IRREDUCIBLE SUBFACTORS OF L(𝔽_{∞}) OF INDEX λ > 4.

DIMITRI SHLYAKHTENKO AND YOSHIMICHI UEDA

Abstract. By utilizing an irreducible inclusion of type III_{q^{2}} factors coming from a free-product type action of the quantum group SU_{q}(2), we show that the free group factor L(𝔽_{∞}) possesses irreducible subfactors of arbitrary index > 4. Combined with earlier results of Radulescu, this shows that L(𝔽_{∞}) has irreducible subfactors with any index value in {4 cos^{2}(π/n) : n ≥ 3} ∪ [4, +∞).

1. Introduction.

Let M be a type II_{1} factor, and let N ⊂ M be a finite index subfactor. Recall that N ⊂ M is called irreducible, if the relative commutant N′ ∩ M is trivial.

In his fundamental paper [8], V.F.R. Jones introduced the notion of index [M : N] of a subfactor N ⊂ M. Related to the index, he introduced two invariants of a factor M:

ℐ(M) = {λ ∈ ℝ_{+} : ∃N ⊂ M subfactor, [M : N] = λ},
𝒞(M) = {λ ∈ ℝ_{+} : ∃N ⊂ M irreduciblesubfactor, [M : N] = λ},

measuring the possible index values of subfactors of M.

In the same paper, Jones showed that for any factor M, the index of a subfactor cannot be arbitrary; in fact, 𝒞(M) ⊂ ℐ(M) ⊂ ℐ = {4 cos^{2}(π/n) : n ≥ 3} ∪ [4, +∞]. Moreover, he produced subfactors of the hyperfinite II_{1} factor R of arbitrary allowable index, thus showing that ℐ(R) = ℐ. However, the subfactors of R with index > 4 that he constructed are not irreducible; to this day, it is still not known whether the equality 𝒞(R) = ℐ holds; in other words, whether every index value can be realized by an irreducible subfactor of the hyperfinite factor. The one general positive result is due to by Popa [12] (see also [14] and [13]), stating that given a number λ ∈ ℐ, there is a (non-hyperfinite) II_{1} factor M, for which λ ∈ 𝒞(M).

Thanks to the theory of free probability, pioneered by Voiculescu in the early 80s (see e.g. [25]), there has been much progress at understanding free group factors L(𝔽_{n}) (Voiculescu’s introduction to [25] states that these are the “best” of the “bad” non-hyperfinite factors). Following this philosophy, Radulescu undertook a study of subfactors of free group factors. Using a particular version of Popa’s construction [16] and by employing free probability and random matrix techniques, he managed to show that for an interpolated free group factor L(𝔽_{t}), 𝒞(L(𝔽_{t})) ∩ [0, 4] = ℐ(L(𝔽_{t})) ∩ [0, 4] = ℐ ∩ [0, 4]. Coupled with his earlier result [15] on the fundamental group of L(𝔽_{∞}) this gave that L(𝔽_{∞}) has (possibly non-irreducible) subfactors of all indices, i.e., ℐ(L(𝔽_{∞})) = ℐ. However, this still left the question about whether subfactors of L(𝔽_{∞}) of index > 4 can be chosen to be irreducible.

The main theorem of this paper is

Theorem. For each λ > 4, there exists an irreducible subfactor of L(𝔽_{∞}) of index λ.

Combined with earlier results of F. Radulescu, this gives:

Corollary. 𝒞(L(𝔽_{∞})) = ℐ(L(𝔽_{∞})) = ℐ.

In fact, more is true: given an allowable index value λ ∈ ℐ, there exists an irreducible subfactor L(𝔽_{∞}) ≅ N ⊂ M ≅ L(𝔽_{∞}) of index λ.

We would like to point out that it is still not known whether every Popa’s λ-lattice [13] (i.e., an abstract system of higher relative commutant of a subfactor) can be realized by a subfactor of some fixed universal II_{1} factor M. The factor M = L(𝔽_{∞}) seems to be a likely candidate, although we don’t know how to prove it; note, however, that Radulescu gave an affirmative answer to this question in the case that the λ-lattice is finite-depth. Also, by the results of this paper, all examples of irreducible subfactors whose λ-lattices come from the representation theory of SU_{q}(2) (see [2], [30], [27]) can be realized as N ⊂ M with N ≅ M ≅ L(𝔽_{∞}).

We don’t know whether our result holds for L(𝔽_{n}), n < ∞.

The strategy to find irreducible subfactors of L(𝔽_{∞}) with index λ > 4 comes from the work of the second-named author [24] on quantum SU_{q}(2) actions on free products of von Neumann algebras, and analysis of resulting Wassermann-type inclusions of type III factors. As was pointed out in [24, Question 2], to prove the main theorem of the present paper, it is sufficient to identify a certain crossed product of a certain von Neumann algebra by SU_{q}(2), which we do in Section 2 by identifying this algebra as a free Araki-Woods factor [18]. For the convenience of the reader, we summarize the main necessary facts from [24] in Section 3.

Acknowledgement.The authors would like to express their sincere gratitude to MSRI and the organizers of the year-long program “Operator Algebras (2000–2001)" held at MSRI for inviting them to the program, where this joint work (although conceived some time ago) was carried out and completed.

D.S. would like to acknowledge support by a National Science Foundation Postdoctoral Fellowship. Y.U. would like to acknowledge support by the Japanese Ministry of Education, Science, Sports and Culture.

2. Amalgamated free products of free group factors with B(H).

In this section, we prove some technical results about amalgamated free products of free group factors with B(H), identifying them as free Araki-Woods factors. The treatment given in this section is more general than strictly speaking necessary for our result, but allows for more succinct and abstract proofs. We encourage the interested reader to look through the appendix to the paper, which can serve as an illustration of the proof given here.

We recall some notation from [20]. Let M be a von Neumann algebra and η : M → M ⊗ B(ℓ^{2}) be a normal completely positive map, given as a matrix η(m) = (η_{ij}(m))_{ij}. Let K be the M-Hilbert bimodule spanned by vectors ξ_{i}, and satisfying

⟨ξ_{i}, mξ_{j}⟩_{M} = η_{ij}(m), ∀m ∈ M

(see [20, Lemma 2.2]).

Let 𝒯 be the universal C^{∗}-algebra generated by M and operators L(ζ) : ζ ∈ K satisfying

L(ζ)^{∗}mL(ζ′) = ⟨ζ, mζ′⟩, ∀m ∈ M, ζ, ζ′ ∈ K.

This is the Toeplitz extension of the Cuntz-Pimsner C^{∗}-algebra associated to the M-bimodule H (see [11]). The operators L(ζ) are called η-creation operators over M.

Set L_{i} = L(ξ_{i}). Let E_{M} denote the canonical conditional expectation from 𝒯 onto M (see [11], [20]). Choose a faithful normal state ϕ_{0} on M and set ϕ = ϕ_{0} ∘ E_{M}. In the GNS representation associated to ϕ, consider the von Neumann algebra

Φ(M, η) = W^{∗}(M, L_{i} + L_{i}^{∗} : i = 1, 2, …).

It can be shown ([20]) that Φ(M, η) depends only on M and (η_{ij}).

Let K^{η} ⊂ K be the ℝ-linear subspace of the Hilbert M-bimodule K, given as the norm closure of the set of vectors

∑_{finite}m_{j}ξ_{j}n_{j}^{∗} + n_{j}ξ_{j}m_{j}^{∗}, m_{j}, n_{j} ∈ M.

Then [20, Proposition 2.19] Φ(M, η) depends only on the inclusion K^{η} ⊂ K. In fact

Φ(M, η) = W^{∗}(M, L(ζ) + L(ζ)^{∗} : ζ ∈ K^{η} ⊂ K) (2.1)

(in the GNS representation associated to ϕ).

In the particular case that M = ℂ, the matrix η_{ij}(1) defines a scalar-valued inner product on K, and K^{η} ⊂ K ends up a particular real subspace. In this case, under the assumption that the state E_{M} = E_{ℂ} is faithful, Φ(M, η) is nothing other than the free Araki-Woods factor Γ(K^{η} ⊂ K)′′ in the notation of Remark 2.6 of [18] (see also [20, Example 3.5]).

A very particular example of such a matrix μ is

μ_{λ} = ( \begin{matrix}1 & −i\frac{λ−1}{λ+1} \\ i\frac{λ−1}{λ+1} & 1 \end{matrix} ) , 0 < λ < 1;

in this case Φ(ℂ, μ_{λ}) is type III_{λ} and is the unique free Araki-Woods factor T_{λ} of this type.

Theorem 2.1. Let H be a finite or infinite dimensional Hilbert space. Let (η_{ij}) : B(H) → B(H)⊗ B(ℓ^{2}) be any normal completely positive map. Assume that E_{B(H)} : Φ(B(H), η) → B(H) is faithful. Then the von Neumann algebra Φ(B(H), η) is stably isomorphic to the von Neumann algebra Φ(ℂ, μ) for some (completely) positive map μ : ℂ → B(ℓ^{2}); moreover, E_{ℂ} : Φ(ℂ, μ) → ℂ is faithful.

Proof. By equation (2.1),

N = Φ(B(H), η) = W^{∗}(B(H), L(ζ) + L(ζ)^{∗} : ζ ∈ K^{η} ⊂ K),

where K and K^{η} are as described above. Let e_{ij}, 0 ≤ i, j < ∞ be a family of matrix units generating B(H); thus

e_{ij}e_{j′i′} = δ_{jj′}e_{ii′}, e_{ij}^{∗} = e_{ji}.

Consider the algebra

P = e_{00}Ne_{00}.

It is clear that P ⊗B(ℓ^{2}) ≅ N ⊗B(ℓ^{2}); thus it is sufficient to prove that P is isomorphic to Φ(ℂ, μ) for some μ. Note that P is generated by the family

e_{0i}(L(ζ) + L(ζ)^{∗})e_{j0}, 0 ≤ i ≤ j, ζ ∈ K^{η};

hence

P = W^{∗}(L(e_{0i}ζe_{j0}) + L^{∗}(e_{0j}ζe_{i0}) : ζ ∈ K^{η}).

Consider the linear space

V_{ℝ} = span_{ℝ}({e_{0i}ζe_{i0} : ζ ∈ K^{η}, i ≥ 0} ∪ {e_{0i}ζe_{j0} + e_{0j}ζe_{i0} : ζ ∈ K^{η}, 0 ≤ i < j}
∪\{\sqrt{−1}(e_{0i}ζe_{j0} − e_{0j}ζe_{i0}) : ζ ∈ K^{η}, 0 ≤ i < j\}) ⊂ K^{η},

as a subspace of the complex linear space

V = span{e_{0i}ζe_{j0} : ζ ∈ K} ⊂ K.

Since for i < j,

(L(e_{0i}ζe_{j0}) + L^{∗}(e_{0j}ζe_{i0})) = \frac{1}{2}L(e_{0i}ζe_{j0} + e_{0j}ζe_{i0}) + L(e_{0i}ζe_{j0} + e_{0j}ζe_{i0})^{∗}

−\frac{1}{2}\sqrt{−1}(L(\sqrt{−1}(e_{0i}ζe_{j0} − e_{0j}ζe_{i0}))

+L(\sqrt{−1}(e_{0i}ζe_{j0} − e_{0j}ζe_{i0}))^{∗}),

it follows that

P = W^{∗}(L(ζ) + L(ζ)^{∗} : ζ ∈ V_{ℝ}).

Since for any ζ, ζ′ ∈ V, e_{00}ζe_{00} = ζ and e_{00}ζ′e_{00} = ζ′, we get that

⟨ζ, ζ′⟩_{B(H)} = ⟨e_{00}ζe_{00}, e_{00}ζ′e_{00}⟩ = e_{00}⟨ζ, ζ′⟩e_{00} ∈ e_{00}ℂ.

It follows that the restriction of the inner product on K to V is scalar-valued; we denote the restriction by ⟨·, ·⟩_{V}. Hence we get an inclusion of a real Hilbert space into a complex Hilbert space

V_{ℝ} ⊂ V.

In particular, for any ζ, ζ′ ∈ V, we have

L(ζ)^{∗}L(ζ′) = ⟨ζ, ζ′⟩_{V} . (2.2)

Denote by θ the state P ∋ m ↦ e_{00}E_{B(H)}(m)e_{00}. Since E_{B(H)} is faithful by assumption, and P (e_{00}me_{00}) = E_{B(H)}(e_{00}me_{00}), we get that θ is a faithful state. Furthermore,

θ(L(ζ_{1}) ⋯ L(ζ_{k})L(ζ_{k+1})^{∗} ⋯ L(ζ_{r})^{∗}) = 0 (2.3)

if r ≠ 0; note that this, together with the relations (2.2) determines θ on P.

Choose now a basis ζ_{i} for V_{ℝ} as a real Hilbert space, and let

μ_{ij} = ⟨ζ_{i}, ζ_{j}⟩_{V} .

Then Φ(ℂ, μ) is generated in the GNS representation associated to E_{ℂ} by l_{i} + l_{i}^{∗}, subject to the relations

l_{i}^{∗}l_{j} = μ_{ij} = ⟨ζ_{i}, ζ_{j}⟩_{V} .

Moreover, ψ = E_{ℂ} is determined by

ψ(l_{i_{1}}l_{i_{2}} ⋯ l_{i_{k}}l_{i_{k+1}}^{∗} ⋯ l_{i_{r}}^{∗}) = 0, r ≠ 0.

Comparing these with (2.2) and (2.3), and using the fact that θ is faithful, we obtain that the map L(ζ_{i})+L(ζ_{i})^{∗} ↦ l_{i} +l_{i}^{∗} extends to an isomorphism P ≅ W^{∗}(l_{i} +l_{i}^{∗}) = Φ(ℂ, μ), which conjugates θ and ψ = E_{ℂ}. It also follows that E_{ℂ} is faithful.

We can now deduce the main technical result needed for further computations; for convenience, we write L(𝔽_{1}) = L(ℤ).

Theorem 2.2. Let H be a separable Hilbert space (finite or infinite-dimensional). Let B ⊂ B(H) be a von Neumann subalgebra, and assume that E : B(H) → B is a normal faithful conditional expectation. For n = 1, 2, … or +∞, consider the reduced amalgamated free product

N = (B(H), E) ∗_{B} (L(𝔽_{n}) ⊗ B, τ ⊗ id).

Then N is stably isomorphic to a free Araki-Woods factor Γ(ℋ_{ℝ}, U_{t})′′.

In particular, if N is type III_{λ}, 0 < λ < 1, then N is isomorphic to the unique type III_{λ} free Araki-Woods factor T_{λ}.

Proof. Let η : B(H) → B(H) ⊗ B(ℓ^{2}) be given by

η(T) = diag(E(T), E(T), …, E(T), 0, 0 …)

(n copies of E(T)). Then by [20, Example 3.3(c)],

Φ(B(H), η) ≅ (B(H), E) ∗_{B} (Φ(B, η|_{B}), E_{B}).

Moreover, by [20, Example 3.3(b)],

(Φ(B, η|_{B}), E_{B}) ≅ (Φ(ℂ, η|_{ℂ1_{B}}) ⊗ B, E_{ℂ} ⊗ id) ≅ (L(𝔽_{n}) ⊗ B, τ ⊗ id).

Hence

Φ(B(H), η) ≅ (L(𝔽_{n}) ⊗ B, τ ⊗ id) ∗_{B} (B(H), E) ≅ N.

Applying Theorem 2.1, we get that

N ⊗ B(ℓ^{2}) ≅ Φ(ℂ, μ) ⊗ B(ℓ^{2})

for some μ. By our identifications above, E_{B(H)} is the canonical conditional expectation from N onto B(H) and is therefore faithful. Hence ϕ = E_{ℂ} : Φ(ℂ, μ) → ℂ is faithful. It follows that Φ(ℂ, μ) ≅ Γ(ℋ_{ℝ}, U_{t})′′ for some real Hilbert space ℋ_{ℝ} and one-parameter group U_{t} (in fact, ℋ_{ℝ} is the real subspace V_{ℝ} constructed in the proof of Theorem 2.1).

If N is type III_{λ}, 0 < λ < 1, it follows from the uniqueness of the type III_{λ} free Araki-Woods factor T_{λ} [18] that N ≅ T_{λ}.

The following theorem can be deduced from the results of Radulescu [17] and Dykema [6], as was done in [18]. We have since found a somewhat shorter argument, which we give below:

Theorem 2.3. Let T_{λ} be the (unique) free Araki-Woods factor of type III_{λ}. Let ϕ be any normal faithful state, so that the modular group σ_{t}^{ϕ} is periodic of period 2π/ log λ. Then the centralizer T_{λ}^{ϕ} is isomorphic to L(𝔽_{∞}). Also, the core of T_{λ} is isomorphic to L(𝔽_{∞}) ⊗ B(H).

Proof. Clearly, only the statement about the centralizer needs to be proved, since the core is isomorphic to the centralizer, tensor B(H), as soon as the centralizer is a factor (see [4]). Also, we can restrict ourselves to dealing with a particular choice of the state ϕ, satisfying the hypothesis of this theorem; indeed, by [4], all centralizers of such states are stably isomorphic. Since by Radulescu’s results [15], the fundamental group of L(𝔽_{∞}) is all of (0, +∞), a II_{1} factor M is stably isomorphic to L(𝔽_{∞}) iff it is actually isomorphic to L(𝔽_{∞}).

Recall [18] that T_{λ} can be described as the free product

L^{∞}[0, 1] ∗ (B(H), θ)

where θ is a normal faithful state on B(H) given by

θ(T) = Tr(DT),

where

D = diag(\frac{1}{1 − λ}, \frac{λ}{1 − λ}, ⋯).

Consider now the completely-positive map η given by η(T) = θ(T), and consider Φ(B(H), η) = W^{∗}(B(H), L + L^{∗}), where L^{∗}T L = η(T) = θ(T) for all T ∈ B(H). By [20, Example 3.3(a)], Φ(B(H), η) ≅ (B(H), θ) ∗ (L^{∞}[0, 1], Lebesgue measure) ≅ (T_{λ}, ϕ_{λ}).

Note that B(H) is generated, as a von Neumann algebra, by a partial isometry v, so that v^{∗}v = 1 and vv^{∗} = diag(0, 1, 1, …). Hence T_{λ} is generated by L+L^{∗} and v. The modular group of ϕ_{λ} acts by fixing L + L^{∗} and sending v to λ^{it}v. The centralizer of the free product state is then isomorphic to the von Neumann algebra generated by all elements of the form

v^{k}(L + L^{∗})(v^{∗})^{k}, (v^{∗})^{l}(L + L^{∗})v^{l}, k, l = 0, 1, …

as well as the projections p_{k} = v^{k}(v^{∗})^{k}.

Let for k ≥ 0

L_{k} = v^{k}L(v^{∗})^{k}

and for −l < 0,

L_{−l} = (v^{∗})^{l}Lv^{l}.

Denote by Ψ the faithful normal conditional expectation

Ψ(m) = ∫_{0}^{2π} σ_{t}^{ϕ_{λ}}(m)dt, Ψ : T_{λ} → T_{λ}^{ϕ_{λ}}.

It follows that any m ∈ T_{λ} can be written as the sum (convergent in L^{2}(T_{λ}, ϕ_{λ}))

m = ∑_{k≥0} v^{k}m_{k} + ∑_{k<0} m_{k}(v^{∗})^{k}, m_{k} = Ψ((v^{∗})^{−k}m), k ≥ 0, m_{k} = Ψ(mv^{−k}), k < 0,

with m_{k} ∈ T_{λ}^{ψ_{λ}}. It follows that the centralizer is linearly spanned by words of the form

v^{k_{1}}(v^{∗})^{l_{1}}Xv^{k_{2}}(v^{∗})^{l_{2}}X ⋯ v^{k_{p}}(v^{∗})^{l_{p}}, ∑ k_{j} − ∑ l_{j} = 0.

where X = L + L^{∗}. Using the notation

X_{k} = L_{k} + L_{k}^{∗}, k ∈ ℤ

and the relation v^{∗}v = 1, v^{k}(v^{∗})^{k} = p_{k} ∈ D, we get that the centralizer is generated by

D and {X_{k} : k ∈ ℤ}.

Note that for all diagonal elements d ∈ D = W^{∗}(p_{k} : k ≥ 0),

L_{k}^{∗}dL_{k′} = v^{k}δ_{kk′}θ((v^{∗})^{k}dv^{k})(v^{∗})^{k},

with the convention that v^{−1} = v^{∗}.

By [22] and [20, Example 2.6], L_{k} are free with amalgamation over D.

Let M_{1} = W^{∗}(D, L_{k} + L_{k}^{∗}, k ≤ 0). Since when k ≤ 0,

L_{k}^{∗}dL_{k} = C_{k}θ(d),

for some nonzero constants C_{k}, we get by [19, Theorem 2.3] that L_{k} is ∗-free from D, and

M_{1} = (D, θ) ∗ (∗_{k≤0}W^{∗}(L_{k} + L_{k}^{∗})).

Since L_{k} + L_{k}^{∗} is semicircular, we get

M_{1} = (D, θ) ∗ L(𝔽_{∞}) ≅ L(𝔽_{∞}),

by the results of Ken Dykema [5].

Now, M = M_{1} ∗_{D} (W^{∗}(L_{k} + L_{k}^{∗}, k > 0)). Since M_{1} is a factor and L_{k} are ω-creation operators over M, where ω_{ij} = δ_{ij}C_{i}p_{i}θ((v^{∗})^{i}dv^{i}), we get by [19, Proposition 5.4] that M ≅ M_{1} ∗ L(𝔽_{t}), where t = ∑_{i} θ(v^{i}(v^{∗})^{i})^{2}. Hence M ≅ L(𝔽_{∞}).

3. Quantum SU_{q} actions.

The main goal of this section is to analyze inclusions of type III factors coming from a minimal free product type action of SU_{q}(2) on a von Neumann algebra.

Let (A, δ) be the Hopf-von Neumann algebra of Woronowicz’s quantum group SU_{q}(2) [29], [28], and denote by h the canonical Haar state on A. Let V be the multiplicative unitary, i.e., the unitary on L^{2}(A) ⊗ L^{2}(A) defined by

V (Λ_{h}(a) ⊗ ζ) = δ(a)(Λ_{h}(1) ⊗ ζ),

and let W be the fundamental unitary, i.e., unitary on L^{2}(A) ⊗ L^{2}(A) defined by

W (η ⊗ Λ_{h}(a)) = δ(a)(η ⊗ Λ_{h}(1)),

where Λ_{h} : A → L^{2}(A) is the canonical injection associated with the Haar state h. The dual Hopfvon Neumann algebra Â is the von Neumann algebra acting on L^{2}(A) generated by elements of the form:

(Id ⊗ ω)(V), ω ∈ B(L^{2}(A))_{∗}.

Let m be an integer or +∞ and set

M = (L(𝔽_{m}), τ) ∗ (A, h)

and let Γ : M → M ⊗ A be the free product of the trivial action of SU_{q}(2) on L(𝔽_{m}), and the left regular representation action δ of SU_{q}(2) on A [24, §3].

Recall that the crossed-product M ⋊_{Γ} SU_{q}(2) is the von Neumann subalgebra of M ⊗B(L^{2}(A)) generated by Γ(M) and ℂ1 ⊗ Â.

For each irreducible finite-dimensional unitary representation π : V_{π} → V_{π} ⊗ A of SU_{q}(2), consider the Wassermann-type inclusion [26] of von Neumann algebras

𝒩 = ℂ1 ⊗ M^{Γ} ⊂ (B(V_{π}) ⊗ M)^{Ad_{π}⊗Γ} = ℳ.

It turns out that the negative (normalized) q-trace τ_{q}^{(π,−)} gives rise to a finite-index conditional expectation ℰ : ℳ → 𝒩 (see [24, eq. (3.6)]). We collect below some facts about this inclusion, most of which are from [24].

Theorem 3.1. Let π be an irreducible representation of SU_{q}(2) and let 𝒩 ⊂ ℳ be as above, taken with the conditional expectation ℰ. Then

(a) 𝒩 and ℳ are type III_{q^{2}} factors and 𝒩 ≅ ℳ;

(b) The inclusion 𝒩 ⊂ ℳ is irreducible;

(c) The index of ℰ is the square of the q-dimension of π, (dim_{q} π)^{2};

(d) Let ϕ denote the restriction of the free product state τ ∗ h to 𝒩 = M^{Γ} ⊂ M. Then the centralizers 𝒩̂ = 𝒩^{ϕ} and ℳ̂ = ℳ^{ϕ∘ℰ} are both factors of type II_{1};

(e) The inclusion 𝒩 ⊂ ℳ is “essentially type II”. More precisely, the restriction of ℰ to the inclusion N ⊂ M is trace-preserving, and

𝒩̂ = 𝒩^{ϕ} ⊂ ℳ^{ϕ∘ℰ} = ℳ̂

is an inclusion of type II_{1} factors with the same index and the same system of higher relative commutants as 𝒩 ⊂ ℳ. In particular, 𝒩̂ ⊂ ℳ̂ is irreducible.

(f) The algebras 𝒩 = M^{Γ} and M ⋊_{Γ} SU_{q}(2) are isomorphic.

Proof. The fact that 𝒩 is type III_{q^{2}} was proved in [24]; this fact also follows from the explicit description of 𝒩 given in the appendix.

For the convenience of the reader we give an expanded proof of the isomorphism 𝒩 ≅ ℳ (see [24, Remark 10]). Let n be the dimension of V_{π} as a vector space. Choose a standard orthonormal system of vectors ξ_{i}, 1 ≤ i ≤ n for the space V_{π} so that the (co)representation π is given by

π(ξ_{j}) = ∑_{i=1}^{n} ξ_{i} ⊗ u_{ij} ∈ V_{π} ⊗ A,

for some u_{ij} ∈ A satisfying ∑_{k} u_{ki}^{∗}u_{kj} = δ_{ij}.

Since 𝒩 = M^{Γ} ⊂ M is type III, we can find isometries S_{i} ∈ M^{Γ}, i = 1, …, n so that ∑ S_{i}S_{i}^{∗} = 1 and S_{i}^{∗}S_{j} = δ_{ij}. Set

v_{j} = ∑_{i=1}^{n} S_{i} · u_{ij} ∈ M^{Γ} · A ⊂ M, H = span_{ℂ}{v_{j} : 1 ≤ j ≤ n}.

Then v_{i}^{∗}v_{j} = δ_{ij} and ∑ v_{i}v_{i}^{∗} = 1. Moreover, since S_{i} ∈ M^{Γ} are fixed by Γ, we get (since Γ is an action and Γ|_{A} = δ)

Γ(v_{j}) = ∑_{i}(S_{j} ⊗ 1) · Γ(u_{ij})
= ∑_{i}(S_{j} ⊗ 1) · δ(u_{ij})
= ∑_{i,k}(S_{j} ⊗ 1) · (u_{ki} ⊗ u_{ij})
= ∑_{i} V_{i} ⊗ u_{ij}.

It follows that the isomorphism H ∋ v_{i} ↦ ξ_{i} ∈ V_{π} is equivariant with respect to the restriction of Γ to H and the (co)representation π on V. It follows that the isomorphism

Φ : B(V_{π}) ⊗ M → M, Φ((q_{ij})_{ij}) = ∑_{ij} v_{i}q_{ij}v_{j}^{∗}

is equivariant with respect to the action Adπ ⊗ Γ on B(V_{π}) ⊗ M and the action Γ on M. Thus Φ restricts to an isomorphism of 𝒩 = M^{Γ} and ℳ = (B(V_{π}) ⊗ M)^{(Adπ⊗Γ)}. Thus (a) holds.

Statements (b)–(e) were proved in [24] (Theorem 8 and the discussion on pp. 43–49).

For part (f), it is known that

M ⋊_{Γ} SU_{q}(2) = (M ⊗ B(L^{2}(A)))Γ̃,

where Γ̃ := Ad(1 ⊗ Σ(W^{∗})) ∘ (Id ⊗ Σ) ∘ (Γ ⊗ Id), and where Σ is the flip map (see [3, Remark 20] for a simple proof, which also works in the W^{∗}-algebraic setting without any change. For the reader’s convenience, we should mention that the definition of “λ(ω)” in that paper involves a typographical error.) Since (M, Γ) contains (A, δ), we see that

(M ⊗ B(L^{2}(A)))Γ̃ ≅ M^{Γ} ⊗ B(L^{2}(A))

(see [23, Lemma 4.2]) so that the crossed-product M ⋊_{Γ} SU_{q}(2) and the fixed-point algebra M^{Γ} are isomorphic to each other, both being of type III.

We would like to point out that the n-th step ℳ_{n} in the basic construction associated to 𝒩 ⊂ ℳ is rather easy to describe:

ℳ_{n} ≅ (B(V_{ρ_{n}}) ⊗ M)^{Adρ_{n}⊗Γ}

where

ρ_{2s+1} = π̂ ⊗ ρ_{2s}
ρ_{2s} = π ⊗ ρ_{2s−1}
ρ_{0} = π,

and π̂ is the canonical dual of π. The conditional expectation ℰ_{n} : ℳ_{n} → ℳ_{n−1} is given by

ℰ_{2s+1} = τ_{q}^{(π̂,−)} ⊗ Id ⊗ ⋯
ℰ_{2s} = τ_{q}^{(π,−)} ⊗ Id ⊗ · · · .

This allows one to compute the λ-lattice of higher relative commutants of 𝒩 ⊂ ℳ (and thus of 𝒩̂ ⊂ ℳ̂) in terms of the representation theory of SU_{q}(2). In particular, one can show that if π is taken to be the fundamental representation, then the principal graph of 𝒩̂ ⊂ ℳ̂ is A_{∞}, and the index is (q + q^{−1})^{2}.

For 0 < λ < 1, let T_{λ} denote the (unique) free Araki-Woods factor of type III_{λ} (see [18] and Section 2).

Proposition 3.2. The factor ((L(𝔽_{∞}), τ)∗(A, h))⋊_{Γ}SU_{q}(2) is isomorphic to the free Araki-Woods factor T_{q^{2}}. Hence ℳ ≅ 𝒩 ≅ T_{q^{2}}.

Proof. It is known that the fundamental unitary W lies in A ⊗ Â′ (acting on L^{2}(A) ⊗ L^{2}(A)), and hence the mapping Φ : x ∈ B(L^{2}(A)) ↦ W (1 ⊗ x)W^{∗} gives rise to an isomorphism from B(L^{2}(A)) onto A ⋊_{δ} SU_{q}(2) = A ⋊_{δ} Â (see e.g. [1]).

Moreover, the crossed product structure and the Haar state h on A give rise to a normal faithful conditional expectation

E : B(L^{2}(A)) = A ⋊_{δ} Â → Â

given by

E(x) = (h ⊗ Id)(W (1 ⊗ x)W^{∗}), x ∈ B(L^{2}(A)).

We claim that

M ⋊_{Γ} SU_{q}(2) ≅ (L(𝔽_{m}) ⊗ Â, τ ⊗ id) ∗Â (B(L^{2}(A)), E).

This isomorphism is obtained by extending the map

Φ : A ⋊_{δ} SU_{q}(2) → B(L^{2}(A))

to

[(L(𝔽_{m}), τ) ∗ (A, h)] ⋊_{Γ} SU_{q}(2) ⊃ A ⋊_{δ} SU_{q}(2)

by mapping elements in L(𝔽_{m}) to the canonical copy of L(𝔽_{m}) inside (L(𝔽_{m}) ⊗ Â, τ ⊗ id) ∗Â (B(L^{2}(A)), E). Since Φ(â) = 1 ⊗ â, â ∈ Â and (h ⊗ Id) ∘ Φ = E, Φ extends to the desired isomorphism (a freeness condition needs to be checked, see [24, Proposition 1] and [25].)

(If instead of a quantum group action, we would have an action of an ordinary group G, the isomorphism above would be the content of the general identity

(R, ϕ) ∗ (S, ψ) ⋊_{id∗α} G = (R ⊗ L(G), ϕ ⊗ id) ∗_{L(G)} (S ⋊_{α} G, E_{G}),

where E_{G} is the canonical L(G)-valued conditional expectation on S ⋊_{α} G associated to the state ψ : S → ℂ.)

Applying Theorem 2.2, and noting that M ⋊_{Γ} SU_{q}(2) is type III_{q^{2}} (Theorem 3.1 (f) and (a)), we find that

ℳ ≅ 𝒩 ≅ (L(𝔽_{m}) ⊗ Â, τ ⊗ id) ∗Â (B(H), E)

is isomorphic to the unique III_{q^{2}} free Araki-Woods factor T_{q^{2}}.

Remark 3.3. Since the free Araki-Woods factor T_{q^{2}} is prime (i.e., cannot be written as a tensor product of two diffuse von Neumann algebras, see [21]), it follows that the inclusion 𝒩 ⊂ ℳ cannot be decomposed as a tensor product of a type II inclusion with a fixed type III factor, although the λ-lattices of the type III and type II inclusions coincide. (This remark can be also obtained from the results of Ge [7]).

Lemma 3.4. With the assumptions of Theorem 3.1 (d), ℳ̂ and 𝒩̂ are both isomorphic to L(𝔽_{∞}).

Proof. Since N̂ = 𝒩^{ϕ} is a factor (Theorem 3.1 (d)) and 𝒩 = M^{Γ} is isomorphic to T_{q^{2}} , 𝒩̂ ≅ L(𝔽_{∞}) (see Theorem 2.3). Similarly, since ℳ ≅ T_{q^{2}} and ℳ̂ = ℳ^{ϕ∘ℰ} is a factor, ℳ̂ ≅ L(𝔽_{∞}).

For a general finite dimensional representation π the factor map structure associated with the type III inclusion ℳ ⊇ 𝒩 (see [9]) can be described as follows. Decompose

π = n_{1} · π_{ℓ_{1}} ⊕ ⋯ ⊕ n_{k} · π_{ℓ_{k}}

into multiples of irreducible spin ℓ representations π_{ℓ} (see [10]) with ℓ_{i} ≠ ℓ_{j} (i ≠ j).

Set X := {1, 2, …, k}×[0, − log q^{2}) and F_{t}(j, s) := (j, s+t), and let (X_{ℳ} , F_{t}^{ℳ} ) and (X_{𝒩} , F_{t}^{𝒩} ) be the flows of weights associated with ℳ and 𝒩, respectively, which are both identified with ([0, − log q^{2}), translation by t). Let us define two factor maps

π^{ℳ} : X → X_{ℳ} = [0, − log q^{2}), π^{𝒩} : X → X_{𝒩} = [0, − log q^{2})

by

π^{ℳ} (j, s) = s − ℓ_{j} log q^{2} mod − log q^{2}, π^{𝒩} (j, s) = s.

These π^{ℳ} , π^{𝒩} describe the factor map structure. As a consequence, we get:

Remark 3.5. The inclusion ℳ ⊇ 𝒩 is of essentially type II if and only if all the set

{ℓ : π_{ℓ} occurs in the decomposition of π}

either consists entirely of half-integers or consists entirely of integers. Under this assumption, all statements in Theorem 3.1 still hold, with the exception of statement (b) (the resulting inclusion is of course no longer irreducible).

This means that our method of constructing irreducible type II_{1} subfactors can be applied to the special standard λ-lattices arising from representations of the quantum group SU_{q}(2) (see [2]).

We are now ready to prove our main result:

Theorem 3.6. For every λ ∈ ℐ, there exists an irreducible subfactor N of M = L(𝔽_{∞}) of index λ, so that N ≅ L(𝔽_{∞}).

Proof. The case λ ∈ ℐ ∩ [0, 4] was obtained by Radulescu in [16].

Let λ > 4, and choose q so that λ = (q + q^{−1})^{2}. Let π be the fundamental representation of SU_{q}(2). By Theorem 3.1 (e) and Lemma 3.4, for each 0 < q < 1, we obtain an irreducible inclusion of type II_{1} factors

L(𝔽_{∞}) ≅ 𝒩̂ = 𝒩^{ϕ} ⊂ ℳ^{ϕ∘ℰ} = ℳ̂ ≅ L(𝔽_{∞})

of index (q + q^{−1})^{2}.

Remark 3.7. In the same way, we see that any irreducible representation π of SU_{q}(2) gives rise to an irreducible subfactor 𝒩̂ ⊂ ℳ̂, 𝒩̂ ≅ ℳ̂ ≅ L(𝔽_{∞}), of index (dim_{q} π)^{2}.

Appendix.

We give here a concrete system of generators for the von Neumann algebra M^{Γ} appearing in the proof of the main theorem. This gives also a concrete example illustrating the proof of Theorem 2.1 works.

Let H be a Hilbert space, given as a direct sum

H = ⨁_{k} H_{k} ⊗ K_{k},

with H_{k} and K_{k} finite-dimensional or infinite-dimensional. Let Â be the subalgebra of B(H) given by

⨁_{k} B(H_{k}) ⊗ Id_{K_{k}}.

For each k, let θ_{k} be a normal faithful state on B(K_{k}) given by

θ_{k}(T) = Tr(D_{k}T),

where D_{k} ∈ B(K_{k}) is a fixed positive matrix. Then

E = ⨁_{k} id ⊗θ_{k}(·)1_{B(K_{k})}

is a normal faithful conditional expectation from B(H) onto Â.

Let d_{k} = dim K_{k}. Choose a basis for K_{k}, so that D_{k} is diagonal, with eigenvalues λ_{1}(k), …, λ_{d(k)}(k). Write e_{ij}(k), 1 ≤ i, j ≤ d(k) for the matrix units generating B(K_{k}), associated to this basis.

Let l_{ij}^{k}(p), p, k = 1, 2, …, 1 ≤ i, j ≤ d(k) be isometries satisfying l_{ij}^{k}(p) · l_{i′j′}^{k′}(p′) = δ_{pp′}δ_{ii′}δ_{jj′}δ_{kk′}1, and generating the Toeplitz extension 𝒯 of the Cuntz algebra. Let ψ be the canonical vacuum state on 𝒯 (see [25]). Consider the tensor product conditional expectation ψ ⊗ E on P = B(L^{2}(𝒯, ψ)) ⊗ B(H).

Let

L_{p} = ∑_{k} ∑_{i,j=1}^{d(k)} \sqrt{λ_{i}(k)}l_{ij}^{k}(p) ⊗ (1_{B(H_{k})} ⊗ e_{ij}(k)) ∈ P

12 DIMITRI SHLYAKHTENKO AND YOSHIMICHI UEDA

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Figure 3.1. Â ⊂ B(H) and L. Here the dotted squares are matrix units of B(H); large solid-line squares are matrix units in B(K_{j}); A ∈ B(H_{1}), B ∈ B(H_{2}), C ∈ B(H_{3}), and d(1) = 3, d(2) = 4 and d(3) = 1.

(see Figure 3.1). Then for all T ∈ B(H), we get

L_{p}^{∗}T L_{q} = δ_{pq} (∑_{k} ∑_{i,j=1}^{d(k)} \sqrt{λ_{i}(k)}l_{ij}^{k}(p) ⊗ (1_{B(H_{k})} ⊗ e_{ij}(k)))^{∗} T

· ∑_{k′} ∑_{i′,j′=1}^{d(k′)} \sqrt{λ_{i′}(k′)}l_{i′j′}^{k′}(p) ⊗ (1_{B(H_{k′})} ⊗ e_{i′j′}(k))

= δ_{pq} ∑_{k} ∑_{i,j}^{d(k)} λ_{i}(k)(1_{B(H_{k})} ⊗ e_{ji}(k))T (1_{B(H_{k})} ⊗ e_{ij}(k))
= δ_{pq}E(T).

Hence by [19, Theorem 2.3], {L_{p} : p = 1, 2, … } are ∗-free with respect to ψ ⊗ E : P → B(H) from B(H) with amalgamation over Â. Moreover, since L_{p}^{∗}aL_{q} = δ_{pq}a for all a ∈ Â, we get that W^{∗}(Â, L_{p} + L_{p}^{∗} : p ≥ 1) ≅ W^{∗}(L_{p} + L_{p}^{∗} : p ≥ 1) ⊗ Â ≅ L(𝔽_{∞}) ⊗ Â (cf. [20, Examples 3.2 and 3.3(b)]). We conclude that M = W^{∗}(B(H), L_{p} + L_{p}^{∗}) ≅ (L(𝔽_{∞}) ⊗ Â, τ ⊗ id) ∗Â (B(H), E).

Now, fix a minimal projection p ∈ B(H_{1}) ⊗ e_{11}(1). Then the compression pMp is generated as a von Neumann algebra by the entries of {L_{p} +L_{p}^{∗} : p ≥ 1} viewed as matrices. These entries have the form \sqrt{λ_{i}(k)}l_{ij}^{k}(p) + \sqrt{λ_{j}(k)}(l_{ji}^{k}(p))^{∗}, k = 1, 2, …, 1 ≤ i ≤ j ≤ d(k), p ≥ 1. The algebra generated by such elements is isomorphic to a free Araki-Woods factor (see [18, Section 5]); the classifying Sd invariant for this type of factors is the multiplicative subgroup of ℝ generated by the ratios λ_{i}(k)/λ_{j}(k), k = 1, 2, …, 1 ≤ i, j ≤ d(k).

We now turn to the particular case of Â arising from a quantum group action, where one can describe the inclusion B(L^{2}(A)) ⊇ Â and the conditional expectation E explicitly. We omit the details of the computation, but give only the consequence for the reader’s convenience.

Let w_{ij}^{(ℓ)}, i, j ∈ I_{ℓ} = {−ℓ, ℓ + 1, …, ℓ}, be the matrix elements of the spin ℓ (∈ (1/2)ℕ_{0}) representation as given in [10]. The Peter-Weyl type theorem says that the set

\{(\frac{1 − q^{2(2ℓ+1)}}{(1 − q^{2})q^{2(ℓ+i)}} )^{1/2} Λ_{h}(w_{ij}^{ℓ})\}_{i,j∈I_{ℓ},ℓ∈(1/2)ℕ_{0}}

forms a complete orthonormal basis of L^{2}(A). Let us denote the matrix units with respect to the basis by e_{(i_{1},j_{1},ℓ_{1})(i_{2},j_{2},ℓ_{2})}’s. The set of elements of the form:

Ê_{i_{1}i_{2}}^{(ℓ)} = ∑_{j∈I_{ℓ}} e_{(i_{1},j,ℓ)(i_{2},j,ℓ)}

forms a complete system of matrix units of Â, and the conditional expectation E is computed as follows:

E (e_{(i_{1},j_{1},ℓ_{1})(i_{2},j_{2},ℓ_{2})}) = δ_{(i_{1},ℓ_{1})(i_{2},ℓ_{2})} \frac{1}{1 + q^{2} + ⋯ + (q^{2})^{2ℓ_{1}}} q^{2(ℓ_{1}−i_{1})}Ê_{j_{1}j_{2}}^{(ℓ_{1})}.

In other words, in the notation of the first part of the appendix, we have dim H_{ℓ} = dim K_{ℓ} = d(ℓ) = 2ℓ + 1, and the eigenvalues of θ_{ℓ} are given by

λ_{i}(ℓ) = \frac{1}{1 + q^{2} + ⋯ + (q^{2})^{2ℓ}} q^{2(ℓ−i)}.

Since the Sd invariant of the associated type III factor is the multiplicative subgroup of positive reals generated by the ratios λ_{i}(ℓ)/λ_{j}(ℓ), we see that in this case Sd = q^{2ℤ}, i.e., the factor has type III_{q^{2}}.

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Department of Mathematics, UCLA, Los Angeles, CA 90095, USA.

  E-mail address: shlyakht@math.ucla.edu

Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan.

  E-mail address: ueda@math.sci.hiroshima-u.ac.jp