}] with respect to horizontal graph G_{2} · G̃_{2} [resp. G̃_{2} · G_{2}]. In other words, A′_{0,∞} ∩ A_{k,∞} = (A′_{0,n} ∩ A_{k,n}) ∩ (A′_{0,n+2} ∩ A_{k,n+2}) for all n ≥ depth.

Proof. Let z^{0} ∈ A′_{0,∞} ∩ A_{k,∞} and set z_{n} = E_{A_{k,n}}(z^{0}). Then z_{n} ∈ A′_{0,n} ∩ A_{k,n}.

• ⟶ • ⟶ ⋯ ⟶ • ⟶  •
↓
⋮
↓
•
↓
M_{k} • (k, n)

z_{n} } M′_{0}

Note that for 2n ≥ depth each z_{2n} is in the copy of k-string field algebra A on G_{1}^{(even)} at the 2n-th place. Define a map φ_{2n} to be the copying of strings from A′_{0,2n} ∩A_{k,2n} to this finite dimensional algebra A. We apply compactness argument to this finite dimensional algebra A as follows. By compactness, we may assume that there is a sequence {n_{j}} such that φ_{2n_{j}}(z_{2n_{j}}) → z, φ_{2n_{j}+2}(z_{2n_{j}+2}) → z′ for some z, z′ ∈ A as j → ∞. (Note that ∥z_{n}∥ is bounded.) Consider the following diagram.

• ⟶ {length 2 \atop •} ⟶ •
↓  ↓
A ⋮  ⋮ A
↓  ↓
• ⟶ {• \atop length 2} ⟶ •

That is, we have two A’s vertically, and two 2-string algebras horizontally. Be-

cause ∥z_{2n} − z_{2n+2}∥_{2} → 0 as n → ∞, we know that z · id^{(2)} = \begin{aligned}& • \\ z & ↓ \\ & • & ⟶_{id^{(2)}} & • \end{aligned} is

identified to id^{(2)} ·z′ = \begin{aligned}• & ⟶^{id^{(2)}} & • \\ ↓ & z′ \\ & & • \end{aligned} via connection W. We write z ·id^{(2)} = id^{(2)} ·z′

for this. We will show that z and z′ are the same as strings in A. Let e be the Jones projection in the upper horizontal string algebra in the above diagram. By flatness of the Jones projection, this e has the same form as the original e in the lower horizontal string algebras after identification via W, that is, e·id_{(k)} = id^{(k)} ·e, where id^{(k)} denotes the identity field of k-strings in A. On the other hand, we can show easily z′ · e = e · z′ under W by induction on k. (The computation for this is same as that for flatness of the Jones projection.) This implies

(z · id^{(2)}) × (id^{(k)} · e) = (id^{(2)} · z′) × (e · id^{(k)}) = e · z′ = z′ · e = (z′ · id^{(2)}) × (id^{(k)} · e),

here “×” means multiplication in the string algebra and “·” means concatenation of strings. Taking a conditional expectation to the left vertical algebra A in the above diagram, we get z = z′ in A. This means that for any j we get

\begin{aligned}& • \\ z & ↓ \\ & • & ⟶^{id^{(2j)}} & • \end{aligned} = \begin{aligned}• & ⟶_{id^{(2j)}} & • \\ ↓ & z \\ & & • \end{aligned}

under identification via W, where id^{(2j)} means the identity in the horizontal algebra for G_{2} · G̃_{2} ⋯ G̃_{2}. Let (f_{x}) be a field of strings on G_{1}^{(even)} corresponding to z. We will show this field is flat with respect to the horizontal graph G_{2} · G̃_{2}. Let x, y be vertices on G_{1}^{(even)}. We have shown that z ·id^{(2j)} = id^{(2j)} ·z under W, which implies

∑_{\begin{matrix}s(ξ_{+})=s(ξ_{−})=x \\ r(ξ_{+})=r(ξ_{−})=y \\ |ξ_{+}|=|ξ_{−}|=2j \end{matrix}}(ξ_{+}, ξ_{−}) · Transp_{x,y}^{(ξ_{+},ξ_{−})}(f_{x}) = ∑_{\begin{matrix}s(ξ)=x,r(ξ)=y \\ |ξ|=2j \end{matrix}}(ξ, ξ) · f_{y}.

Thus for two paths ξ_{+}, ξ_{−} on G_{2} with even length 2j from x to y, we get Transp_{x,y}^{(ξ_{+},ξ_{−})}(f_{x}) = δ_{ξ_{+},ξ_{−}}f_{y}, which shows flatness of the field (f_{x}) by definition. We denote by z ∈ A′_{0,2n}∩A_{k,2n} the elements denoted by this flat field. It is clear that z^{0} = lim_{j} z_{2n_{j}} = z.

Conversely, if we have a flat field z = (f_{x}) of k-strings on G_{1}^{(even)} with respect to horizontal graph G_{2} · G̃_{2}, this can be embedded in A′_{0,2n} ∩ A_{k,2n} for all sufficiently large n with the same form of strings. We denote this by z. It is trivial that z ∈ A′_{0,∞} ∩ A_{k,∞}.

The statement for G_{3}^{odd} is proved similarly. Q.E.D.

Note that the above theorem contains so-called Wenzl’s lemma as a particular case. Indeed, suppose G_{1}^{(even)} has a vertex x_{0} which has only one edge from x_{0}.

Because the restriction of the fields:(f_{x})_{x} ↦ f_{x_{0}} is an algebra isomorphism by flatness, we get A′_{0,∞} ∩ A_{1,∞} ⊂ String_{x_{0}}^{(1)}G_{1} = C.

The theorem says that if a connection is flat, we get the same graph back via the Galois functor, which is the Range Theorem in [O1].

Here we show another method of a proof based on amenability of the horizontal graph in the sense that E_{(id^{(n−1)}·Flat_{m})′}(ẽ_{n−1}) converges to a scalar τ̃ in L^{2}-norm as m → ∞ for all n, where Flat_{m} denote the field algebra of flat m-stings on the horizontal algebra with respect to vertical transport, ẽ_{n} is the Jones projection on the horizontal string algebra and τ̃ = β_{G_{2}}^{−2}.

Suppose z ∈ A′_{0,∞} ∩A_{k,∞} and set z_{n} = E_{A_{k,n}}(z) again. Now for n ≥ depth, z_{n} ∈ A_{k,n} can be written as ∑_{finite} a_{i}ẽ_{n−1}b_{i}, where a_{i}, b_{i} ∈ A_{k,n−1}, ẽ_{n−1} ∈ A′_{k,n−2} ∩A_{k,n} as well as ẽ_{n−1} ∈ A′_{0,n−2} ∩ A_{0,n} (by flatness of the Jones projection). Note that z_{n−1} = E_{A_{k,n−1}}(z_{n}) = τ̃ ∑ a_{i}b_{i}. If x ∈ id^{(n−1)} · Flat_{m}, then x can be regarded as an element in A′_{k,n−1} ∩ A_{k,n−1+m} by flatness, hence we get a_{i}x = xa_{i}, that is, a_{i}, b_{i} ∈ (id^{(n−1)} · Flat_{m})′. Now we get by amenability that

∥z − z_{n}∥_{2} ≥ ∥E_{(id^{(n−1)}·Flat_{m})′}(z − z_{n})∥_{2}
= ∥z − E_{(id^{(n−1)}·Flat_{m})′}(z_{n})∥_{2}
= ∥z − ∑ a_{i}E_{(id^{(n−1)}·Flat_{m})′}(e_{n−1})b_{i}∥_{2}
→ ∥z − τ̃ ∑ a_{i}b_{i}∥_{2}, as m → ∞
= ∥z − z_{n−1}∥_{2}.

This implies ∥z − z_{n}∥_{2} ≥ ∥z − z_{n−1}∥_{2} ≥ 0. Because the sequence {∥z − z_{n}∥_{2}}_{n} is an increasing sequence of positive numbers converging to 0, all the terms must be 0. Then we get the conclusion as in the proof of the theorem above.

We get the amenability for a finite graph is obtained by Perron-Frobenius theory, because we can write down an explicit formula for E_{(id^{(n−1)}·Flat_{m})′}(ẽ_{n−1}) as in [O2]. The last part of [J] shows the lack of amenability for A_{∞}. This amenability is related to random walk and path ergodicity. (See [O1, O2].)

II.7. Fourier transform and convolutions

Flatness is an algebraic property. Flat fields form an algebra; if f, g are flat, then the product fg is also flat. If f ∈ Flat^{(n)}, then E_{Flat}_{(n−1)}(f) ∈ Flat^{(n−1)}. Concatenations of flat fields are also flat; if f, g are flat, then the concatenation f · g is also flat. Here the concatenation is defined by (ρ_{+}, ρ_{−}) · (η_{+}, η_{−}) = (ρ_{+} · η_{+}, ρ_{−} ·η_{−}) and extended linearly. The embedding of String^{(n)} into String^{(n+k)} is a concatenation by id^{(k)}. The map f ↦ id^{(1)} · f − f · id^{(1)} is similar to Lie derivation.

For group case, we get String_{∗}^{(2)} = ℓ^{∞}(G). In this case, rotation with 90^{∘} on this algebra gives the Fourier transform. We remark that rotation of a flat field is also a flat field, but we need a number (μ(s(ρ))/μ(r(ρ)))^{1/2} to get a flat field. This is an analogue of reversing the orientation: dx_{i} → \frac{d}{dx_{i}}.

•
•
→
•
•

The identity id^{(2)} is given by ∑_{|ξ|=|η|=1}(ξ · η, ξ · η), thus the rotation is given by the formula ∑(coefficient)(ξ · ξ̃, η̃ · η). This is β^{−1} times the Jones projection.

Thus, the Jones projection is equal to the Fourier transform of id^{(2)} up to a scalar. For the group case, L^{∞}(G) is spanned by f_{g}, a projection corresponding to g ∈ G. In this case, the Jones projection is f_{1}. Now the left regular representation algebra L(G) = C[G] is generated by λ_{g}, a translation corresponding to g ∈ G. Then the Jones projection is |G|^{−1} ∑_{g} λ_{g}.

For the tower M_{0} ⊂ M_{1} ⊂ M_{2} ⊂ M_{3} and x ∈ M′_{1} ∩ M_{3}, the Fourier transform F(x) is given by F(x) = E_{M_{2}}(xe_{0}e_{1}) ∈ M′_{0} ∩ M_{2}. Here E_{M_{2}} is an analogue to the integral, and e_{0}e_{1} is an analogue to the coefficient for the integration. Example. If M_{1} = M_{0}⋊G, the crossed product by an outer action, then M′_{0}∩M_{2} = L^{∞}(G), M′_{1} ∩ M_{3} = L(G), M′_{0} ∩ M_{3} = B(L^{2}(G)).

Suppose f = (f_{+}^{1} · f_{+}^{2}, f_{−}^{1} · f_{−}^{2}), g = (g_{+}^{1} · g_{+}^{2}, g_{−}^{1} · g_{−}^{2}) ∈ String^{(2n)} with |f_{+}^{1}| = |f_{+}^{2}| = |f_{−}^{1}| = |f_{−}^{2}| = |g_{+}^{1}| = |g_{+}^{2}| = |g_{−}^{1}| = |g_{−}^{2}| = n. Then the operation (f, g) → δ_{f_{+}^{2},g̃_{+}^{1}}δ_{f_{−}^{2},g̃_{−}^{1}}(f_{+}^{1} · g_{+}^{2}, f_{−}^{1} · g_{−}^{2}) with a certain coefficient defines the convolution. The convolution of flat fields is again flat. The convolution in L^{∞}(G) is given by f_{g}∗f_{h} = f_{gh}, that is, the Fourier transform converts the convolution into the multiplication, as expected.

Let x, y ∈ M′_{0} ∩ M_{2} = End(_{N}M_{N}) = N′ ∩ (M ⊗_{N} M). What is x ∗ y ? Note that the product of a ⊗_{N} b and c ⊗_{N} d is given by aE_{N}(bc) ⊗_{N} d as compositions of endomorphisms. The convolution is given by (a ⊗_{N} b) ∗ (c ⊗_{N} d) = ca ⊗_{N} bd. Though (a⊗_{N} b)^{∗} = b^{∗} ⊗_{N} a^{∗}, another involutions is given by (a⊗_{N} b)^{∘} = a^{∗} ⊗_{N} b^{∗}. In case M′_{0} ∩ M_{3} is a factor, these give a Kac algebra structure. (See Herman-Ocneanu for discrete Kac algebra and Herman-Nest-Ocneanu for type III cases.) The Haar weight on M′_{0} ∩M_{2} is given by restriction of a weight giving a conditional expectation. The algebras M′_{0} ∩ M_{2} and M′_{1} ∩ M_{3} are dual to each other.

For a subfactor M_{0} ⊂ M_{1}, there is a problem whether there is any intermediate subfactor P, M_{0} ⊂ P ⊂ M_{1}. A subfactor is “simple” if there is no intermediate subfactor, as an analogue of a simple group.

Theorem. There is a one-to-one correspondence between intermediate subfactors M_{0} ⊂ P ⊂ M_{1} and non-zero projections p ∈ Proj(M′_{0} ∩ M_{2}) such that F(p) is a scalar multiple of a projection.

The trivial intermediate subfactors M_{0} ⊂ M_{0} ⊂ M_{1} and M_{0} ⊂ M_{1} ⊂ M_{1} correspond to the Jones projection and 1, respectively.

Choosing a Jones projection e_{−1} ∈ M_{0} for the downward basic construction, we get F(p) = E_{M_{0}}(pe_{−1}e_{0}). For example, if p = e_{0}, then E_{M_{0}}(e_{0}e_{−1}e_{0}) = τE_{M_{0}}(e_{0}) = τ^{2}, and if p = 1, then E_{M_{0}}(e_{−1}e_{0}) = τe_{−1}.

Example. Set M = M_{0} ⊂ M ⊗ M_{k}(C) = M_{1}. Then M_{2} = M ⊗ M_{k}(C) ⊗ M_{k}(C) and M′_{0} ∩ M_{2} = M_{k}(C) ⊗ M_{k}(C). The explicit formulas for the product and the convolution in this situation are left as an exercise. Writing H_{k} = C^{k}, we can write M′_{0} ∩ M_{2} = H_{k} ⊗ H̄_{k} ⊗ H_{k} ⊗ H̄_{k}. Then the Fourier transform is a rotation of these four factors. The Fourier transform maps the identity ∑ ξ_{i} ⊗ ξ̄_{i} ⊗ ξ_{j} ⊗ ξ̄_{j} to ∑ ξ̄_{i} ⊗ ξ_{j} ⊗ ξ̄_{j} ⊗ ξ_{i} = ∑ e_{ij} ⊗ e_{ji}, which is the Jones projection up to a scalar. In this case, intermediate subfactors are of the form M ⊂ M ⊗ Mat_{l} ⊂ M ⊗ Mat_{k}, k = lm. A projection in Mat_{k} ⊗ Mat_{k} have the property that F(p) is a projection up to a scalar exactly when p = P_{Mat_{l}}. In this case, F(p) = P_{Mat}′_{l} = P_{Mat_{m}}.

II.8. Knot invariant The braid group B_{n} has generators g_{1},…, g_{n−1} and the relations

g_{i}g_{i+1}g_{i} = g_{i+1}g_{i}g_{i+1},
g_{i}g_{j} = g_{j}g_{i},  |i − j| ≥ 2.

The group B_{n} is a semidirect product of free groups and S_{n} as follows. Let P_{n} be the set of the braids which do not permute endpoints. (These are called pure braids.) Then P_{n} = F_{n−1} ⋉ F_{n−2} ⋉ ⋯, and we have an exact sequence

0 → P_{n} → B_{n} → S_{n} → 0.

We get a link by taking a closure of a braid. By Alexandar’s theorem, this procedure is surjective. Markov’s theorem asserts that closures of two braids give the same link if and only if the two braids are equivalent under the equivalence relation generated by the following two Markov moves.

I. v · w ∼ w · v,
II. v · g_{n}^{±1} ∼ v,  v ∈ B_{n}.

We choose complex numbers A, B so that the map π : g_{i} ↦ Ae_{i} + B gives a homomorphism from ⟨g_{i}⟩_{i} to ⟨1, e_{1}, e_{2},…⟩.

Then the trace tr on String^{(n)}, suitably normalized, yields a knot invariant. Markov move I corresponds to the trace property, and Markov move II to conditional expectation property.

III. Central sequences and asymptotic inclusions.

III.1. Asymptotic inclusions

Let

M_{0} ⊂ M_{1} ⊂ M_{2} ⊂ ⋯ ⊂ M_{∞} = \overline{⋃ M_{i}}^{w}.

We call the subfactor M_{0} ∨ (M′_{0} ∩ M_{∞}) ⊂ M_{∞} asymptotic inclusion. Theorem. The principal graph for the asymptotic inclusion (for a subfactor with finite depth) is the connected component of ∗ = _{M}M_{M} in the following.

(_{M}(X_{1})_{M},_{M}(X_{2})_{M})_{•}

↓

MX_{M}^{•}

Here X_{1}, X_{2}, X are M-M bimodules and the number of arrows is a multiplicity of X in X_{1} ⊗_{M} X_{2}. Example. Consider the following graph A_{4}.

A_{4} : 1\φ/φ\1

Here β = φ = 1.618 ⋯ with 1 + φ = φ^{2}. In this case, we have two kinds of M-M bimodules M and P with the multiplication rules: M ⊗ M = M, M ⊗ P ≅ P ⊗ M ≅ P, P ⊗ P ≅ M ⊗ P. Thus the graph for the asymptotic inclusion is as follows.

(M, M)_{∗}  (M, P)_{·}  (P, M)_{·}  (P, P)_{·}
M^{·}  P^{·}

This graph is D_{6}.

Using e_{−1} ∈ M′_{−2} ∩ M_{0} and e_{−1}M_{0}e_{−1} ≅ e_{−1}M_{−2}, M_{−2} is identified with id^{(2)} ·Flat. Strings in M_{∞} are strings of arbitrary length in both upward and downward direction. The trace tr in M_{∞} is given by tr(ρ) = δ_{ρ_{+},ρ_{−}}β^{−|ρ|}μ(r(ρ))μ(s(ρ)). Multiplication in M_{∞} is done by transport and the usual multiplication. We get

[M_{∞} : M_{0} ∨ (M′_{0} ∩ M_{∞})] = ∑_{X:irreducible}(dim_{N}X_{N})^{2}.

Example. In the above example of A_{4}, we get [M_{∞}, M_{0} ∨ (M′_{0} ∩ M_{∞})] = 1 + φ^{2} = 4 cos^{2}(π/10). (Note that the Coxeter number for D_{6} is 10.)

Setting A_{k,l} = M′_{k} ∩ M_{l}, we get a commuting square

A_{−n,0} ∨ A_{0,n} ⊂ A_{−n,n}
∩  ∩
A_{−m,0} ∨ A_{0,m} ⊂ A_{−m,m},

where 0 ≤ n ≤ m. This commuting square approximates M_{0} ∨ (M′_{0} ∩ M_{∞}) ⊂ M_{∞}.

III.2. Central sequences

Let N ⊂ M be a subfactor of the hyperfinite II_{1} factor with finite index and finite depth. Let ω ∈ βN \ N be a free ultrafilter over N. Set

M^{ω} = {(x_{n})_{n} | x_{n} ∈ M, sup ∥x_{n}∥ < ∞}/ ∼,
M_{ω} = M′ ∩ M^{ω} = {(x_{n})_{n} | [x_{n}, y] → 0 for all y ∈ M}/ ∼,

where ∼ denotes the equivalence relation meaning the termwise difference goes to zero when n → ω. We get the inclusion N_{ω} ∩ M_{ω} = N_{ω} ∩ M′ ⊂ M_{ω}.

Example. Here G denotes a finite group. If M = N ⋊ G, then N_{ω} ∩ M^{ω} = (N_{ω})^{G} and the conditional expectation from M_{ω} onto N_{ω} ∩ M_{ω} is given by (x_{n})_{n} ↦ (E_{N}(x_{n}))_{n}.

If N = M^{G} for an outer action of G, then N_{ω} ∩M_{ω} = (M_{ω})^{G} and the conditional expectation from M_{ω} onto N_{ω} ∩ M_{ω} is again given by (x_{n})_{n} ↦ (E_{N}(x_{n}))_{n}, where E_{N}(X) = E_{M^{G}}(x) = |G|^{−1} ∑_{g} α_{g}(x).

Choose a tunnel:

⋯ ⊂ M_{−n} ⊂ ⋯ ⊂ M_{0} ⊂ M_{1} ⊂ ⋯ M_{∞}.

The Jones projection e_{k} satisfies e_{k} ∈ M′_{k−1} ∩ M_{k+1}. Because (M_{0})_{ω} ∩ (M_{−1})_{ω} = M′_{0} ∩ M_{−1}^{ω} and e_{−1} ∈ M_{0}, we get M′_{0} ∩ (M_{−1})^{ω} ⊂ M′_{0} ∩ (M_{−2})^{ω}. Similarly, we get M′_{0}∩(M_{−1})^{ω} ⊂ M′_{0}∩(M_{−3})^{ω} and so on. Thus we get (M_{0})_{ω}∩(M_{−1})_{ω} = ⋂_{k}(M_{−k})^{ω}. We also need the following theorem.

Central Freedom Lemma. Let M be a type II_{1} von Neumann algebra and P ⊂ N ⊂ M. If P is hyperfinite and P ∩ P′ ⊂ N′ ∩ N, then

(P′ ∩ N^{ω})′ ∩ M^{ω} = P ∨ (N′ ∩ M)^{ω}.

Note that the inclusion “⊃” in the above is trivial. By these, we can show the following. Theorem. A subfactor (M_{0})_{ω} ∩ (M_{−1})_{ω} ⊂ (M_{0})_{ω} has the same Galois invariant as the asymptotic inclusion M_{0} ∨ (M′_{0} ∩ M_{∞}) ⊂ M_{∞}.

III.3. Group case

We work on crossed product algebras by finite group actions in detail.

Let M = N ⋊ G. For simplicity, assume G = S_{3}. Then the principal graph for the central sequence algebras are obtained by (σ_{1}, σ_{2}) ∈ Irr(G × G) as follows.

·^{(1,1)}  ·^{(ε,ε)}  ·^{(σ,σ)}  ·  ·  ·  ·  ·  ·
·_{1}  ·_{ε}  ·_{σ}

For a subfactor M^{G} ⊂ M, the central sequence algebra has the same invariant as the original subfactor, but not for N ⊂ N ⋊ G. For N = R ⋊ H ⊂ R ⋊ G = M, with H a subgroup of G, we get a principal graph from Irr(G) and Irr(H). Thus, if M = N ⋊ G, then the subfactor N_{ω} ∩ M_{ω} ⊂ M_{ω} has the same invariant as P ⋊ G ⊂ P ⋊ (G × G), where P is some factor and G sits in G × G diagonally.

Note that we have a tower

M = N ⋊ G = ⟨N, λ_{g}⟩ ⊂ M_{1} = ⟨M, f_{g}⟩ ⊂ M_{2} = ⟨M_{1}, ρ_{g}⟩.

Here the action of λ_{g}, ρ_{g} is given by Ad(λ_{g})(f_{h}) = f_{gh} and Ad(ρ_{g})(f_{h}) = f_{hg^{−1}}. For the extension

N_{ω} ∩ M_{ω} ⊂ M_{ω} ⊂ ⟨M_{ω}, f_{1}⟩,

we get ⟨M_{ω}, f_{1}⟩ ⊂ (N′∩M_{1}^{ω})^{Ad(λ_{g}·ρ_{g})} because [M_{ω}, λ_{g}] = 0 (λ_{g} ∈ M), [M_{ω}, ρ_{g}] = 0, and Ad(λ_{g} · ρ_{g})(f_{1}) = f_{g1g^{−1}} = f_{1}. The central freedom lemma implies the action Ad(λ_{g} · ρ_{g}) is outer, and we get ⟨M_{ω}, f_{1}⟩ = (N′ ∩ M_{1}^{ω})^{Ad(λ_{g}·ρ_{g})} because these two have the same index in N′ ∩ M_{1}^{ω}. Note that G × G acts by Ad(λ_{g} · ρ_{h}). Then

(N′ ∩ M_{1}^{ω})^{Ad(λ_{g}·ρ_{h})} = (N′ ∩ M^{ω})^{Ad(λ_{g})} = M′ ∩ M^{ω} = M_{ω}.

Setting P = N′ ∩ M_{1}^{ω}, we get an outer action Ad(λ_{g} · ρ_{h}) on P. We have a tower

N_{ω} ∩ M_{ω} ⊂ M_{ω} = P^{G×G} ⊂ ⟨M_{ω}, f_{1}⟩ = P^{G},

where G sits in G × G diagonally.

Furthermore, G acts on N_{ω} outerly, so there is a partition of unity r_{g} ∈ N_{ω}, ∑_{g} r_{g} = 1, α_{g}(r_{h}) = r_{gh} by non-commutative Rohlin’s lemma. We set p_{g,h} = r_{g} · f_{gh}, then this is a partition of unity and, we get

Ad(λ_{k} · ρ_{l})(p_{g,h}) = r_{kg}f_{kghl^{−1}} = p_{kg,hl^{−1}}.

We can find a family of unitaries λ̄_{g,h} ∈ (N′∩M_{1}^{ω})^{G×G} = M_{ω} such that Ad(λ̄_{g,h})(p_{k,l}) = p_{kg^{−1},hl} and λ̄_{g,h}λ̄_{k,l} = λ̄_{gk,hl}. Set R = {p_{1,1}}′ ∩ M_{ω}. We see R ⊂ N_{ω}. Indeed, if x ∈ M_{ω} satisfies [x, r_{1}f_{1}] = 0, then [x, f_{1}] = [x, ∑_{g} Ad(λ_{g} · ρ_{g})(f_{1}r_{1})] = 0 by Ad(λ_{g} · ρ_{g})(x) = x, hence x ∈ N_{ω} ∩ M_{ω}. Now M_{ω} is expressed as ⟨R, λ̄_{g,h}⟩, the crossed product R ⋊ (G × G). Because

Ad(λ̄_{g,g})(f_{1}) = Ad(λ̄_{g,g})(∑_{k} p_{k,k^{−1}}) = ∑_{k} p_{kg^{−1},gk^{−1}} = f_{1},

we get λ̄_{g,g} ∈ {f_{1}}′ ∩ M_{ω} = N_{ω} ∩ M_{ω}. Hence we get

⟨R, λ̄_{g,g}⟩ ⊂ N_{ω} ∩ M_{ω} ⊂ ⟨R, λ̄_{g,h}⟩ = M_{ω}.

But by the equality

[⟨R, λ̄_{g,h}⟩ : M_{ω} ∩ N_{ω}] = [⟨R, λ̄_{g,h}⟩ : ⟨R, λ̄_{g,g}⟩] = |G|,

we get

⟨R, λ̄_{g,g}⟩ = N_{ω} ∩ M_{ω}.

Hence the central sequence algebras are of the form R ⋊ G ⊂ R ⋊ (G × G), where G sits in G × G diagonally.

IV. Computation of paragroups of small order IV.1. Main result for index< 4

By Perron-Frobenius theory, since the matrix Δ has non-negative entries, Δ has a unique non-negative eigenvector. This eigenvalue is equal to the operator norm of Δ and also equal to [M : N]^{1/2}, hence we get [M : N] = ∥Δ∥^{2}. If [M : N] < 4, then ∥Δ∥ < 2, and by a result of Kronecker, all the self-adjoint irreducible matrices Δ which have non-negative integer entries and ∥Δ∥ < 2 are known. (Notice that if G, H are graphs, G ⊊ H, and G is finite, then ∥Δ_{G}∥ < ∥Δ_{H}∥, since ∥Δ_{G}∥ is an eigenvalue of μ_{G}.)

Here we have List 1 of graphs with eigenvectors for the eigenvalue 2.

Ã_{n} :  1/1 \1  Δμ = 2μ,

D̃_{n} :  1\ 1/ 2—2⋯ 2—2/1 \1  Δμ = 2μ,

Ẽ_{6} :  1—2—1 ; 2 ; 3^{|}^{|}—2—1  Δμ = 2μ,

Ẽ_{7} :  1—2—3—2 ; 4^{|}—3—2—1  Δμ = 2μ,

Ẽ_{8} :  1—2—3—4—5—3 ; 6^{|}—4—2  Δμ = 2μ,

A_{∞} :  1—2—3—4—5 ⋯  Δμ = 2μ,

D_{∞} :  1\ 1/ 2—2—2 ⋯  Δμ = 2μ,

A_{−∞,∞} :  ⋯ 1—1—1—1—1 ⋯  Δμ = 2μ,

Theorem. The only graphs which do not contain a graph in List 1 are finite and are in List 0.

A_{n} : ·— · — ⋯ —·,

D_{n} : ·\ ·/ ·—·—⋯ —·,

E_{6} : ·— · — · ; ·^{|} — · —·,

E_{7} : ·— · — · ; ·^{|} — · — · —·,

E_{8} : ·— · — · ; ·^{|} — · — · — · —·,

Proof. Let G be a graph which does not contain any in List 1. Because it does not contain A_{−∞,∞}, it is finite. Because it does not contain Ã_{n}, there are no cycles. There are no 4-points and at most one 3-point because it does not contain D̃_{n}. If there is a 3-point, the three legs from it are not too long because G does not contain Ẽ_{6}, Ẽ_{7}, Ẽ_{8}. Q.E.D.

Thus if [M : N] < 4, then [M : N]^{1/2} is equal to an eigenvalue of one in List 0, which is 2 cos \frac{π}{(Coxeter number)}. Here the Coxeter number is defined to be n + 1 for A_{n}, 2n − 2 for D_{n}, 12, 18, 30 for E_{6}, E_{7}, E_{8}, respectively. For A_{n} having vertices v_{1}, v_{2},…, v_{n}, the vector μ defined by μ(v_{k}) = sin k \frac{π}{n + 1} has an eigenvalue 2 cos \frac{π}{n + 1} because

sin \frac{(k − 1)π}{n + 1} + sin \frac{(k + 1)π}{n + 1} = 2 cos \frac{π}{n + 1} sin \frac{kπ}{n + 1}.

(This suggests a relation between a minimal model in conformal field theory (CFT) and a paragroup, which is the name for the invariant appearing from the Galois functor.)

The only graphs which do not contain strictly a graph in List 1 are those in List 1 and List 0. This means that the other graphs have norm bigger than 2.

We have seen the Galois functor assigning a graph to a subfactor N ⊂ M. If the graph is infinite, then the Perron-Frobenius eigenvector is actually a part of the obtained data, since it is not unique.

The problem here is finding all the possible invariants for subfactor with index<

Note that the Perron-Frobenius eigenvalue of the adjacency matrix of the graph must be less than 2, hence each graph is one of Dynkin diagrams of type A, D, E. Any paragroup (in duality form) has a distinguished initial point ∗. First we have

End(_{M}M_{M}) ≅ Z(M) = C,
End(_{N}M_{M}) ≅ N′ ∩ End(M_{M}) ≅ N′ ∩ M = C.

We have a distinguished homomorphism in Hom(_{M}M_{M},_{N}M ⊗_{M} M_{M}). This is like an axiom for a unit. For the biunitarity axiom, the number of paths for making rows and columns are equal, and the Perron-Frobenius eigenvectors must match at the common vertices. This implies that the dual graph is equal to the principal graph in each case. (In general cases index≥ 4, this does not hold any more as shown by the following example S_{5}/S_{4}.)

·^{1}  ·^{4}  ·^{5}  ·^{6}  ·^{5}  ·^{4}  ·^{1}
·^{\sqrt{5}}  ·^{3\sqrt{5}}  ·^{2\sqrt{5}}  ·^{3\sqrt{5}}  ·^{\sqrt{5}}
·_{1}  ·_{4}  ·_{3}  ·_{8}  ·_{2}  ·_{3}  ·_{4}  ·_{1}

The paragroups with [M : N] < 4 are of the following form:

⋯
Ĝ Ĝ
⋯  ⋯
G  G
⋯

where G and Ĝ are the same as a graph, and one of Dynkin diagrams of type A, D, E. For each A_{n}, there is only one connection on it. For D_{n}, there are two connections, but there is only one up to a graph isomorphism (of switching two endpoints next to the triple point). For E_{6}, E_{7}, E_{8}, there are two connection. But (geometrical) flatness of the connection is not satisfied by the connections on D_{odd}, E_{7}. Then the conclusion is that there is one subfactor for A_{n}, D_{2n}, and two subfactors for E_{6}, E_{8}, which are opposite conjugate but not conjugate to each other. Notice similarity to CFT. Also note that End(_{N}M_{M}) ≅ N′ ∩ M and N′ ∩ M = C is equivalent to that the vertex ∗ has the only one neighbour.

IV.2. Concrete computations of connections

We determine the connection in each case. (cf. IRF model.) Recall that we have a gauge for the connection coming from the choice of edges.

We now work on an example A_{5}. We label each vertex as follows.

1 •  2 •  3 •
4 •  5 •  9 •  10•
• 6  • 7  • 8

Fixing vertices 1,6, we get the following connection values by the rule of renormalization.

W ( \begin{matrix}4 & ↙ & 1 & ↘ & 9 \\ & ↘ & 6 & ↙ \end{matrix} ) = 1, W ( \begin{matrix}4 & ↗ & 1 & ↘ & 9 \\ ↘ & 6 & ↗ \end{matrix} ) = \sqrt{\frac{1}{\sqrt{3}}} · \sqrt{\frac{1}{\sqrt{3}}} · 1 = \frac{1}{\sqrt{3}}.

Then fix vertices 1,7 next. By changing the choice of (4, 7), we may assume that the connection for the left below is 1.

W ( \begin{matrix}4 & ↙ & 1 & ↘ & 9 \\ & ↘ & 7 & ↙ \end{matrix} ) = 1, W ( \begin{matrix}4 & ↗ & 1 & ↘ & 9 \\ ↘ & 7 & ↗ \end{matrix} ) = \begin{matrix}√ & \overline{2} \\ √ & \overline{3} \end{matrix}.

Fixing vertices 2,6, we get the following similarly.

W ( \begin{matrix}4 & ↙ & 2 & ↘ & 9 \\ & ↘ & 6 & ↙ \end{matrix} ) = 1, W ( \begin{matrix}4 & ↗ & 2 & ↘ & 9 \\ ↘ & 6 & ↗ \end{matrix} ) = \begin{matrix}√ & \overline{2} \\ √ & \overline{3} \end{matrix}.

Fixing vertices 4,9, we get a 2 × 2-matrix. By unitarity of ( \begin{matrix}\frac{1}{\sqrt{3}} & \begin{matrix}√ & \overline{2} \\ √ & \overline{3} \end{matrix} \\ \begin{matrix}√ & \overline{2} \\ √ & \overline{3} \end{matrix} & ∗ \end{matrix} ), we get

W ( \begin{matrix}4 & ↗ & 2 & ↘ & 9 \\ ↘ & 7 & ↗ \end{matrix} ) = −\frac{1}{\sqrt{3}} and hence W ( \begin{matrix}4 & ↙ & 2 & ↘ & 9 \\ & ↘ & 7 & ↙ \end{matrix} ) = −\frac{1}{2}.

Fixing 5,9, and using a gauge for (5, 2), we get

W ( \begin{matrix}5 & ↗ & 2 & ↘ & 9 \\ ↘ & 7 & ↗ \end{matrix} ) = 1, W ( \begin{matrix}5 & ↙ & 2 & ↘ & 9 \\ & ↘ & 7 & ↙ \end{matrix} ) = \frac{\sqrt{3}}{2}.

Fixing 4,10, and using a gauge for (7, 10), we get

W ( \begin{matrix}4 & ↗ & 2 & ↘ & 10 \\ ↘ & 7 & ↗ \end{matrix} ) = 1, W ( \begin{matrix}4 & ↙ & 2 & ↘ & 10 \\ & ↘ & 7 & ↙ \end{matrix} ) = \frac{\sqrt{3}}{2}.

Fixing 7,2, we get a unitary 2×2-matrix and W ( \begin{matrix}5 & ↙ & 2 & ↘ & 10 \\ & ↘ & 7 & ↙ \end{matrix} ) = \frac{1}{2} similarly.

This computation works for all A_{n}, and we get a unique connection, up to gauge choice, on each A_{n}. The Perron-Frobenius eigenvector is given by k ↦ s(k)/s(1), where a function s(k) is defined by s(k) = sin \frac{kπ}{n + 1}.

If one meets a 3-point, as in D_{n}, E_{n}, then the previous argument works up to the 3-point. At the 3-point, one has a 3 × 3-matrix (a_{ij}). The values of |a_{ij}| are completely determined. (See IV.4.) For each row and column, we have a freedom of gauge. We can, for example, make the first row and the first column real.

( \begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & x & y \\ a_{31} & z & t \end{matrix} ) , a_{11}, a_{12}, a_{13}, a_{21}, a_{31} ∈ R_{+}.

We need to choose x, y. (Then z, t are determined by unitarity.) Because |x|, |y| are given, and we have an equation −xa_{12} − ya_{13} = a_{21}a_{11}, if we call v = −a_{12}x ∈ C, w = −a_{13}y ∈ C, c = a_{21}a_{11} ∈ R_{+}, then our equation is v + w = c, with |v|, |w| given.

|w| |v|
· c

The above picture shows that this problem has at most two solutions (conjugate to each other). We have three cases; There are no solutions; There is one (self-conjugate) solution; There are two solutions (conjugate to each other). The computation is not difficult for D_{n}. We shall display a solution for each A_{n}, D_{n}, E_{n}. Note that if we find two conjugate solutions, then they are the only solution because we have at most one 3-point. (This was noted independently by Pasquier.)

Use an identification of the four graphs. (The identification is not canonical, but we choose one. Consider Z_{3} ≅ Ẑ_{3} for example.) For a vertex v, call μ(v) the Perron-Frobenius eigenvector. Let N be the Coxeter number of the graph and β the Perron-Frobenius eigenvalue 2 cos π/N. We set ε = i exp(πi/2N). Our formula for W is

W ( \begin{matrix}k & ↙ & i & ↘ & l \\ & ↘ & j & ↙ \end{matrix} ) = δ_{kl}ε + ρδ_{ij}ε̄, ρ = (\frac{μ(k)μ(l)}{μ(i)μ(j)})^{1/2} .

Note that this satisfies the renormailization rule. We check unitarity as follows.

∑_{l} W ( \begin{matrix}k & ↙ & i & ↘ & l \\ & ↘ & j & ↙ \end{matrix} ) \overline{W ( \begin{matrix}m & ↙ & i & ↘ & l \\ & ↘ & j & ↙ \end{matrix} )} =?.

We consider two cases.

Case 1. (i ≠ j.) We get ? = ∑_{l} δ_{kl}εδ_{ml}ε̄ = δ_{km}.

Case 2. (i = j.) We get

? = ∑_{l} δ_{kl}εδ_{ml}ε̄+ ∑_{l} δ_{kl}ε^{2} \frac{μ(l)^{1/2}μ(m)^{1/2}}{μ(i)}

+ ∑_{l} \frac{μ(k)^{1/2}μ(l)^{1/2}}{μ(i)} δ_{ml}ε̄^{2} + ∑_{l} \frac{μ(k)^{1/2}μ(m)^{1/2}μ(l)}{μ(i)^{2}}

= δ_{km} + \frac{μ(k)^{1/2}μ(m)^{1/2}}{μ(i)} (ε^{2} + ε̄^{2} + \frac{∑_{l} μ(l)}{μ(i)}).

But now ε^{2}+ε̄^{2}+ \frac{∑_{l} μ(l)}{μ(i)} = ε^{2}+ε̄^{2}+β = 0 by the definition of ε. The other unitarity can be verified similarly. This computation also works even if N = ∞ for Ã, D̃, Ẽ. If we take a complex conjugate on a connection, we get an anti-automorphism. (cf. Yang-Baxter equation.)

The graph D_{n} with ∗-distinguished point has a Z_{2} symmetry. This switches W and W̄.

Example. The graph D_{4} corresponds to the group Z_{3}. In this case, a connection W is given by a unitary 3 × 3-matrix

W = \frac{1}{\sqrt{3}} ( \begin{matrix}1 & 1 & 1 \\ 1 & α & ᾱ \\ 1 & ᾱ & α \end{matrix} ) ,

where α ∈ C satisfies α + α^{2} + 1 = 0. The symmetry switches the second and the third columns.

For D_{n}, there is only one solution, up to graph automorphism. For E_{6}, E_{7}, E_{8}, we have two anti-isomorphic but non-isomorphic solutions.

IV.3. Flatness

Our result on flatness of the connections given above is as follows.

Theorem. The connections defined above for A_{n} are flat. The connections for D_{2n} are flat, and for D_{4k} two connections are conjugate to each other by switching the endpoints, and connections for D_{4k+2} are self-conjugate. Connections for E_{6}, E_{8} are flat. Connections for E_{7} and D_{odd} are not flat.

The result for D_{odd} means that composition table of bimodule multiplications for D_{odd} is impossible.

Consider D_{6} for example.

D_{6} : 1—2—3—4 /5 \5′

For flatness, we only need to check

W ( \begin{matrix}∗ & ⟶^{ξ} & 5 & ⟵^{ξ} & ∗ \\ ^{η}↓ & & ↓ & & ↓^{η′} \\ 5 & ⟶_{ζ} & x & ⟵_{ζ′} & 5′ \end{matrix} ) = 0,

where ξ = η = (1—2—3—4—5), η′ = (1—2—3—4—5′). The reason is as follows. The string algebras A_{k,0}, k ≤ 3, are generated by Jones projections and A_{k,0}, k ≥ 5, are generated by A_{k−1,0} and the Jones projection, thus we may assume |ξ| = 4 in the above diagram. Moreover, if r(ξ) = 1, 2, 3, 4, then the formula is valid again by flatness of the Jones projections. Hence we may assume that ξ = η = (1—2—3—4—5) or ξ = η = (1—2—3—4—5′), and by

(1—2—3—4—5, 1—2—3—4—5) + (1—2—3—4—5′, 1—2—3—4—5′) ∈ ⟨e_{i}⟩,

we need consider only the former case. Because the string (ξ, ξ) ∈ A_{5,0} commutes with horizontal Jones projections again by their flatness, it is enough to assume ξ = η and eta′ = (1—2—3—4—5′). Using that

W ( \begin{matrix}∗ & ⟶^{ξ} & 5 & ⟵^{ξ} & ∗ \\ ^{η}↓ & & ↓ & & ↓^{η′} \\ 5 & ⟶_{ζ} & x & ⟵_{ζ′} & 5′ \end{matrix} ) + W ( \begin{matrix}∗ & ⟶^{ξ} & 5′ & ⟵^{ξ} & ∗ \\ ^{η}↓ & & ↓ & & ↓^{η′} \\ 5 & ⟶_{ζ} & x & ⟵_{ζ′} & 5′ \end{matrix} ) = 0,

obtained by flatness of the Jones projections, and that W (2×2-cell) can be adjusted to be real by choice of gauges, we can show the desired formula.

IV.4. Non-amenability of the E_{10} subfactor

It is known that the possible values of the Perron-Frobenius eigenvalues for (possibly infinite) graphs are given by the following picture. (See Appendix I of [GHJ].)

2 cos π/n

———————————————

1  2_{(Ã,D̃,Ẽ,A_{∞},D_{∞})} \sqrt{φ + 1/φ}  (1 + φ = φ^{2})

The interval between 2 and \sqrt{φ + 1/φ}, where φ is the golden ratio, is given by the following picture.

————————————— 2 2.006 ⋯

The first value above 4 is given by E_{10}, and the first accumulation point is given by E_{∞}:

E_{∞} : ·— · — · ; ·^{|} — ⋯ ∞,

The second accumulation point and the value \sqrt{φ + 1/φ} are given by the following, respectively.

·— · — · — · ; ·^{|} — ⋯ ∞,

Here we give a proof for the theorem on non-amenability of the E_{10}-subfactor in II.5. Proof. First note that E_{10} is the only finite graph with the given Perron-Frobenius eigenvalue. (See Proposition I.3.4 in Appendix I of [GHJ].) If the graph is infinite, then an eigenvalue is greater than or equal to the norm of the graph, so the possibility of the principal graph now is limited to one of E_{10}, A_{∞}, D_{∞}. Then the lower part of the paragroup (the dual graph) should be the same as the principal graph because the Perron-Frobenius eigenvectors must match. Thus it is enough to eliminate possibilities of E_{10}, D_{∞}. For this purpose, suppose the principal graph has a triple point a, and the dual graph a corresponding triple point a′. We show that the biunitarity axiom gives a contradiction. Set β = [M : N]^{1/2} > 2. Let (c_{ij})_{i,j=1,2,3}

be the 3 × 3-unitary matrix given by W ( \begin{matrix}b_{i} & ↙ & a & ↘ & b′_{j} \\ & ↘ & a′ & ↙ \end{matrix} ). If i ≠ j, then we can

apply the biunitarity axiom to a 1 × 1-matrix and use the renormalization rule to

get |c_{ij}| = \frac{μ(b_{i})^{1/2}μ(b′_{i})^{1/2}}{μ(a)^{1/2}μ(a′)^{1/2}} . By

|c_{11}|^{2} + |c_{12}|^{2} + |c_{13}|^{2} = 1,
μ(b′_{2}) + μ(b′_{3}) = βμ(a) − μ(b′_{1}),

we get

|c_{11}|^{2} = \frac{μ(a)^{2} + μ(b_{1})μ(b′_{1}) − βμ(a)μ(b_{1})}{μ(a)^{2}}.

Applying the same formula to c_{11}, c_{21}, c_{31}, we get μ(b_{1}) = μ(b′_{1}), and hence

|c_{11}|^{2} = 1 − \frac{βμ(b_{1})}{μ(a)} + (\frac{μ(b_{1})}{μ(a)})^{2} .

Setting x_{i} = μ(b_{i})/μ(a), we get

(|c_{ij}|) = ( \begin{matrix}\sqrt{1 − βx_{1} + x_{1}^{2}} & \sqrt{x_{1}x_{2}} & \sqrt{x_{1}x_{3}} \\ \sqrt{x_{2}x_{1}} & \sqrt{1 − βx_{2} + x_{2}^{2}} & \sqrt{x_{2}x_{3}} \\ \sqrt{x_{3}x_{1}} & \sqrt{x_{3}x_{2}} & \sqrt{1 − βx_{3} + x_{3}^{2}} \end{matrix} ) .

Because c_{11}c̄_{21} + c_{12}c̄_{22} + c_{13}c̄_{23} = 0, x_{i} > 0 and β > 2, we get

x_{3} = \sqrt{x_{1}x_{3}}\sqrt{x_{2}x_{3}}/\sqrt{x_{1}x_{2}}
≤ \sqrt{1 − βx_{1} + x_{1}^{2}} + \sqrt{1 − βx_{2} + x_{2}^{2}}
≤ 1 − x_{1} + 1 − x_{2},

which implies 2 < β = x_{1} + x_{2} + x_{3} ≤ 2, a contradiction. Q.E.D.

Note that the above proof eliminates a large class of graphs for small index values> 4. (That is, if the upper graph and the lower graphs are the same, it has no cycle of length 4 and the Perron-Frobenius eigenvalue is bigger than 2, then there is no biunitary connection on this system.)

IV.5. Paragroups for index> 4

Recall Jones’ construction of subfactors ⟨e_{i}⟩ in the string algebra of a graph G at a vertex x. (See [GHJ].) Okamoto [Ok] showed that [String(G) : ⟨e_{i}⟩] < ∞ if and only if the graph G is one of A, D, E, A_{∞}, D_{∞}. He also computed principal graphs for these. The smallest index value arising in this way is 3 + \sqrt{3} and this is the smallest known value> 4 for which a flat connection exists, in other words, this is a subfactor with finite depth with the smallest known index value> 4. It is also conjectured that this is actually the smallest value above 4 as an index value of an irreducible subfactor with finite (or amenable) depth.

References

[GHJ] F. Goodman, P. de la Harpe, & V. F. R. Jones, “Coxeter graphs and towers of algebras”, MSRI publications 14, Springer, 1989.

[HSO] U. Haagerup & J. Schou, with an appendix by A. Ocneanu, in preparation.

[J1] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1–15.

[J2] V. F. R. Jones, A converse to Ocneanu’s theorem, J. Operator Theory 10 (1983), 61–63.

[O1] A. Ocneanu, Quantized group string algebras and Galois theory for algebras, in “Operator algebras and applications, Vol. 2 (Warwick, 1987),” London Math. Soc. Lect. Note Series Vol. 136, Cambridge University Press, 1988, pp. 119–172.

[O2] A. Ocneanu, Graph geometry, quantized groups and nonamenable subfactors, Lake Tahoe Lectures, June–July, 1989.

[Ok] S. Okamoto, Invariants for subfactors arising from Coxeter graphs, preprint, Pennsylvania State University.

[PP] M. Pimsner & S. Popa, Entropy and index for subfactors, Ann. Scient. Éc. Norm. Sup. 19 (1986), 57–106.

[R] Ph. Roche, Ocneanu cell calculus and integrable lattice models, Comm. Math. Phys. 127 (1990), 395–424.

[W] H. Wenzl, Hecke algebras of type A and subfactors, Invent. Math. 92 (1988), 345–383.

Index adjacency matrix - - - 12,13,48 Alexandar - - - - - - - - - - - 39 amenable - - - - - - - 34,35,58 asymptotic inclusion - - 40,42 basis - - - - - - - - - - - - - 2,3,4 Bernoulli shift - - - - - - - - 18 biunitarity - - 10,15,18,25,27,48 Boltzmann weight - - - - - - - 7 braid - - - - - - - - - - - - - - 38 Bratteli diagram - - - - - 20,21 Cayley graph - - - - - - - - - 29 cell - - - - - - - - - - - - - - - - 7 central freedom lemma - 42,43 central sequence - - - 18,43,45 CFT - - - - - - - - - - - - - - 48 commuting square - - - - 18,41 completely positive - - - - - 26 connection - - - - - - - - 7,49,50 convolution - - - - - - - - - - 37 coupling constant - - - - - - - 2 Coxeter number - - - 41,48,53 depth - - - - - - - - - - - - - - - 5 Dynkin diagram - - - - - 48,49 field of strings - - - - - - - - 26 finite depth - - - - - 5,40,41,58 flat - - - - - - 26,31,33,49,54,58 Fourier transform - - - - 36,37 Frobenius - - - - - - - - - - - - 5 Galois - - - - - - - - - 1,2,19,48 gauge - - - - - - - - - - - - - 7,50 GNS construction - - - - - - 14 golden ratio - - - - - - - - - - 56 granifold - - - - - - - - - - - - 13 harmonic - - - - - - - - - - 14,23 Herman - - - - - - - - - - - - - 3 hyperfinite II_{1} factor - - - - 14 intertwiner - - - - - - 1,5,6,7,22 IRF - - - - - - - - - - - - - - - 50 irreducible bimodule - 1,2,4,6 Jones index - - - - - - - - - - - 2 Jones projection 18,23,28,36,42 Kac algebra - - - - - - - - - - 37 KK-theory - - - - - - - - - - - 3 knot invariant - - - - - - - - 39 Kronecker - - - - - - - - - - - 45 Laplacian - - - - - - - - - 13,23 Mackey machine - - - - - - - - 8 Markov move - - - - - - - - - 39 Murray-von Neumann - - - 14 non-amenable - - - - - - - 29,57 Okamoto - - - - - - - - - - - - 58 paragroup - - - - - - - 48,49,57 partition function - - - - - - 23 Pasquier - - - - - - - - - - - - 53 path - - - - - - - - - - - - - - 13 Perron-Frobenius 35,45,48,52,56 Peter-Weyl - - - - - - - - - - 10 Pimsner-Popa - - - - - - - - - - 3 Radon-Nikodym - - - - - - - - 2 Range Theorem - - - - - - - - 34 renormalization - - - 9,28,50,53 Roche - - - - - - - - - - - - - - 19 Sauvageot - - - - - - - - - - - 30 simple subfactor - - - - - - - 37 standard - - - - - - - - - - - - 31 string - - - - - - - - - 13,20,25,41 string algebra - - - - - - - 15,18 transport - - - - - - - - - 25,41 tunnel - - - - - - - - - - - - - 42 Wenzl - - - - - - - - - - - - - 34 Yang-Baxter equation - - - - 54