CANONICAL EXTENSION OF ENDOMORPHISMS OF TYPE III FACTORS

MASAKI IZUMI

Abstract. We extend the notion of the canonical extension of automorphisms of type III factors to the case of endomorphisms with finite statistical dimensions. Following the automorphism case, we introduce two notions for endomorphisms of type III factors: modular endomorphisms and Connes-Takesaki modules. Several applications to compact groups of automorphisms and subfactors of type III factors are given from the viewpoint of ergodic theory.

1. Introduction

The canonical extension of an automorphism of a type III factor gives rise to a natural isomorphism from the automorphism group of the type III factor into that of the crossed product by the modular automorphism group. Although the notion itself had been known to specialists before, (e.g. [12]), the first systematic analysis of the canonical extension was proposed and accomplished by U. Haagerup and E. Størmer. In [21], [22], they introduced several classes of automorphisms of von Neumann algebras and investigated their structure using the canonical extension.

One of the purposes of the present notes is to generalize the notion of the canonical extension to an endomorphism whose image has a finite index [38], [60], [41]. Our analysis is based on techniques of the common continuous decomposition of inclusions and endomorphisms of type III factors, mainly developed by Ph. Loi [47], T. Hamachi-H. Kosaki [25], [26], [27], R. Longo [49], [50], and H. Kosaki-R. Longo [44]. In [63], J. E. Roberts introduced an action of the dual object of a compact group on a von Neumann algebra, now called a Roberts action, which is a functor from the category of the finite dimensional unitary representations to that of the endomorphisms of the von Neumann algebra. As several invariants of discrete group actions on type III factors can be captured by the canonical extension [72], [65], it is natural to expect that the canonical extension of endomorphisms plays an important role in the analysis of compact group actions through the dual Roberts actions. Indeed, we obtain several new results on minimal actions of compact groups on type III factors. Study of minimal actions of compact groups was initiated by S. Popa and A. Wassermann using subfactor techniques developed in [61]. Interested readers are recommended to consult [62] for related topics.

We introduce two new notions for endomorphisms corresponding to extended modular automorphisms and the Connes-Takesaki module in the automorphism case [11]: modular endomorphisms and the Connes-Takesaki module of endomorphisms. As in the case of extended modular automorphisms, a modular endomorphism carries a unitary group valued cocycle of the flow of weights, and it gives rise to a Roberts action of the dual object of the essential range of the cocycle, the notion developed by G. Mackey and R. Zimmer [56], [80]. The dual actions of Roberts actions consisting of modular endomorphisms provide a large variety of new examples of minimal actions of compact groups on type III_{0} factors. Also, we give a strong constraint for the possible types of an algebra-fixed point algebra pair for a minimal action of a compact semisimple Lie group using these two notions.

Work supported by Mathematical Sciences Research Institute.

In [80], Zimmer introduced the notion of extensions of ergodic transformation groups with relatively discrete spectrum, which is a relative version of the notion of ergodic transformations with pure point spectrum. It turns out that a non-commutative generalization of this notion is closely related to modular endomorphisms (Theorem 5. 11), which may be expected because both objects are described in terms of cocycles of ergodic transformation groups with values in compact groups. In particular, we characterize a subfactor, not necessarily of a finite index, coming from an ergodic extension of the flow of weights in terms of a group-subgroup subfactor (c.f. [23], [24], [25], [26], [71]).

Our original purpose for introducing the canonical extension of endomorphisms is to settle a problem left unsolved in [31]. When a type III factor and its subfactor with a finite index have the common flow of weights, their “type II principal graph” makes sense via the common crossed product decomposition [47], [44]. In [31], we characterized the situation where the type II principal graph does not coincide with the principal graph of the original subfactor for the case of III_{λ}, λ ≠ 0 factors. Namely, the two graphs do not coincide if and only if some power of the canonical endomorphism for the inclusion contains a non-trivial modular automorphism. It would be tempting to conjecture that the same statement should be true in the type III_{0} case if the modular automorphism is replaced with an extended modular automorphism. However, it turns out that the right counterpart for our purpose is a modular endomorphism (Theorem 3.7). One can observe a similar phenomenon in the recent work of T. Masuda [57] (c.f. [43]).

Final part of this work was finished while the author stayed at MSRI, and he would like to thank them for their hospitality. He also would like to thank T. Hamachi and H. Kosaki for stimulating discussions. Some of the main results in this paper were announced in [32].

2. Canonical Extension

First, we briefly summarize notation used in this paper. Our basic references are [67] and [68] for Tomita-Takesaki theory, [41] for index theory of type III factors,

[35] for sector theory and infinite index inclusions, and [80] for cocycles of ergodic transformation groups. Undefined terms and notations used in this paper should be found in these references. We always assume that von Neumann algebras have separable preduals, Hilbert spaces are separable, and locally compact groups are second countable. Automorphisms and endomorphisms of von Neumann algebras are always assumed to be ∗-preserving and unital. For a von Neumann algebra M, we denote by Aut(M), Int(M), and End(M) the sets of automorphisms, inner automorphisms, and endomorphisms respectively. We denote by Out(M) and Sect(M) the sets of unitary equivalence classes of automorphisms and endomorphisms respectively. A member in Sect(M) is called a sector. For ρ, σ ∈ End(M), the intertwiner space from ρ to σ is defined by

(ρ, σ) = {V ∈ M; V ρ(x) = σ(x)V }.

When M is a factor and ρ is irreducible, that is, ρ(M)′ ∩ M = ℂ1, (ρ, σ) is a Hilbert space with the inner product

⟨V_{1}, V_{2}⟩1 = V_{2}^{∗}V_{1}.

Every action of a topological group on a von Neumann algebra M is assumed to be continuous with respect to the u-topology in Aut(M). For a group action α on M, M^{α} denotes the fixed point subalgebra of M under the action α. When M acts on a Hilbert space H, we denote by π_{α} the representation of M on L^{2}(G, H) defined by

π_{α}(x)ξ(g) = α_{g^{−1}}(x)ξ(g), ξ ∈ L^{2}(G, H).

We denote by l(g) and r(g) the left and right regular representations of G respectively. For a weight φ on M or an operator valued weight from M to a subalgebra, we denote by 𝔪_{φ} and 𝔫_{φ} the domain of φ and the left ideal corresponding to 𝔪_{φ} as usual.

Let φ be a faithful normal semifinite weight on M. We denote by M̃ = M ⋊_{σ^{φ}} ℝ the crossed product of M by the modular automorphism group {σ_{t}^{φ}}_{t∈ℝ}, which is the von Neumann algebra generated by π_{σ^{φ}}(M) and the implementing one-parameter unitary group {λ^{φ}(t)}_{t∈ℝ}, where λ^{φ}(t) = 1⊗l(t). We often omit π_{σ^{φ}} when there is no possibility of confusion. We denote by θ and τ the dual action of σ^{φ} and the natural trace constructed from the dual weight of φ on M̃ and the generator of {λ^{φ}(t)}_{t∈ℝ}. Then, the triple (M̃, θ, τ) does not depend on the choice of φ under the following identification

λ^{ψ}(t) = [Dψ : Dφ]_{t}λ^{φ}(t),

where ψ is another faithful normal weight on M and {[Dψ : Dφ]_{t}}_{t∈ℝ} is the Connes cocycle derivative. As for the relationship between Aut(M) and Aut(M̃), we have the following [22, Proposition 12.1]:

Proposition 2.1. Let M be a von Neumann algebra and φ be a faithful normal semifinite weight on M. Then, for every automorphism α ∈ Aut(M), there exists a unique automorphism α̃ ∈ Aut(M) satisfying

α̃(x) = α(x), x ∈ M,
α̃(λ^{φ}(t)) = [Dφ · α^{−1} : Dφ]_{t}λ^{φ}(t).

Moreover, the map α ↦ α̃ is a homomorphism from Aut(M) into Aut(M̃).

α̃ is called the canonical extension of α. The goal of this section is to introduce the canonical extension of endomorphisms with finite statistical dimensions generalizing the automorphism case.

Let M be an infinite factor. We denote by End(M)_{0} the set of endomorphisms of M with finite statistical dimensions. For ρ ∈ End(M)_{0}, we denote by E_{ρ}, ϕ_{ρ}, and d(ρ), the minimal conditional expectation from M onto the image of ρ [29], the standard left inverse of ρ, and the statistical dimension of ρ, respectively: that is, ϕ_{ρ} = ρ^{−1} ·E_{ρ} and d(ρ) = [M : ρ(M)]_{0}^{1/2}, where [M : ρ(M)]_{0} = Index E_{ρ} is the minimum index of M ⊃ ρ(M). We denote by Sect(M)_{0} the unitary equivalence classes of the endomorphisms in End(M)_{0}. Sect(M) has three natural operations, forming a direct sum, composing two endomorphisms (regarded as a product), and taking conjugation. The statistical dimensions are additive, multiplicative, and invariant under these three operations on Sect(M)_{0} respectively [29], [50], [39], [53]. Let M ⊃ N be an inclusion of type III factors with a finite index. Then, we regard Ñ as a subalgebra of M̃ through the identification λ^{ω·E}(t) = λ^{ω}(t), where E is the minimal expectation from M onto N and ω is a faithful normal semifinite weight on N. We denote by Ẽ the extension of E to M̃ leaving λ^{φ_{0}·E}(t) fixed, which is a normal conditional expectation from M̃ onto Ñ [44]. Even when M ⊃ N has an infinite index, if E is a unique normal conditional expectation from M onto N, we use the same convention.

Before introducing the canonical extension, we need some preparation.

Definition 2.2. Let M be an infinite factor. A pair (φ, ρ) consisting of a faithful normal semifinite weight φ on M and ρ ∈ End(M)_{0} is said to be an invariant pair if the following hold:

φ · ρ = d(ρ)φ,
φ · E_{ρ} = φ.

Note that the above conditions are equivalent to d(ρ)φ·ϕ_{ρ} = φ, thanks to ρ·ϕ_{ρ} = E_{ρ} and ϕ_{ρ} · ρ = id.

Lemma 2.3. Let M be a type III factor. Then, the following hold:

(1) If φ is a faithful normal semifinite weight on M and ρ ∈ End(M)_{0}, then φ · ϕ_{ρ} is a faithful normal semifinite weight on M.

(2) If (φ, ρ) is an invariant pair, then ρ commutes with σ_{t}^{φ} for t ∈ ℝ.

(3) Let ψ be a dominant weight (see [11] for the definition). Then, for every sector in Sect(M)_{0}, there exists a representative ρ ∈ End(M)_{0} such that (ψ, ρ) is an invariant pair.

(4) Let ψ be a dominant weight on M and ρ ∈ End(M)_{0}. If (ψ, ρ) is an invariant pair, then there exist irreducible ρ_{i} ∈ End(M)_{0} and isometries V_{i} ∈ (ρ_{i}, ρ) i = 1, 2, ⋯, n such that each V_{i} belongs to the centralizer M_{ψ} and

ρ(x) = ∑_{i=1}^{n} V_{i}ρ_{i}(x)V_{i}^{∗}, x ∈ M.

Moreover, each (ψ, ρ_{i}) is an invariant pair.

(5) Let ψ be a dominant weight and ρ_{1}, ρ_{2} ∈ End(M)_{0}. If (ψ, ρ_{1}) and (ψ, ρ_{2}) are invariant pairs, then (ρ_{1}, ρ_{2}) is in the centralizer M_{ψ}.

Proof. (1). Since φ · ρ^{−1} is a faithful normal semifinite weight on ρ(M), so is φ · ϕ_{ρ} = φ · ρ^{−1} · E_{ρ} on M.

(2). Let N = ρ(M) and φ_{0} be the restriction of φ to N, which is a faithful normal semifinite weight on N by assumption. Then, we have

φ_{0} · ρ = d(ρ)φ,
φ_{0} · E_{ρ} = φ.

Regarding ρ as an isomorphism from M to N, we get

ρ · σ_{t}^{φ} = σ_{t}^{φ_{0}} · ρ = σ_{t}^{φ} · ρ.

(3). This is [35, Lemma 2.12].

(4). Let N = ρ(M) and ψ_{0} be the restriction of ψ to N. Then, we have common continuous decomposition [47], [44]

M = M_{ψ} ⋊_{θ_{0}} ℝ ⊃ N = N_{ψ_{0}} ⋊_{θ_{0}} ℝ,

such that

M ∩ N′ = (M_{ψ} ∩ N′_{ψ_{0}})^{θ_{0}}.

Therefore, each non-zero projection p ∈ M ∩ N′ is equivalent to 1 in M_{ψ} (more strongly in M_{ψ}^{θ_{0}}), and there exist isometries V_{i} ∈ M_{ψ}, i = 1, 2, ⋯, n such that each p_{i} = V_{i}V_{i}^{∗} is a minimal projection in M ∩ N′ and {p_{i}}_{i=1}^{n} forms a partition of unity. Setting ρ_{i}(x) = V_{i}^{∗}ρ(x)V_{i}, we get the first assertion. Since the minimal expectation for p_{i}Mp_{i} ⊃ Np_{i} is given by E_{ρ}(x)p_{i}/E_{ρ}(p_{i}), we have

E_{ρ_{i}}(x) = \frac{d(ρ)}{d(ρ_{i})}V_{i}^{∗}ρ · ϕ_{ρ}(V_{i}xV_{i}^{∗})V_{i}. = \frac{d(ρ)}{d(ρ_{i})}ρ_{i} · ϕ_{ρ}(V_{i}xV_{i}^{∗}), x ∈ M,

thanks to the local index formula for the minimal conditional expectation [29] and the fact that Ad(V_{i}^{∗}) is an isomorphism from p_{i}Mp_{i} ⊃ N_{p_{i}} to M ⊃ ρ_{i}(M). This implies

ϕ_{ρ_{i}}(x) = \frac{d(ρ)}{d(ρ_{i})}ϕ_{ρ}(V_{i}xV_{i}^{∗}), x ∈ M.

Thus,

d(ρ_{i})ψ · ϕ_{ρ_{i}} = d(ρ)ψ · ϕ_{ρ}(V_{i} · V_{i}^{∗}) = φ(V_{i} · V_{i}^{∗}).

Since V_{i} is in the centralizer of ψ, this implies that (ψ, ρ_{i}) is an invariant pair.

(5). Thanks to (4), the general case is reduced to the case where ρ_{1} and ρ_{2} are irreducible, and we make this assumption. If (ρ_{1}, ρ_{2}) = 0, we have nothing to prove, and so we assume ρ_{1} = Ad(u) · ρ_{2} for some unitary u ∈ M. Since σ_{t}^{ψ} commutes with ρ_{1} and ρ_{2}, σ_{t}^{ψ}(u) is proportional to u, and there exists a scalar c > 0 such that σ_{t}^{ψ}(u) = c^{it}u. However, the KMS condition and ψ · ρ_{1} = ψ · ρ_{2} imply c = 1, which shows u ∈ M_{ψ}.

Now we introduce the canonical extension of an endomorphism.

Theorem 2.4. Let M be a type III factor and φ be a faithful normal semifinite weight on M. Then, for every ρ ∈ End(M)_{0}, there exists a unique endomorphism ρ̃ ∈ End(M̃) satisfying

ρ̃(x) = ρ(x), x ∈ M
ρ̃(λ^{φ}(t)) = d(ρ)^{it}[Dφ · ϕ_{ρ} : Dφ]_{t}λ^{φ}(t).

Under the identification λ^{ψ}(t) = [Dψ : Dφ]_{t}λ^{φ}(t), ρ̃ does not depend on the choice of the weight φ.

Proof. Let ψ be a dominant weight. First we assume that (ψ, ρ) is an invariant pair, and N and ψ_{0} are as in the proof of Lemma 2.3, (4). Thanks to Lemma 2.3, (2), we may consider ρ an isomorphism from M onto N intertwining σ^{ψ} and σ^{ψ_{0}}. Therefore, ρ extends to an isomorphism ρ̃ from M̃ onto Ñ sending λ^{ψ}(t) to λ^{ψ_{0}}(t). Since ψ satisfies ψ = ψ_{0} · E_{ρ}, we may regard Ñ as a subalgebra of M̃ identifying λ^{ψ_{0}}(t) with λ^{ψ}(t). We show that ρ̃ satisfies the required property. Indeed,

ρ̃(λ^{φ}(t)) = ρ̃([Dφ : Dψ]_{t}λ^{ψ}(t)) = ρ([Dφ : Dψ]_{t})λ^{ψ}(t).

Regarding ρ as an isomorphism from M onto N again, we have

ρ([Dφ : Dψ]_{t}) = [Dφ · ρ^{−1} : Dψ · ρ^{−1}]_{t} = [Dφ · ρ^{−1} · E_{ρ} : Dψ · ρ^{−1} · E_{ρ}]_{t},

and so

ρ̃(λ^{φ}(t)) = [Dφ · ϕ_{ρ} : Dψ · ϕ_{ρ}]_{t}λ^{ψ}(t) = d(ρ)^{it}[Dφ · ϕ_{ρ} : Dφ]_{t}λ^{φ}(t).

Now, we treat the general case. Thanks to Lemma 2.3, (3), it suffices to show the statement for ρ_{1} = Ad(u) · ρ, where u is a unitary in M, and ρ and ψ are as before. Indeed,

Ad(u) · ρ̃(λ^{φ}(t)) = d(ρ)^{it}uσ_{t}^{φ·ϕ_{ρ}}(u^{∗})λ^{φ·ϕ_{ρ}}(t)
= d(ρ)^{it}[Dφ · ϕ_{ρ} · Ad(u^{∗}) : Dφ · ϕ_{ρ}]_{t}λ^{φ·ϕ_{ρ}}(t)
= d(ρ)^{it}[Dφ · ϕ_{ρ_{1}} : Dφ]_{t}λ^{φ}(t).

Therefore, ρ̃_{1} := Ad(u) · ρ̃ has the desired property.

The same formula of the Connes cocycle as above also appears in [52].

We call the above ρ̃ the canonical extension of ρ. When, α is an automorphism, the left inverse ϕ_{α} is nothing but the inverse of α. Therefore, our definition of the canonical extension generalizes that in the automorphism case.

Let M ⊃ N be properly infinite von Neumann algebras. Then, the standard representation of M is automatically standard for N and the product of two modular conjugation J_{N}J_{M} makes sense. The canonical endomorphism γ ∈ End(M) for the inclusion M ⊃ N is defined by γ(x) = J_{N}J_{M}xJ_{M}J_{N} ∈ N [48]. In general, modular conjugations depend on the choice of natural cones. However, a different choice of modular conjugations amounts to only a perturbation of γ by an inner automorphism of N. Therefore, we call any endomorphism in End(M) of the form Ad(u) · γ with a unitary u ∈ N the canonical endomorphism as well.

Proposition 2.5. Let M be a type III factor and ρ, μ ∈ End(M)_{0}. Then, the following hold:

(1) (ρ, μ) ⊂ (ρ̃, μ̃).

(2) ρ̃ · μ = ρ̃· μ̃.

(3) Let N be a subfactor of M with a finite index and γ be the canonical endomorphism for M ⊃ N. Then, γ̃ is the canonical endomorphism for M̃ ⊃ Ñ.

(4) τ · ρ̃ = d(ρ)τ.

(5) ρ̃· θ_{t} = θ_{t} · ρ̃.

Proof. In the proof of Theorem 2.4, we have already shown that for ρ ∈ End(M)_{0} and a unitary u ∈ M, we have

{ \atop (Ad(u) · ρ)} = Ad(u) · ρ̃.

Thus, in order to prove the statements we may replace ρ and μ with unitary equivalent endomorphisms. Thanks to Lemma 2.3, (3), we may and do assume that (ψ, ρ) and (ψ, μ) are invariant pairs, where ψ is a dominant weight. Therefore, we have

ρ̃(λ^{ψ}(t)) = μ̃(λ^{ψ}(t)) = λ^{ψ}(t).

In this situation, (2), (4), and (5) are obvious and (1) follows from Lemma 2.3, (5).

To prove (3), we consider the common continuous decomposition of M ⊃ N. We may and do assume that ψ is of the form ψ_{0} · E, where ψ_{0} is a dominant weight on N and E is the minimal expectation from M to N. Then, we have the common continuous decomposition

M = M_{ψ} ⋊_{θ_{0}} ℝ ⊃ N = N_{ψ_{0}} ⋊_{θ_{0}} ℝ.

Let τ_{0} be the trace whose dual weight is ψ. Then, thanks to [49, Section 4], the canonical endomorphism of M_{ψ} ⊃ N_{ψ_{0}} scales τ_{0} by [M : N]_{0}. Let J_{M_{ψ}} and J_{N_{ψ_{0}}} be the modular conjugations of M_{ψ} and N_{ψ_{0}}, and γ_{0} be the corresponding canonical endomorphism of M_{ψ} ⊃ N_{ψ_{0}}. To define the canonical endomorphism of M ⊃ N and M̃ ⊃ Ñ, we utilize the modular objects of these algebras naturally coming from the modular objects of M_{ψ} and N_{ψ_{0}} through the crossed products as computed in [18, Lemma 2.8]. Let γ and γ_{1} be such canonical endomorphisms of M ⊃ N and M̃ ⊃ Ñ respectively. Then, (ψ, γ) is an invariant pair [31, Section 3]. On the other hand, applying [31, Lemma 3.1] twice and using [18, Lemma 2.8] for the computation of Δ_{ψ} and Δ_{ψ_{0}}, we know that γ_{1} is an extension of γ leaving λ^{ψ}(t) invariant. Therefore, γ_{1} coincides with the canonical extension of γ.

3. Modular endomorphisms

Let M be a type III factor and α be an automorphism of M. In [22, Proposition 5.4], it was shown that α̃ is inner if and only if α is the composition of an inner automorphism with an extended modular automorphism. We adopt this property as the definition of an endomorphism counterpart of an extended modular automorphism.

Definition 3.1. Let M be a type III factor and ρ ∈ End(M)_{0}. We say that ρ is a modular endomorphism if ρ̃ is an inner endomorphism, that is, there exist isometries {V_{i}}_{i=1}^{n} ⊂ M̃ with mutually orthogonal ranges and ∑_{i=1}^{n} V_{i}V_{i}^{∗} = 1 such that

ρ̃(x) = ∑_{i=1}^{n} V_{i}xV_{i}^{∗}, x ∈ M̃.

The system of isometry {V_{i}}_{i=1}^{n} satisfying the above condition is called an implementing system for ρ̃.

We denote by End(M)_{m} the set of modular endomorphisms of M. Thanks to Proposition 2.5, (1), an endomorphism unitary equivalent to a modular endomorphism is again a modular endomorphism, and so it makes sense to call a sector modular if its representatives are modular endomorphisms. We denote by Sect(M)_{m} the set of sectors of modular endomorphisms. It is also easy to see from Proposition 2.5, (1), (2) that End(M)_{m} is closed under forming a direct sum and a product of finitely many elements. What is not clear for the moment is whether End(M)_{m} is closed under conjugation and irreducible decomposition, which will be shown later. The number n of the isometries of the implementing system in Definition 3.1 is nothing but d(ρ), which also will be shown later.

Let M be as above and Z(M̃) be the center of M̃. Then, the restriction of θ to Z(M̃) is an ergodic action of the real number group ℝ. Thus, by Mackey’s point realization theorem [54], there exist a standard Borel space X_{M}, a probability measure μ_{M} on X_{M}, and an ergodic flow F^{M} on (X_{M}, μ_{M}) such that Z(M̃) = L^{∞}(X_{M}, μ_{M}) and θ_{t}(f)(ω) = f(F_{−t}^{M}ω) for f ∈ L^{∞}(X_{M}, μ_{M}), ω ∈ X_{M}, t ∈ ℝ. (For simplicity, we often omit F^{M} and denotes F_{−t}^{M}ω just by ω · t). F^{M} is called the smooth flow of weights [11].

Let K be a compact group. A Borel map c : X_{M} × ℝ ⟶ K is said to be a cocycle of F^{M} if for fixed s, t ∈ ℝ, the cocycle relation

c(ω, s)c(ω · s, t) = c(ω, s + t)

holds for almost all ω ∈ X_{M}. Two cocycles agree on the outside of a null set are identified as usual. We denote by Z^{1}(F^{M}, K) the set of all K-valued cocycles of F^{M}. Two cocycles c and c′ are said to be equivalent (or cohomologous) if there exists a Borel map a : X_{M} ⟶ K such that for fixed t ∈ ℝ

c(ω, t) = a(ω)c′(ω, t)a(ω · t)^{−1}

holds for almost all ω ∈ X_{M}. A cocycle equivalent to a constant function e ∈ K is said to be a coboundary. We denote by H^{1}(F^{M}, K) the set of equivalence classes of K-valued cocycles of F^{M}. For the significance of cocycles from the viewpoint of Mackey’s notion of virtual groups, readers are refered to [56] and [80].

As Connes and Takesaki showed that the group of extended modular automorphisms divided by the inner automorphism group is isomorphic to H^{1}(F^{M}, U(1)) [11], we show that Sect(M)_{m} is “isomorphic” to the union ⋃_{n=1}^{∞} H^{1}(F^{M}, U(n)) of the unitary group U(n)-valued cohomology classes.

Let ρ ∈ End(M)_{m} and {V_{i}}_{i=1}^{n} be an implementing system for ρ̃. Since (id, ρ̃) is globally preserved by θ, c(t)_{ij} := V_{i}^{∗}θ_{t}(V_{j}) belongs to Z(M̃). c(t) = (c(t)_{ij}) is regarded as a U(n)-valued Borel function on X_{M} × ℝ.

Lemma 3.2. Let ρ and c(t) be as above. Then,

(1) c is a U(n)-valued cocycle of F^{M}, whose cohomology class depends only on the sector of ρ.

(2) Let c′ be a cocycle equivalent to c. Then, there exists an implementing system {V′_{i}}_{i=1}^{n} for ρ̃ such that c′(t)_{ij} = V′_{i}^{∗}θ_{t}(V′_{j}).

Proof. (1). The cocycle relation of c follows from

∑_{j=1}^{n} c(s)_{ij}θ_{s}(c(t)_{jk}) = ∑_{j=1}^{n} V_{i}^{∗}θ_{s}(V_{j}V_{j}^{∗}θ_{t}(V_{k})) = c(s + t)_{ik}.

Let {W_{i}}_{i=1}^{m} ⊂ M̃ be another implementing system for ρ̃, and set a_{ij} = V_{i}^{∗}W_{j}. Since a(t) = (a(t)_{ij}) is a matrix valued function that is unitary, we have n = m. Setting b(t)_{ij} = W_{i}^{∗}θ_{t}(W_{j}), we get

c(t)_{ij} = V_{i}^{∗}θ_{t}(V_{j}) = ∑_{k=1,l}^{n} V_{i}^{∗}W_{k}W_{k}^{∗}θ_{t}(W_{l}W_{l}^{∗}V_{j}) = ∑_{k=1,l}^{n} a_{ik}b(t)_{kl}θ_{t}(a_{jl}^{∗}),

which shows that b and c are equivalent. Let u be a unitary of M and ρ_{1} = Ad(u) · ρ. Then, {uV_{i}}_{i=1}^{n} is an implementing system for ρ̃_{1}. Since u is fixed by θ_{t}, ρ_{1} gives the same cohomology class as ρ.

(2). We take a U(n)-valued function f satisfying

c(t)_{ij} = ∑_{k,l=1}^{n} f_{ik}c′(t)_{kl}θ_{t}(f_{jl}^{∗}).

Then, V′_{i} := ∑_{j=1}^{n} f_{ji}V_{j} has a desired property.

We introduce a map

δ_{m} : Sect(M)_{m} ⟶ ⋃_{n=1}^{∞} H^{1}(F^{M}, U(n))

sending [ρ] to [c] in the above lemma.

As described in [80, Section 2], Z^{1}(F^{M}, U(n)) has natural three operations in analogous to the unitary representation theory of compact groups: direct sum c ⊕ c′, tensor product c ⊗ c′ and complex conjugate \overline{c}. H^{1}(F^{M}, U(n)) inherits these operations and we use the same notation for the cohomology classes as well.

Theorem 3.3. Let δ_{m} be as above and ρ_{1}, ρ_{2} ∈ End(M)_{m}. Then,

(1) δ_{m} is a bijection.

(2) We have δ_{m}([ρ_{1}] ⊕ [ρ_{2}]) = δ_{m}([ρ_{1}]) ⊕ δ_{m}([ρ_{2}]).

(3) We have δ_{m}([ρ_{1}][ρ_{2}]) = δ_{m}([ρ_{1}]) ⊗ δ_{m}([ρ_{2}]).

(4) [ρ_{1}] is the conjugate sector of [ρ_{2}] if and only if δ_{m}([ρ_{1}]) = \overline{δ_{m}([ρ_{2}])}. In particular, Sect(M)_{m} is closed under conjugation.

(5) δ_{m} is grade preserving in the sense that δ_{m}([ρ]) ∈ H^{1}(F^{M}, U(n)) if and only if d(ρ) = n.

Proof. (1). First we show that δ_{m} is injective. Let ρ_{1}, ρ_{2} ∈ End(M)_{m} satisfying δ_{m}([ρ_{1}]) = δ_{m}([ρ_{2}]). Then, thanks to Lemma 3.2, (2), there exist implementing systems {V_{i}^{(1)}}_{i=1}^{n} and {V_{i}^{(2)}}_{i=1}^{n} for ρ̃_{1} and ρ̃_{2} respectively such that they give the same cocycle. We set u = ∑_{i=1}^{n} V_{i}^{(1)}V_{i}^{(2)}^{∗}, which is a unitary in M̃^{θ} = M satisfying ρ_{1} = Ad(u) · ρ_{2}. This shows that δ_{m} is injective.

Next we show that δ_{m} is surjective. Let ψ be a dominant weight on M and M = M_{ψ} ⋊_{θ_{0}} ℝ be the continuous decomposition. We take the trace τ_{0} on M_{ψ} whose dual weight is ψ, and take the implementing one-parameter unitary group {u(s)}_{s∈ℝ} for θ_{0}. We assume that M_{ψ} acts on a Hilbert space H. Then, the Takesaki duality theorem [58, Chapter I] implies that there exists an isomorphism Φ from M̃ to M_{ψ} ⊗ B(L^{2}(ℝ)) such that

Φ(x) = π_{θ_{0}}(x), x ∈ M_{ψ},
Φ(u(s)) = 1 ⊗ l(s),
Φ(λ^{ψ}(t)) = 1 ⊗ m(t),

where l(s) is the left regular representation of ℝ and m(t) is the multiplication operator of e^{−its}. Moreover, θ_{t} is identified with (θ_{0,t} ⊗Ad(r(t))) under Φ, or more precisely we have Φ · θ_{t} = (θ_{0,t} ⊗ Ad(r(t))) · Φ. Using Φ, we identify the flow (Z(M̃), θ) with

(Z(M_{ψ} ⊗ B(L^{2}(ℝ))), θ_{0} ⊗ Ad(r)) ≅ (Z(M_{ψ}), θ_{0}).

Let c ∈ Z^{1}(F^{M}, U(n)) be a given cocycle, where L^{∞}(X_{M}, μ_{M}) is understood as the point realization of Z(M_{ψ}). We take a system of isometries {W_{i}}_{i=1}^{n} in M_{ψ} satisfying ∑_{i=1}^{n} W_{i}W_{i}^{∗} = 1 and set

W (t) = ∑_{i,j=1}^{n} c(t)_{ij}W_{i}θ_{0,t}(W_{j}^{∗}) ∈ M_{ψ}.

Then, W (t) is a unitary satisfying

W (s + t) = ∑_{i,j,k=1}^{n} c(s)_{ik}θ_{0,s}(c(t)_{kj})W_{i}θ_{0,s+t}(W_{j}^{∗})

= ∑_{i,j,k=1}^{n} c(s)_{ik}W_{i}θ_{0,s}(W_{k}^{∗}c(t)_{lj}W_{l}θ_{0,t}(W_{j}^{∗}))
= W (s)θ_{0,s}(W (t)),

which means that W (t) is a θ_{0}-cocycle. Since θ_{0} is stable [11, Theorem III.5.1], every θ_{0}-cocycle is a coboundary and there exists a unitary w such that W (t) = w^{∗}θ_{0,t}(w). Let V_{i} = wW_{i}. Then, V_{i} is an isometry in M_{ψ} satisfying ∑_{i=1}^{n} V_{i}V_{i}^{∗} = 1 and

θ_{0,t}(V_{i}) = ∑_{j=1}^{n} c(t)_{ji}V_{j}.

For x ∈ M_{ψ} we set,

ρ_{0}(x) = ∑_{i=1}^{n} V_{i}xV_{i}^{∗},

ϕ_{0}(x) = \frac{1}{n} ∑_{i=1}^{n} V_{i}^{∗}xV_{i},

and E_{0} = ρ_{0}·ϕ_{0}. Then, ρ_{0} is an endomorphism of M_{ψ} commuting with θ_{0,t}, ϕ_{0} is a left inverse of ρ_{0} satisfying nτ_{0} ·ϕ_{0} = τ_{0}, and E_{0} is a τ_{0}-preserving conditional expectation from M_{ψ} onto the image of ρ_{0}. We introduce ρ ∈ End(M) that is an extension of ρ_{0} to M leaving u(t) invariant. Indeed, such ρ exists because the following hold:

Φ(ρ_{0}(x)) = ∑_{i=1}^{n}(V_{i} ⊗ 1)Φ(x)(V_{i}^{∗} ⊗ 1), x ∈ M_{ψ},

Φ(u(t)) = ∑_{i=1}^{n}(V_{i} ⊗ 1)Φ(u(t))(V_{i}^{∗} ⊗ 1).

In a similar way, we define a left inverse ϕ of ρ and a conditional expectation E from M onto the image of ρ by the relations

Φ(ϕ(x)) = \frac{1}{n} ∑_{i=1}^{n}(V_{i}^{∗} ⊗ 1)Φ(x)(V_{i} ⊗ 1), x ∈ M,

and E = ρ · ϕ. Note that ρ, ϕ, and E are extensions of ρ_{0}, ϕ_{0}, and E_{0} respectively.

We claim that [M : ρ(M)]_{0} = n^{2}, ϕ = ϕ_{ρ}, E = E_{ρ}, and (ψ, ρ) is an invariant pair. Indeed, it suffices to show Index E = [M : ρ(M)]_{0} = n^{2}, and the other claims can be shown easily. We define T ∈ M_{ψ} by

T = \sqrt{n} ∑_{i=1}^{n} V_{i}V_{i}.

Then, for x ∈ M_{ψ} and u(t), we have

T^{∗}E(T xu(t)) = T^{∗}E_{0}(T x)u(t) = \frac{1}{n} ∑_{i,j=1}^{n} T^{∗}V_{i}V_{j}^{∗}T xV_{j}V_{i}^{∗}u(t)

= ∑_{i,j=1}^{n} V_{i}^{∗}V_{j}xV_{j}V_{i}^{∗}u(t) = xu(t),

which shows {T } is a one-element Pimsner-Popa basis and Index E = T^{∗}T = n^{2} [60], [78]. To show that E is minimal, it suffices to show T^{∗}xT = n^{2}E(x) for all x ∈ M ∩ρ(M)′ [78, Theorem 2.12.3]. Thanks to the Connes-Takesaki relative commutant theorem [11, Theorem II.5.1], for x ∈ M ∩ ρ(M_{ψ})′ we have

x_{ij} := V_{i}^{∗}xV_{j} ∈ M ∩ M′_{ψ} = Z(M_{ψ}),

and

x = ∑_{i,j=1}^{n} V_{i}x_{ij}V_{j}^{∗}.

Thus,

T^{∗}xT = n ∑_{i=1}^{n} x_{ii} = n^{2}E(x),

which shows the claims.

Since (ψ, ρ) is an invariant pair, ρ̃ is an extension of ρ leaving λ^{ψ}(t) invariant. Therefore, we have

Φ(ρ̃(x)) = ∑_{i=1}^{n}(V_{i} ⊗ 1)Φ(x)(V_{i}^{∗} ⊗ 1), x ∈ M̃,

which shows that ρ is a modular endomorphism carrying c.

(2) and (3) are easy, and (5) follows from the above proof.

(4). Assume that δ_{m}([ρ_{1}]) = [c] and δ_{m}([ρ_{2}]) = [\overline{c}]. Let \{V_{i}\}_{i=1}^{n} and \{W_{i}\}_{i=1}^{n} be implementing systems for ρ̃_{1} and ρ̃_{2} satisfying

θ_{t}(V_{i}) = ∑_{j=1}^{n} c(t)_{ji}V_{j},

θ_{t}(W_{i}) = ∑_{j=1}^{n} \overline{c}(t)_{ji}W_{j}.

We define two isometries R and S by

R = \frac{1}{\sqrt{n}} ∑_{i=1}^{n} V_{i}W_{i},

S = \frac{1}{\sqrt{n}} ∑_{i=1}^{n} W_{i}V_{i}.

R and S belong to M̃^{θ} = M satisfying

R ∈ (id, ρ_{1} · ρ_{2})
S ∈ (id, ρ_{2} · ρ_{1})

S^{∗}ρ_{2}(R) = R^{∗}ρ_{1}(S) = \frac{1}{n}.

Thus, [ρ_{1}] = \overline{[ρ_{2}]} [50]. Since δ_{m} is a bijection, this also shows that [ρ] ∈ Sect(M)_{m} implies \overline{[ρ]} ∈ Sect(M)_{m} and δ_{m}(\overline{[ρ]}) = \overline{δ_{m}([ρ])}.

The following is a useful criterion for an irreducible endomorphism to be modular.

Proposition 3.4. Let M be a type III factor.

(1) Let ρ ∈ End(M)_{0} be an irreducible endomorphism. If there exists a non-zero element in (id, ρ̃), ρ is a modular endomorphism.

(2) Every irreducible component of a modular endomorphism is again a modular endomorphism. In particular Sect(M)_{m} is closed under irreducible decomposition.

Proof. (1). First, we show that (id, ρ̃) contains at least one isometry if (id, ρ̃) ≠ {0}. Let PI(id, ρ̃) be the set of partial isometries in (id, ρ̃). Using the polar decomposition, we know that PI(id, ρ̃) is not empty. We introduce an order into PI(id, ρ̃) as follows: For two element v_{1}, v_{2} ∈ PI(id, ρ̃), we say that v_{2} dominates v_{1} if the following holds

v_{1} = v_{2}v_{1}^{∗}v_{1}.

Note that when this is the case, v_{1}^{∗}v_{1} ∈ Z(M̃), and v_{2}^{∗}v_{1} = v_{1}^{∗}v_{1}, v_{2}v_{1}^{∗} = v_{1}v_{1}^{∗} hold. We show that PI(id, ρ̃) is an inductively ordered set with this order. Let T be a totally ordered subset of PI(id, ρ̃). Since {vv^{∗}}_{v∈T} and {v^{∗}v}_{v∈T} are increasing nets of projections in Z(M̃) and M̃, they converge to projections in Z(M̃) and in M̃ respectively in strong topology. We claim that the net {v}_{v∈T} converges in strong ∗ topology. Let v_{1}, v_{2} ∈ T with v_{2} dominating v_{1}, and ξ be a vector in the Hilbert space that M̃ acts on. Then,

||(v_{2} − v_{1})ξ||^{2} = ⟨(v_{2}^{∗}v_{2} − v_{1}^{∗}v_{1})ξ, ξ⟩,
||(v_{2}^{∗} − v_{1}^{∗})ξ||^{2} = ⟨(v_{2}v_{2}^{∗} − v_{1}v_{1}^{∗})ξ, ξ⟩.

Therefore, the strong ∗ limit exists, and it gives a majorant of T , which shows that PI(id, ρ̃) is inductively ordered. Now, we apply Zorn’s lemma to PI(id, ρ̃) and take a maximal element V ∈ PI(id, ρ̃). We show that V is an isometry. Suppose V^{∗}V ≠ 1. Since V^{∗}V ∈ Z(M̃) and θ acts on Z(M̃) ergodically, there exists some t ∈ ℝ such that θ_{t}(V^{∗}V)(1 − V^{∗}V) ≠ 0. Therefore, V_{1} := V + θ_{t}(V)(1 − V^{∗}V) ∈ PI(id, ρ̃) dominates V, which is contradiction. Thus, V is an isometry.

Let {V_{λ}}_{λ∈Λ} be a maximal set of isometries in (id, ρ̃) with mutually orthogonal ranges (such a set exists thanks to Zorn’s lemma again.), and let P = ∑_{λ∈Λ} V_{λ}V_{λ}^{∗}. If P = 1, we are done, and so we assume P ≠ 1. Let PI_{P^{⊥}}(id, ρ̃) be the subset of elements in PI(id, ρ̃) with range projections orthogonal to P. We claim that there exists a non-zero element in PI_{P^{⊥}}(id, ρ̃). Indeed, since (ρ̃, ρ̃)^{θ} = (ρ, ρ) = ℂ1, there exists some t ∈ ℝ such that (1 − P)θ_{t}(P) ≠ 0, which shows that there exists non-zero x ∈ (id, ρ̃) with (1 − P)x ≠ 0. Using the polar decomposition, we get the claim. We introduce an order into PI_{P^{⊥}}(id, ρ̃) as before and take a maximal element W. Since {V_{λ}}_{λ∈Λ} is a maximal set, W is not an isometry, and we set W^{∗}W = z ≠ 1. We claim (1−P)(1−z) = 0. Indeed, by maximality of W, (1−P)(1−z)x = (1−P)x(1−z) = 0 holds for x ∈ (id, ρ̃), and θ_{t}((1 − P)(1 − z))x = 0 for x ∈ (id, ρ̃) as well for all t ∈ ℝ because (id, ρ̃) is globally invariant under θ. Thus, we get

(⋁_{t∈ℝ}θ_{t}((1 − P)(1 − z)))x = 0, x ∈ (id, ρ̃).

Since (ρ̃, ρ̃)^{θ} = ℂ1, this implies (1 − P)(1 − z) = 0.

Let V′_{λ} = V_{λ}(1 − z). Then, {V′_{λ}}_{λ∈Λ} have mutually orthogonal ranges such that

∑_{λ∈Λ} V′_{λ}V′_{λ}^{∗} = P (1 − z) = (1 − z).

Thus, we get

(M̃(1 − z) ⊃ ρ̃(M̃)(1 − z)) ≅ ((B(ℓ^{2}(Λ)) ⊗ M̃(1 − z)) ⊃ (ℂ1 ⊗ M̃(1 − z))).

Since there exists a conditional expectation from M̃ to ρ̃(M̃) satisfying the Pimsner-Popa inequality [27], [60], Λ is a finite set, and we identify Λ with {1, 2, ⋯, n}. We take t ∈ ℝ satisfying z_{1} := θ_{t}(z)(1 − z) ≠ 0, and set Ω ⊂ X_{M} to be the Borel subset corresponding z_{1}. Let R_{i} = V′_{i}z_{1}, S_{i} = θ_{t}(V_{i})z_{1}, i = 1, 2, ⋯ n, and S_{n+1} = θ_{t}(W)z_{1}. Then, {R_{i}}_{i=1}^{n} ⊂ (id, ρ̃) and {S_{i}}_{i=1}^{n+1} ⊂ (id, ρ̃) satisfy

R_{i}^{∗}R_{j} = δ_{i,j}z_{1}, i, j = 1, 2, ⋯ n,

∑_{i=1}^{n} R_{i}R_{i}^{∗} = z_{1},

S_{i}^{∗}S_{j} = δ_{i,j}z_{1}, i, j = 1, 2, ⋯, n + 1,

∑_{i=1}^{n+1} S_{i}S_{i}^{∗} ≤ z_{1}.

We set f = (f_{ij}) = (R_{i}^{∗}S_{j}), which is regarded as an n by n+1 matrix-valued function on Ω. However f satisfies

∑_{i=1}^{n} f_{ij}^{∗}f_{ik} = δ_{j,k}z_{1}, j, k = 1, 2, ⋯, n + 1,

which is contradiction. Thus, we conclude P = 1, and

ρ̃(x) = ∑_{λ∈Λ} V_{λ}xV_{λ}^{∗}.

(2). Let ρ be a modular endomorphism with an implementing system {V_{i}}_{i=1}^{n} for ρ̃, and σ be an irreducible component of ρ. We take a non-zero element T ∈ (σ, ρ). Thanks to Proposition 2.5, (1), T^{∗}V_{i} belongs to (id, σ̃), i = 1, 2, ⋯, n. Since T^{∗} = ∑_{i=1}^{n} T^{∗}V_{i}V_{i}^{∗}, there exists some i such that T^{∗}V_{i} is non-zero. Thus, σ is a modular endomorphism thanks to (1).

Though it is shown in the above that Sect(M)_{m} is closed under irreducible decomposition, we have not given a criterion for a modular endomorphism to be irreducible, or that of how to decompose it when reducible, in terms of ergodic theory yet. We discuss this issue now.

Let X be a standard Borel space, μ be a probability measure on X, and G be a locally compact group ergodically acting on (X, μ) as a non-singular transformation group. For a compact group K, we define a K-valued cocycle c as in the case of the flow. We denote by K_{c} the closed subgroup generated by the image of c. We collect necessary results on cocycles from R. Zimmer’s fundamental paper [80, Part I. Section 3] in the next theorem in order to introduce a few notions for cocycles. Though Zimmer treated only the measure preserving case in [80], his argument works for the non-singular case with a formal modification (see also [15]):

Theorem 3.5 (R. Zimmer). Let the notations be as above, and c ∈ Z^{1}(G, K). Then, the following hold:

(1) There exists a closed subgroup H ⊂ K such that c is equivalent to an H-valued cocycle c′, and c′ is never equivalent to c′′ with K_{c′′} a proper subgroup of H. H is uniquely determined up to conjugacy. We call such c′ a minimal cocycle and call H the minimal subgroup of c.

(2) Let K_{1} be a compact group containing K. Then, if c is a minimal cocycle as a K-valued cocycle, it is the case as a K_{1}-valued cocycle.

(3) Let c be a minimal K-valued cocycle such that K = K_{c}, and π be a continuous homomorphism from K to a compact group K_{1}. Then, π·c is a minimal cocycle.

(4) Let c be a minimal K-valued cocycle with K = K_{c}, and (π_{1}, H_{π_{1}}), (π_{2}, H_{π_{2}}) be finite dimensional unitary representations of K. If a is a Borel map from X to Hom(H_{1}, H_{2}) such that for every fixed k,

a(ω)π_{2}(c(ω, k)) = π_{1}(c(ω, k))a(ω · k),

holds for almost all ω ∈ X, then a is constant almost everywhere with the value in (π_{1}, π_{2}).

We say that a closed subgroup K ⊂ U(n) is irreducible if the defining representation of K on ℂ^{n} is irreducible.

Corollary 3.6. Let M be a type III factor and ρ be a modular endomorphism with δ_{m}([ρ]) = [c]. Then, ρ is irreducible if and only if the minimal subgroup of c is irreducible.

Proof. Thanks to theorem 3.3, it suffices to show that the minimal subgroup of c is irreducible if and only if c is never equivalent to a direct sum of two cocycles. Assume that the minimal subgroup of c is irreducible. If c were equivalent to c_{1} ⊕ c_{2}, c_{i} ∈ Z^{1}(G, U(n_{i})), i = 1, 2, with n_{1}+n_{2} = n, c could be considered as a U(n_{1})×U(n_{2})- valued cocycle, and the minimal subgroup of c would be conjugate to a subgroup of U(n_{1}) × U(n_{2}) in U(n) thanks to Theorem 3.5, (2). This is contradiction and c is never equivalent to a direct sum of two cocycles. The converse also follows from Theorem 3.5, (2).

We end this section with giving an answer to the problem about the type II and type III principal graphs described in Introduction.

Let M ⊃ N be an inclusion of type III factors of a finite index with the minimal expectation E. We say that M and N has the common flow of weights if Z(Ñ) = Z(M̃). When this is the case, we have common central decomposition

(M̃ ⊃ Ñ) = ∫_{X_{M}}^{⊕} (M̃(ω) ⊃ Ñ(ω))dμ_{M}(ω),

and we could define the type II principal graph for that of M̃(ω) ⊃ Ñ(ω) for ω in a conull set. However, it would be cumbersome to treat “measurable field of subfactors” (though it should not be too hard to do so, c.f. [79].) Instead, we formulate the problem using a global term. Let

N ⊂ M ⊂ M_{1} ⊂ ⋯ M_{n} ⊂ ⋯

be the tower for N ⊂ M. Then, using the minimal conditional expectation in each step, we have the “tower for the core inclusion” [44]:

Ñ ⊂ M̃ ⊂ M̃_{1} ⊂ ⋯ M̃_{n} ⊂ · · · .

We still denote by θ the natural extension of θ to M̃_{n} leaving the Jones projections fixed. Then, we have

M_{n} ∩ M′ = (M̃_{n} ∩ M̃′)^{θ}.

We say that graph change occurs if Z(M̃)∨(M_{n}∩M′) does not coincides with M̃_{n}∩M̃′ for some n. Let E_{n} be the minimal expectation from M_{n} onto M_{n−1} and Ẽ_{n} be the natural extension of E_{n} to M̃_{n}, which is a conditional expectation from M̃_{n} onto M̃_{n−1}. Then, using E_{n+1} and Ẽ_{n+1}, we can show that Z(M̃) ∨ (M_{n} ∩ M′) ≠ M̃_{n} ∩ M̃′ implies Z(M̃) ∨ (M_{n+1} ∩ M′) ≠ M̃_{n+1} ∩ M̃′. In a similar way using downward basic construction and mirroring J_{M} · J_{M}, JM̃ · JM̃, we can also show that graph change occurs for M ⊃ N if and only if it occurs for M_{1} ⊃ M.

A modular endomorphism is said to be non-trivial if it is irreducible and not equivalent to identity.

Theorem 3.7. Let M ⊃ N be an inclusion of type III_{0} factors with a finite index and with the common flow of weights. Then, graph change occurs for M ⊃ N if and only if a non-trivial modular endomorphism appears in some power of the canonical endomorphism γ for M ⊃ N.

Proof. Assume that graph change occurs for N ⊂ M. Taking sufficiently large n, we may assume Z(M̃) ∨ (M_{2n} ∩ M′) ≠ M̃_{2n} ∩ M̃′. Let γ be the canonical endomorphism for M ⊃ N. Then, thanks to Proposition 2.5, this is equivalent to

(γ̃^{n}, γ̃^{n}) ≠ (γ^{n}, γ^{n}) ∨ Z(M̃).

Therefore, either of the following two holds: (i) γ^{n} contains two irreducibles ρ_{1}, ρ_{2} ∈ End(M)_{0} such that [ρ_{1}] ≠ [ρ_{2}] and (ρ̃_{1}, ρ̃_{2}) ≠ {0}. (ii) γ^{n} contains an irreducible ρ ∈ End(M)_{0} such that (ρ̃, ρ̃) ≠ Z(M̃). Indeed, let {p_{i}}_{i=1}^{n} be a partition of unity consisting of minimal projections in M ∩ γ^{n}(M)′. If

X ∈ (γ̃^{n}, γ̃^{n}) \ (γ^{n}, γ^{n}) ∨ Z(M̃),

there exist some i and j such that

p_{i}Xp_{j} ∈ (γ̃^{n}, γ̃^{n}) \ (γ^{n}, γ^{n}) ∨ Z(M̃).

If the irreducible components corresponding to p_{i} and p_{j} are different, (i) occurs. If they are the same, (ii) occurs.

In the case (i), the Frobenius reciprocity implies (id, { \atop ρ_{2} · \overline{ρ}_{1}}) ≠ \{0\} (though M̃ is not a factor, the argument in [33, Section 2] works). Thanks to Lemma 3.3, this shows that ρ_{2} · \overline{ρ}_{1}, which is contained in γ^{2n}, contains a modular endomorphism. Since ρ_{2} · \overline{ρ}_{1} does not contain identity, it is indeed a non-trivial one.

In the case (ii), we take X ∈ (ρ̃, ρ̃) \ Z(M̃). Let R ∈ (id, \overline{ρ} · ρ), \overline{R} ∈ (id, ρ · \overline{ρ}) be isometries satisfying the usual relation [50]

R^{∗}\overline{ρ}(\overline{R}) = \overline{R}^{∗}ρ(R) = \frac{1}{d(ρ)}.

Then, X\overline{R} ∈ (id, ρ̃ · \overline{ρ}). Let σ be an irreducible sector contained in ρ · \overline{ρ} that is not equivalent to identity, and \{Y (σ)_{i}\}_{i=1}^{k} be a basis of (σ, ρ · \overline{ρ}). If Y (σ)_{i}^{∗}X\overline{R} ∈ (id, σ̃) is not zero for some i, we are done thanks to Lemma 3.3. Thus, we assume Y (σ)_{i}^{∗}X\overline{R} = 0 for every σ not equivalent to identity. This would imply that \overline{R}\overline{R}^{∗}X\overline{R} = X\overline{R}. However, since ρ(R) commutes with X, we would get

X = d(ρ)ρ(R^{∗})X\overline{R} = d(ρ)ρ(R^{∗})\overline{R}\overline{R}^{∗}X\overline{R} = \overline{R}^{∗}X\overline{R} ∈ Z(M̃),

which is contradiction. Thus, ρ·\overline{ρ} contains a non-trivial modular endomorphism, and so does γ^{2n}.

The converse can be shown easily.

Remark 3.8. For a type III_{λ}, 0 < λ ≤ 1 factor, the flow is a transitive and every U(n)-valued cocycle is equivalent to a direct sum of U(1)-valued cocycles that come from homomorphisms from the stabilizer subgroup of a point [56], ℤ for λ ≠ 1 and ℝ for λ = 1. This means that every modular endomorphism is decomposed into usual modular automorphisms. Therefore, the above theorem generalizes the previous one obtained in [31, Theorem 3.5].

There exist plenty of examples in the type III_{0} case, where higher dimensional nontrivial modular endomorphisms appear in the canonical endomorphisms (see Section 5 and Proposition A.5 ).

4. Connes-Takesaki modules

For a type III factor M, we denote by Aut(F^{M}) the set of automorphisms of Z(M̃) that commute with the restriction of θ to Z(M̃). For an automorphism α of M, the restriction of α̃ to Z(M̃) belongs to Aut(F^{M}), which is called the Connes-Takesaki module of α. This correspondence gives a homomorphism

mod : Aut(M) ⟶ Aut(F^{M}),

which is called the fundamental homomorphism. (Though the original definition of Connes and Takesaki looks different from ours [11], this is an equivalent description of it.) The purpose of this section is to introduce the Connes-Takesaki module for endomorphisms.

Definition 4.1. Let M be a type III factor and ρ ∈ End(M)_{0}. We say that ρ has a Connes-Takesaki module if M ⊃ ρ(M) has the common flow of weights. When ρ has a Connes-Takesaki module, we denote by mod(ρ) the restriction of ρ̃ to Z(M̃). mod(ρ) ∈ Aut(F^{M}) is called the Connes-Takesaki module of ρ.

We denote by End(M)_{CT} (respectively End(M)_{tr}) the set of endomorphisms in End(M)_{0} with Connes-Takesaki modules (respectively trivial Connes-Takesaki modules). Note that mod(ρ) depends only on the sector of ρ. We denote by Sect(M)_{CT} and Sect(M)_{tr} the corresponding subsets of Sect(M)_{0}.

Proposition 4.2. Let M be a type III factor. Then,

(1) If ρ_{1}, ρ_{2} ∈ End(M)_{0} have Connes-Takesaki modules, then so does ρ_{1} · ρ_{2} and mod(ρ_{1} · ρ_{2}) = mod(ρ_{1}) · mod(ρ_{2}).

(2) Let ρ, ρ_{i} ∈ End(M)_{0}, i = 1, 2, ⋯, n, satisfying [ρ] = ⊕_{i=1}^{n}[ρ_{i}]. Then, ρ has a Connes-Takesaki module if and only if each ρ_{i} does and mod(ρ_{i}) does not depend on i. When it is the case, mod(ρ) = mod(ρ_{i}).

(3) ρ ∈ End(M)_{0} has a Connes-Takesaki module if and only if the conjugate \overline{ρ} does. When it is the case, mod(\overline{ρ}) = mod(ρ)^{−1}.

(4) Every modular endomorphism has a trivial Connes-Takesaki module.

(5) Let N ⊂ M be a subfactor with a finite index. If M and N have the common flow of weights, then the canonical endomorphism γ for M ⊃ N has a trivial Connes-Takesaki module.

Proof. (1). This is trivial.

(2). We take isometries V_{i} ∈ (ρ_{i}, ρ) satisfying

ρ(x) = ∑_{i=1}^{n} V_{i}ρ_{i}(x)V_{i}^{∗}, x ∈ M.

Assume ρ ∈ End(M)_{CT}. Thanks to Lemma 2.4, (1), we have

ρ̃_{i}(z) = V_{i}^{∗}ρ̃(z)V_{i} = ρ̃(z), z ∈ Z(M̃),

which shows that ρ_{i} ∈ End(M)_{CT} for every i and mod(ρ_{i}) does not depend on i. Next we assume that ρ_{i} ∈ End(M)_{CT} for every i and mod(ρ_{i}) does not depend on i, which is denoted by ν. Then,

ρ̃(z) = ∑_{i=1}^{n} V_{i}ρ̃_{i}(z)V_{i}^{∗} = ∑_{i=1}^{n} ν(z)V_{i}V_{i}^{∗} = ν(z), z ∈ Z(M̃),

which shows ρ ∈ End(M)_{CT}.

(3). Since M ⊃ \overline{ρ}(M) is the dual inclusion of M ⊃ ρ(M), the first part is obvious. The second part follows from (1) and (2) because ρ · \overline{ρ} contains identity.

(4). This follows from the definition of modular endomorphisms.

(5). This follows from Proposition 2.5, (3).

Let M ⊃ N be an inclusion of type III_{λ}, 0 < λ < 1 factors with a finite index, and E be the minimal expectation from M onto N. We say that M ⊃ N has common discrete decomposition if there exist an inclusion of type II_{∞} factors P ⊃ Q with a finite index, a minimal expectation E_{0} from P to Q preserving the trace, and a trace scaling automorphism θ_{0} of P globally preserving Q such that

(M ⊃ N) ≅ (P ⋊_{θ_{0}} ℤ ⊃ Q ⋊_{θ_{0}} ℤ).

Note that in [47], common discrete decomposition with respect to any faithful normal expectation is discussed, while here we consider the minimal expectation only.

The following lemma is standard and we omit the proof.

Lemma 4.3. Let M ⊃ N be an inclusion of type III_{λ}, 0 < λ < 1 factors with a finite index, and E be the minimal expectation from M to N. Then, the following are equivalent:

(1) M ⊃ N has common discrete decomposition.

(2) M ⊃ N has the common flow of weights.

(3) For some (and any) generalized trace ψ_{0} on N (see [5] for the definition, it is also called λ-trace in [67]), ψ_{0} · E is a generalized trace on M.

Corollary 4.4. Let M be a type III_{λ}, 0 < λ < 1, factor, and ρ ∈ End(M)_{0}. Then,

(1) If ρ is irreducible, ρ has a Connes-Takesaki module.

(2) M ⊃ ρ(M) has common discrete decomposition if and only if ρ has a Connes-Takesaki module, which is further equivalent to that every irreducible component of ρ has the same Connes-Takesaki module.

Proof. (1). This follows from the condition (3) of Lemma 4.3. (2). This follows from Proposition 4.2, (2) and Lemma 4.3.

Let M be a type III_{λ} factor, 0 < λ < 1, and ψ be a generalized trace on M. We set T = −2π/ log λ. Then, the modular automorphism group {σ_{t}^{ψ}}_{t∈ℝ} has period T and we consider σ^{ψ} an action of 𝕋 := ℝ/T ℝ. We denote by M̂ the crossed product M ⋊_{σ^{ψ}} 𝕋, which is a type II_{∞} factor generated by a copy of M and the implementing unitary representation {λ_{d}^{ψ}(t)} of 𝕋. If φ is another weight on M satisfying [Dφ : Dψ]_{T} = 1, {[Dφ : Dψ]_{t}} is a σ^{ψ}-cocycle as a 𝕋 action, and we identify [Dφ : Dψ]_{t}λ_{d}^{ψ}(t) with λ_{d}^{φ}(t). We denote by τ_{d} and θ_{d} the natural trace on M̂ and the dual action of σ^{ψ} respectively. In analogous to the canonical extension, we can construct a functor from End(M)_{tr} to End(M̂)_{0}. Note that thanks to Proposition 4.2, End(M)_{tr} is a full subcategory of End(M)_{0}.

Proposition 4.5. Let M be a type III_{λ} factor and ψ be a generalized trace. Then, for every ρ ∈ End(M)_{tr}, there exists a unique endomorphism ρ̂ ∈ End(M̂) satisfying

ρ̂(x) = ρ(x), x ∈ M
ρ̂(λ_{d}^{ψ}(t)) = d(ρ)^{it}[Dψ · ϕ_{ρ} : Dψ]_{t}λ_{d}^{ψ}(t).

Moreover, for ρ, μ ∈ End(M)_{tr}, the following hold:

(1) (ρ, μ) ⊂ (ρ̂, μ̂).

(2) ρ̂ · μ = ρ̂· μ̂.

(3) Let N be a subfactor of M with a finite index and γ be the canonical endomorphism for M ⊃ N. If M ⊃ N has common discrete decomposition, then, γ̂ is the canonical endomorphism for M̂ ⊃ N̂.

(4) τ_{d} · ρ̂ = d(ρ)τ_{d}.

(5) ρ̂· θ_{d} = θ_{d} · ρ̂.

Proof. Since the center of M̃ is generated by λ^{ψ}(T), we have ρ̃(λ^{ψ}(T)) = λ^{ψ}(T) by assumption, which implies

[Dd(ρ)ψ · ϕ_{ρ} : Dψ]_{T} = 1.

Therefore, there exists a unitary u ∈ M such that d(ρ)ψ · ϕ_{ρ} = ψ · Ad(u) thanks to [5, Théorème 4.3.2]. This means that (ψ, Ad(u) · ρ) is an invariant pair. The rest of the proof is the same as those of Theorem 2.4 and Proposition 2.5 if the dominant weight is replaced with the generalized trace.

We end this section with a remark on the type II_{∞} case.

Remark 4.6. Let M be a type II_{∞} factor and τ be a faithful normal semifinite trace on M. Then, the canonical extension and the Connes-Takesaki modules make sense for M as well. We introduce a scalar valued module Mod(ρ) of ρ ∈ End(M)_{0} by

τ · ρ = d(ρ)Mod(ρ)τ.

Note that every inclusion of type II_{∞} factors with a finite index comes from the tensor product of a common type I factor and an inclusion of II_{1} factors. Therefore, the restriction of τ to the image of ρ is a semifinite trace and the above definition makes sense. The center of M̃ is generated by λ^{τ}(t), and we have

ρ̃(λ^{τ}(t)) = d(ρ)^{it}(\frac{dτ · ϕ_{ρ}}{dτ})^{it}λ^{τ}(t) = Mod(ρ)^{−it}(\frac{dτ · E_{ρ}}{dτ})^{it}λ^{τ}(t).

This means that ρ has a Connes-Takesaki module if and only if E_{ρ} is trace preserving. When this is the case, we have

mod(ρ)(λ^{τ}(t)) = Mod(ρ)^{−it}λ^{τ}(t).

Therefore, Proposition 4.2, (2) implies that M ⊃ ρ(M) is an extremal inclusion (see [60] for the definition) if and only if each irreducible component of ρ has the same Mod.

5. Minimal Actions of Compact Groups

In this section, we investigate the structure of minimal actions of compact groups on type III factors applying our machinery to the corresponding Roberts actions. Modular endomorphisms and the Connes-Takesaki modules for endomorphisms provide new invariants for minimal actions of compact groups through Roberts actions.

5.1. Minimal actions as dual actions. Let K be a compact group and Rep(K) be the tensor category of finite dimensional unitary representations of K. Roughly speaking, a Roberts action of the dual of K on a von Neumann algebra N is a functor from a (sufficiently big) subcategory R of Rep(K), (called a ring in [58]), to End(N)_{0}, where the set of arrows from ρ_{1} ∈ End(N)_{0} to ρ_{2} ∈ End(N)_{0} is (ρ_{1}, ρ_{2}). For the precise definition of the Roberts action and the crossed product by it, we refer to [63] and [58] (see also [13]). For each equivalence class of irreducible representations of K, we choose a representative (π, H_{π}), and denote by K̂ the collection of them. We always assume that R contains every member of K̂ and its complex conjugate representation. We often omit to specify R in a Roberts action, and say “a Roberts action of K̂” if there is no possibility of confusion (although it is a bit sloppy terminology). For (π_{1}, H_{π_{1}}), (π_{2}, H_{π_{2}}) ∈ R, we denote by (π_{1}, π_{2}) the set of K-homomorphisms from H_{π_{1}} to H_{π_{2}}, that is, the set of arrows from π_{1} to π_{2}.

Let M be a factor and α be an action of K on M. α is said to be minimal if α is faithful and the relative commutant M ∩ M^{α}′ of the fixed point subalgebra M^{α} is trivial [62]. Note that when α is minimal in our sense, the crossed product M ⋊_{α} K is automatically a factor isomorphic to M^{α} ⊗ B(L^{2}(K)). Indeed, to show this, we may assume that M^{α} is infinite by taking tensor product with a type I factor if necessary. Then, a similar argument as in [2, Lemma III.3.4] implies that for each irreducible representation π of K, there exists a Hilbert space H_{π} in M that is globally invariant under α such that the restriction of α to H_{π} is equivalent to π. Therefore, [63, Proposition 6.9] implies the claim.

More strongly, [63, Theorem 6.5] shows that M is the crossed product N ⋊_{β} K̂ by some Roberts action β, and that α is its dual action β̂, where N = M^{α}. The relationship between β_{π} and H_{π} is as follows: Let {W (π)}_{i=1}^{d(π)} be an orthonormal basis for H_{π}, where d(π) is the dimension of π. Then, β_{π} ∈ End(N)_{0} is given by

β_{π}(x) = ∑_{i=1}^{d(π)} W (π)_{i}xW (π)_{i}^{∗}, x ∈ N.

On the other hand, H_{π} is recovered from β_{π} via

H_{π} = {W ∈ M; W x = β_{π}(x)W, x ∈ N}.

Note that β is uniquely determined up to equivalence defined in [63, Definition 5.6].

We will use these notations throughout this section whenever a minimal action α of a compact group K on a factor M is given.

Lemma 5.1. Let N be an infinite factor, and β be a Roberts action of K̂ on N. Then, the dual action of β is minimal if and only if β_{π} is irreducible for every irreducible representation (π, H_{π}) ∈ K̂.

Proof. We set M = N ⋊_{β} K̂, α = β̂, and

E(x) = ∫_{K} α_{k}(x)dm_{K}(k), x ∈ M,

where m_{K} is the Haar measure of K. Let {W (π)_{i}}_{i=1}^{d(π)} be as above. For x ∈ M, we define x(π)_{i} ∈ N by

x(π)_{i} = d(π)E(W (π)_{i}x).

Then, x has the following formal expansion (see [35, Section 3]):

x“ = ” ∑_{(π,H_{π})∈K̂} ∑_{i=1}^{d(π)} W (π)_{i}^{∗}x(π)_{i}.

More precisely, the above summation converges in the GNS Hilbert space topology with respect to an α-invariant normal state, and {x(π)_{i}} completely determines x. Using this expansion, we can see that x ∈ M ∩ N′ if and only if x(π)_{i} ∈ (id, β_{π}). Therefore, α is minimal if and only if β_{π} does not contain identity for every non-trivial irreducible representation π. However, the latter is equivalent to the statement that β_{π} is irreducible for very irreducible π thanks to the Frobenius reciprocity for the usual compact group representations and that of sectors [34].

Let {u(k)}_{k∈K} be an α-cocycle. We denote by α^{u} the perturbation Ad(u(k)) · α_{k} of α by the cocycle u. α is called stable if every α-cocycle is a coboundary.

The following statements are probably well-known among specialists, and actually some of them already exist in the literature. However, since we cannot find complete proofs in the literature, we provide them here for convenience of readers.

Proposition 5.2. Let α be a minimal action of a compact group K on a factor M. We denote by E the unique normal conditional expectation from M onto M^{α} obtained by taking average over K. Then, the following hold:

(1) Every cocycle perturbation of α is again minimal.

(2) If M is of type II_{1}, so is M^{α}. α is stable in this case [76, Theorem 12].

(3) If M is of type II_{∞}, so is M^{α}. α is stable in this case.

(4) If M is of type III_{0}, M^{α} is of type III. α is stable in this case (more generally, α is stable whenever M^{α} is of type III).

(5) M is of type III_{λ}, λ ≠ 0 and M^{α} is of type II, if and only if the center of α_{K} contains the modular automorphism group {σ_{t}^{φ·E}}_{t∈ℝ}, for some faithful normal semifinite weight φ on M^{α} (which is actually a trace if this is the case). α is not stable in this case.

Proof. (1). First we claim that the second dual action α ⊗ Ad(r) on M ⊗ B(L^{2}(K)) is minimal whenever α is so. To show the claim, we may assume that M^{α} is infinite as usual. Thus, we have a Hilbert space H_{π} in M for each (π, H_{π}) ∈ K̂ as before. This implies that there exists a Hilbert space H in M such that the restriction of α to H is equivalent to the regular representation. Therefore, (M, α) is conjugate to (M ⊗B(L^{2}(K)), α⊗Ad(r)), which shows the claim. Let α^{u} be a cocycle perturbation of α. Since M ⋊_{α} K ⊂ M ⊗ B(L^{2}(K)) is an irreducible inclusion and there exists a projection p ∈ M ⋊_{α} K such that

(M^{α^{u}} ⊂ M) ≅ (p(M ⋊_{α} K)p ⊂ p(M ⊗ B(L^{2}(K)))p),

which shows that α^{u} is minimal.

(2),(3). The first part is easy. Stability in these two cases (and the case (4) as well) follows from (1) and Connes’ 2 by 2 matrix trick [9].

(4). We show that if M^{α} is of type II, M is either of type II or of type III_{λ}, λ ≠ 0. Let τ be a faithful normal semifinite trace on M^{α}. Then, the centralizer M_{τ·E} is an intermediate subfactor between M^{α} and M, and in particular M_{τ·E} is a factor. Therefore, the Connes spectrum Γ(σ^{τ·E}) coincides with the Arveson spectrum Sp(σ^{τ·E}) [5], [67]. Thus, M is either of type II or type III_{λ}, λ ≠ 0 depending on the period of the modular automorphism group {σ_{t}^{τ·E}}_{t∈ℝ}.

(5). Note that part of the proof has been already done in the above. Assume that α_{K} contains {σ_{t}^{φ·E}}_{t∈ℝ}, for some faithful normal semifinite weight φ on M^{α}. Then, the centralizer M_{φ·E} is an intermediate subfactor of M^{α} ⊂ M. Let F be the restriction of E to M_{φ·E}, which is a normal conditional expectation from M_{φ·E} to M^{α}. Since the restriction of φ · E to M_{φ·E} is nothing but φ · F and it is semifinite, M_{φ·E} is of type II, and so is M^{α} as well because σ^{φ·F} is trivial. This finishes the proof of the first statement.

Now, we show that α is not stable in the above situation with type III M . First we assume that M^{α} is of type II_{∞}. Let p ∈ M^{α} be a non-zero finite projection, and V ∈ M be an isometry satisfying V V^{∗} = p. We set u(k) = V^{∗}α_{k}(V), which is an α-cocycle. It is easy to show that M^{α^{u}} is finite and u is not a coboundary. When M^{α} is of type II_{1}, a similar construction identifying M with a corner of M ⊗ B(H) works.

Remark 5.3. In the above situation, assume that M^{α} is of type II_{∞} with a faithful normal trace τ. Then, the type of M (or the period of {σ^{τ·E}}_{t∈ℝ}) is completely determined by Mod(β_{π}) for the Roberts action β satisfying M = M^{α} ⋊_{β} K̂, where Mod is the scalar Connes-Takesaki module introduced in Remark 4.6. Indeed, as in [35, Section 3], we can obtain the action of the modular automorphism group on H_{π} as follows:

σ_{t}^{τ·E}(W) = Mod(β_{π})^{it}W, W ∈ H_{π}.

It is possible to show that the canonical extension gives continuous homomorphism from Aut(M) to Aut(M̃) in the u-topologies in the same spirit of the Cones-Takesaki’s proof of continuity of the fundamental homomorphism [11]. However, we just mention here that when α is an action of a locally compact group G on M with a faithful normal invariant state, then it is particularly easy to show that G ∋ g ↦ α̃_{g} is continuous. On the other hand, when β is a Roberts action of K̂ on a factor N, Proposition 2.5 shows that β extends to β̃ on Ñ via the canonical extension.

Lemma 5.4. Let α be an action of a compact group K on a factor M, and E be the normal conditional expectation from M onto N := M^{α} given by the average of α_{k} over K. We regard Ñ as a subalgebra of M̃ identifying λ^{φ_{0}}(t) with λ^{φ_{0}·E}(t), where φ_{0} is a faithful normal semifinite weight on Ñ. Then,

(1) The fixed point subalgebra of M̃ under α̃ coincides with Ñ.

(2) When α is a minimal action that is the dual action of a Roberts action β of K̂ on N. Then, M̃ is naturally isomorphic to Ñ ⋊β̃ K̂.

Proof. (1). We set φ = φ_{0} · E. Let Ẽ be the normal conditional expectation from M̃ onto the fixed point subalgebra of M̃ under α̃ obtained by average of α̃_{k} over K. Since α̃_{k} leaves λ^{φ}(t) fixed for all k ∈ K and t ∈ ℝ, the image of Ẽ coincides with Ñ.

(2). Let ψ_{0} be a dominant weight on N, and ψ = ψ_{0} ·E. For each β_{π}, (π, H_{π}) ∈ K̂, we may and do assume that (ψ_{0}, β_{π}) is an invariant pair thanks to Lemma 2.3, (3). Then, the modular automorphism group {σ_{t}^{ψ}}_{t∈ℝ} acts on H_{π} trivially [35, Section 3] and β̃_{π} leaves λ^{ψ_{0}}(t) invariant. Therefore, β̃_{π} is given by

β̃_{π}(x) = ∑_{i=1}^{d(π)} W (π)_{i}xW (π)_{i}^{∗}, x ∈ Ñ.

Thus, (1) and [63, Theorem 6.5] imply that M̃ = Ñ ⋊β̃ K̂ and α̃ is the dual action of β̃.

5.2. A Roberts action of modular endomorphisms. The purpose of this subsection is to show how a system of modular endomorphisms gives rise to a Roberts action, and to see the structure of its dual action. There are two options to do so, (of course they are essentially the same), and we start with the one using the minimal subgroup of a cocycle.

Let N be a type III factor and D ⊂ End(N)_{m} be a countable family of modular endomorphisms of N. By adding new endomorphisms to D if necessary, we may and do assume the following conditions:

(i) D is closed under taking a product of any two members in D. (ii) For every pair ρ_{1}, ρ_{2} ∈ D there exists ρ ∈ D such that [ρ_{1}] ⊕ [ρ_{2}] = [ρ]. (iii) If σ ∈ End(N)_{0} is contained in ρ ∈ D, there exists an endomorphism in D that

is equivalent to σ. (iv) Every ρ ∈ D has conjugate (up to equivalence) in D. For each ρ ∈ D we choose c_{ρ} ∈ Z^{1}(F^{N}, U(d(ρ))) satisfying δ_{m}([ρ]) = [c_{ρ}], and set

K_{1} = ∏_{ρ∈D} U(d(ρ)),

c = (c_{ρ})_{ρ∈D} ∈ Z^{1}(F^{N}, K_{1}),

where U(n) is the unitary group. Thanks to Theorem 3.5, we may and do assume that c is already a minimal cocycle. We denote by K the minimal subgroup of c.

Let (π_{ρ}, H_{π_{ρ}}) be the representation of K on H_{π_{ρ}} = ℂ^{d(ρ)} that is the projection onto the ρ-component. Then, we have c_{ρ} = π_{ρ} · c. We denote by R the collection of (π_{ρ}, H_{π_{ρ}}), ρ ∈ D.

Now, we construct a Roberts action β of R on N sending π_{ρ} to ρ. Thanks to Lemma 3.2, (2), we can choose an implementing system {V (ρ)_{i}}_{i=1}^{d(ρ)} for ρ̃ giving c_{ρ}. Let K_{ρ} be the linear span of {V (ρ)_{i}}_{i=1}^{d(ρ)}, which is a Hilbert space in Ñ. For ρ_{1}, ρ_{2} ∈ D, we denote by K_{ρ_{2}}K_{ρ_{1}}^{∗} the linear span of elements of the form V_{2}V_{1}^{∗}, V_{1} ∈ K_{1}, V_{2} ∈ K_{2}, which is naturally identified with Hom(K_{ρ_{1}}, K_{ρ_{2}}) and Hom(H_{π_{ρ_{1}}}, H_{π_{ρ_{2}}}). Since (ρ_{1}, ρ_{2}) is the fixed point subset of K_{ρ_{2}}K_{ρ_{1}}^{∗} under θ, Theorem 3.5, (4) implies that the above identification carries (ρ_{1}, ρ_{2}) ⊂ (ρ̃_{1}, ρ̃_{2}) onto (π_{ρ_{1}}, π_{ρ_{2}}) ⊂ Hom(H_{π_{ρ_{1}}}, H_{π_{ρ_{2}}}). Therefore, we define a functor β setting β_{T}, T ∈ (π_{ρ_{1}}, π_{ρ_{2}}) to be the corresponding element in (ρ_{1}, ρ_{2}). Note that since β maps every irreducible object to an irreducible object, Lemma 5.1 implies that the resulting crossed product gives irreducible inclusion of factors N ⊂ N ⋊_{β} K̂ =: M, and the dual action α := β̂ is minimal.

Another way to construct (M, α, K) from D is to apply the Doplicher-Roberts duality theorem [13, Theorem 7.1] to D (see also [14]). For this purpose, we have to specify the permutation symmetry E(ρ_{1}, ρ_{2}) for every pair ρ_{1}, ρ_{2} ∈ D. Indeed, we set

E(ρ_{1}, ρ_{2}) := ∑_{i=1}^{d(ρ_{1})} ∑_{j=1}^{d(ρ_{2})} V (ρ_{2})_{j}V (ρ_{1})_{i}V (ρ_{2})_{j}^{∗}V (ρ_{1})_{i}^{∗}.

Note that E(ρ_{1}, ρ_{2}) belongs to Ñ^{θ} = N and it does not depend on the choice of implementing systems. It is a routine work to show that E(ρ_{1}, ρ_{2}) is a permutation symmetry for D, and that the resulting crossed product by D and the dual action constructed in [13] are the same as M and α above.

In general, D does not uniquely determine either the compact group K or the Roberts action β [36]. (When K is abelian, the obstruction is nothing but the usual H^{2}(K̂, 𝕋) obstruction for uniqueness of liftings of the K̂-kernel with a trivial 3 cocycle [37], [70].) Whenever we deal with Roberts actions consisting of modular endomorphisms in the rest of this paper, we always assume that we make the above choice of β on the arrows (or equivalently, the above choice of the permutation symmetries).

Theorem 5.5. Let the notations be as above. Then,

(1) M̃ is generated by Ñ and the relative commutant M̃∩ Ñ′, and M̃∩ Ñ′ coincides with the center Z(M̃) of M. The smooth flow of weights F^{M} of M is given by the skew product (X_{N} ×_{c} K, μ_{N} ×m_{K}), that is: X_{M} = X_{N} ×K, μ_{M} = μ_{N} ×m_{K}, and the flow is given by

(ω, k) · t = (ω · t, kc(ω, t)), t ∈ ℝ, k ∈ K, ω ∈ X_{N}.

The factor map from X_{M} onto X_{N} corresponding to the inclusion Z(Ñ) ⊂ Z(M̃) is the projection onto the first component.

(2) The Connes-Takesaki module mod(α_{k}) of α_{k}, k ∈ K is given by

mod(α_{k})(f)(ω, l) = f(ω, k^{−1}l), f ∈ L^{∞}(X_{M}, μ_{M}), (ω, l) ∈ X_{M}.

Proof. Before starting the proof, we simplify the notation a little. We may and do assume K̂ ⊂ R thanks to the assumption (iii). For ρ = β_{π}, (π, H_{π}) ∈ K̂, we simply denote K_{π} = K_{ρ} and V (π)_{i} = V (ρ)_{i}.

(1) We take H_{π} ∈ M and an orthonormal basis {W (π)_{i}}_{i=1}^{d(π)} as before, where {W (π)_{i}}_{i=1}^{d(π)} is transformed by α in the same way as the canonical basis of H_{π} = ℂ^{d(π)}. Let E be the unique normal conditional expectation from M onto N, and Ẽ be the natural extension of E to M̃ that is a normal conditional expectation from M̃ onto Ñ. As in the proof of Lemma 5.1, for x ∈ M̃ we set

x(π)_{i} = d(π)Ẽ(W (π)_{i}^{∗}x).

Then, x is in M̃ ∩ Ñ′ if and only if x(π)_{i} ∈ (id, β̃_{π}) for every (π, H_{π}) ∈ K̂ and every i = 1, 2, ⋯, d(π). This is further equivalent to that there exists z(π)_{ij} ∈ Z(Ñ) such that

x(π)_{i} = ∑_{j=1}^{d(π)} z(π)_{ij}V (π)_{j},

for every (π, H_{π}) ∈ K̂ and every i = 1, 2, ⋯, d(π). Note that thanks to Proposition 4.2, (4) and Lemma 5.4, W (π)_{i} commutes with every element in Z(Ñ). Therefore, in the same way as in [35, Lemma 3.8], we can conclude that M̃ ∩ Ñ′ is generated by Z(Ñ) and {f(π_{ij})}_{π,i,j}, where f(π_{ij}) = W (π)_{i}^{∗}V (π)_{j}. Since M̃ is generated by Ñ and {W (π)_{i}}_{π,i}, it is also generated by Ñ and {f(π_{ij})}_{π,i,j} as well. Thus, to prove Z(M̃) = M̃ ∩ Ñ′, it suffices to show that {f(π_{ij})}_{π,i,j} generate a commutative von Neumann algebra.

Let B be the von Neumann algebra generated by {f(π_{ij})}_{π,i,j}. We show that there exists a ∗ isomorphism from L^{∞}(K) onto B sending π_{ij} to f(π_{ij}). For this, it suffices to check that both π_{ij} and f(π_{ij}) have the same algebraic relations because we have Ẽ(f(π_{ij})) = m_{K}(π_{ij}) = δ_{π,1}, where

m_{K}(ξ) = ∫_{K} ξ(k)dm_{K}(k), ξ ∈ L^{∞}(K).

First, we show f(π_{ij})^{∗} = f(\overline{π}_{ij}). Let R_{π} ∈ (id, β\overline{π} · β_{π}) and R\overline{π} ∈ (id, β_{π} · β\overline{π}) be isometries defined by

R_{π} = \frac{1}{\sqrt{d(π)}} ∑_{i=1}^{d(π)} V (\overline{π})_{i}V (π)_{i},

R\overline{π} = \frac{1}{\sqrt{d(π)}} ∑_{i=1}^{d(π)} V (π)_{i}V (\overline{π})_{i}.

Then, the definition of the crossed product by a Roberts action implies that we also have

R_{π} = \frac{1}{\sqrt{d(π)}} ∑_{i=1}^{d(π)} W (\overline{π})_{i}W (π)_{i},

R\overline{π} = \frac{1}{\sqrt{d(π)}} ∑_{i=1}^{d(π)} W (π)_{i}W (\overline{π})_{i}.

Thus, we get

f(π_{ij})^{∗} = V (π)_{j}^{∗}W (π)_{i} = \sqrt{d(π)}V (π)_{j}^{∗}W (\overline{π})_{i}^{∗}R_{π} = \sqrt{d(π)}W (\overline{π})_{i}^{∗}β̃\overline{π}(V (π)_{j}^{∗})R_{π}
= W (\overline{π})_{i}^{∗}V (\overline{π})_{j} = f(\overline{π})_{ij}.

For the same reason, we have

E(π, σ) = ∑_{i=1}^{d(π)} ∑_{a=1}^{d(σ)} V (σ)_{a}V (π)_{i}V (σ)_{a}^{∗}V (π)_{i}^{∗} = ∑_{i=1}^{d(π)} ∑_{a=1}^{d(σ)} W (σ)_{a}W (π)_{i}W (σ)_{a}^{∗}W (π)_{i}^{∗},

which implies

f(π_{ij})f(σ_{ab}) = W (π)_{i}^{∗}V (π)_{j}W (σ)_{a}^{∗}V (σ)_{b} = W (π)_{i}^{∗}W (σ)_{a}^{∗}β̃_{σ}(V (π)_{j})V (σ)_{b}
= W (π)_{i}^{∗}W (σ)_{a}^{∗}V (σ)_{b}V (π)_{j} = W (π)_{i}^{∗}W (σ)_{a}^{∗}E(π, σ)V (π)_{j}V (σ)_{b}
= W (σ)_{a}^{∗}W (π)_{i}^{∗}V (π)_{j}V (σ)_{b} = f(σ_{ab})f(π_{ij}).

Thus, B is abelian. For the same reason again, there exist T_{(π,i),(σ,a)}^{(ν,s)} ∈ (β_{ν}, β_{π} · β_{σ}) such that

V (π)_{i}V (σ)_{a} = ∑_{(ν,H_{ν})∈K̂} ∑_{s=1}^{d(ν)} T_{(π,i),(σ,a)}^{(ν,s)}V (ν)_{s},

W (π)_{i}W (σ)_{a} = ∑_{(ν,H_{ν})∈K̂} ∑_{s=1}^{d(ν)} T_{(π,i),(σ,a)}^{(ν,s)}W (ν)_{s},

which implies

f(π_{ij})f(σ_{ab}) = ∑_{(ν,H_{ν})∈K̂} ∑_{s,t=1}^{d(ν)}⟨T_{(π,j),(σ,b)}^{(ν,t)}, T_{(π,i),(σ,a)}^{(ν,s)}⟩f(ν_{st}).

Note that thanks to the Peter-Weyl theorem, T_{(π,i),(σ,a)}^{(ν,s)} is obtained by

T_{(π,i),(σ,a)}^{(ν,s)} = d(ν)E(W (π)_{i}W (σ)_{a}W (ν)_{s}^{∗}).

Thus, we have

⟨T_{(π,j),(σ,b)}^{(ν,t)}, T_{(π,i),(σ,a)}^{(ν,s)}⟩
= d(ν)^{2} ∫_{K} E(W (ν)_{s}W (σ)_{a}^{∗}W (π)_{i}^{∗}α_{g}(W (π)_{j}W (σ)_{b}W (ν)_{t}^{∗}))dm_{K}(g)

= d(ν)^{2} ∑_{r=1}^{d(ν)} ∫_{K} π_{ij}(g)σ_{ab}(g)\overline{ν_{rt}(g)}E(W (ν)_{s}W (ν)_{r}^{∗})dm_{K}(g)

= d(ν) ∫_{K} π_{ij}(g)σ_{ab}(g)\overline{ν_{st}(g)}dm_{K}(g).

This shows that

π_{ij}σ_{ab} = ∑_{(ν,H_{ν})∈K̂} ∑_{s,t=1}^{d(ν)}⟨T_{(π,j),(σ,b)}^{(ν,t)}, T_{(π,i),(σ,a)}^{(ν,s)}⟩ν_{st}

holds as well thanks to the Peter-Weyl theorem again. Therefore, there exists a desired isomorphism from L^{∞}(K) onto B.

Let φ_{0} be a faithful normal state of Ñ whose restriction is given by the measure μ_{N}, and we set φ = φ_{0} · Ẽ. Then, for z ∈ Z(Ñ), f ∈ B, we have

φ(zf) = φ_{0}(zẼ(f)) = φ(z)φ(f).

This shows that Z(M̃) = Z(Ñ) ⊗ B. θ acts on f(π_{ij}) as

θ_{t}(f(π_{ij})) = W (π)_{i}^{∗}θ_{i}(V (π)_{j}) = ∑_{l=1}^{d(π)} W (π)_{i}^{∗}V (π)_{l}π_{lj}(c(t)) = ∑_{l=1}^{d(π)} f(π_{il})π_{lj}(c(t)),

which shows that the flow is given by the skew product.

(2) Since α̃ acts on V (π)_{i} trivially, we have

α̃_{k}(f(π_{ij})) = α_{k}(W (π)_{i}^{∗})V (π)_{j} = ∑_{l=1}^{d(π)} \overline{π_{li}(k)}f(π_{lj}) = ∑_{l=1}^{d(π)} π_{il}(k^{−1})f(π_{lj}).

This means that the Connes-Takesaki module of α_{k} is given as in the statement.

Definition 5.6. Let N be a type III factor, K be a compact group, and c ∈ Z^{1}(F^{N}, K) be a minimal cocycle with K_{c} = K. For each (π, H_{π}) ∈ K̂, we choose ρ_{π} ∈ End(N)_{m} such that δ_{m}([ρ_{π}]) = [π · c]. Let D be the subset of End(N)_{m} generated by {ρ_{π}}_{(π,H_{π})∈K̂} and satisfying the above conditions (i)-(iv). We denote by N ⊗_{c} L^{∞}(K) the above crossed product N ⋊_{β} K̂ and call it the skew product of N and L^{∞}(K) by the cocycle c. We call β̂ the dual action of the skew product.

Note that (N ⊗_{c} L^{∞}(K), β̂) depends only on the class [c] ∈ H^{1}(F^{N}, K).

Our next task is to show the “converse” of the above theorem, and we need some preparation.

Lemma 5.7. Let α be an action of a locally compact group G on a von Neumann algebra M, and H be the standard Hilbert space of M. Let J̃ be the modular conjugation of M ⋊_{α} G := π_{α}(M) ∨ (ℂ ⊗ l(G)′′) acting on L^{2}(G, H) as given by [18, Lemma 2.8]. Then, we have

J̃((M ⋊_{α} G) ∩ π_{α}(M)′)J̃ = (M ⊗ B(L^{2}(G))) ∩ (M ⋊_{α} G)′.

Proof. Let J be the modular conjugation of M on H. Then, J̃ is given by

(J̃ξ)(g) = Δ_{G}(g)^{−1/2}u(g)^{∗}Jξ(g), ξ ∈ L^{2}(G, H),

where Δ_{G} is the modular function of G and u(g) is the canonical implementation of G on H. Direct computation yields

J̃π_{α}(x)J̃ = JxJ ⊗ ℂ, x ∈ M,
J̃(u(g) ⊗ r(g))J̃ = 1 ⊗ l(g), g ∈ G.

Thus, we get

J̃((M ⋊_{α} G) ∩ π_{α}(M)′)J̃ = J̃((M ⊗ B(L^{2}(G))) ∩ (u ⊗ r)(G)′ ∩ π_{α}(M)′)J̃
= (J̃((M′ ⊗ ℂ) ∨ (u ⊗ r)(G)′′ ∨ π_{α}(M))J̃)′
= (π_{α}(M) ∨ (ℂ ⊗ l(G)′′) ∨ (M′ ⊗ ℂ))′
= (M ⋊_{α} G)′ ∩ (M ⊗ B(L^{2}(G))).

Let M ⊃ N be an irreducible inclusion of factors with a normal conditional expectation E. We assume that M̃ is generated by Ñ and M̃∩Ñ′. Then, the restriction of Ẽ to M̃ ∩ Ñ′ is a conditional expectation onto Z(Ñ). For a, b ∈ M̃ ∩ Ñ′, Ẽ satisfies the trace property Ẽ(ab) = Ẽ(ba). Indeed, for x ∈ Ñ in the domain of τ, we have

τ(Ẽ(ab)x) = τ(abx) = τ(axb) = τ(bax) = τ(Ẽ(ba)x),

which shows Ẽ(ab) = Ẽ(ba).

Lemma 5.8. Let N be a type III factor and A be a von Neumann algebra whose center includes Z(Ñ). We assume that there exist an ergodic ℝ-action ϑ on A extending the flow of Z(Ñ), and a faithful normal conditional expectation ϵ from A onto Z(Ñ) such that θ_{t} · ϵ = ϵ · ϑ_{t}, t ∈ ℝ and ϵ(xy) = ϵ(yx) for x, y ∈ A. Then, there exists a unique irreducible inclusion M ⊃ N of type III factors with a normal conditional expectation E such that M̃ = (M̃ ∩ Ñ′) ∨ Ñ and

(A ⊃ Z(Ñ), ϑ, ϵ) ≅ (M̃ ∩ Ñ′ ⊃ Z(Ñ), θ|_{M̃∩Ñ′}, Ẽ|_{M̃∩Ñ′}).

Proof. First, we show uniqueness of M. Let M be a von Neumann algebra including N with a unique conditional expectation E from M onto N. We assume that M̃ is generated by M̃ ∩ Ñ′ and Ñ. Then, the linear span of (M̃ ∩ Ñ′)Ñ is dense in M̃ by assumption. For a faithful normal state φ on Ñ and a ∈ M̃ ∩ Ñ′, x ∈ Ñ, we have

φ · Ẽ(ax) = φ(Ẽ(a)x).

This means that the structure of (M̃ ⊃ Ñ, Ẽ, θ) is uniquely determined by (Ñ, θ|Ñ) and

(M̃ ∩ Ñ′ ⊃ Z(Ñ), θ|_{M̃∩Ñ′}, Ẽ|_{M̃∩Ñ′}).

On the other hand, (M̃ ⊃ Ñ, Ẽ, θ) completely determine the original inclusion via the Takesaki duality theorem. Therefore, such M as in the statement is unique if it exists.

We set L = Ñ ⊗_{Z(Ñ)} A as in [64], [69]. More precisely, let H be the standard Hilbert space of Ñ. We introduce an inner product into the algebraic tensor product H ⊙_{Z(Ñ)} A by

⟨ξ ⊙ a, η ⊙ b⟩ = ⟨ϵ(b^{∗}a)ξ, η⟩, a, b ∈ A, ξ, η ∈ H.

Let H_{1} be the completion of H ⊙_{Z(Ñ)} A, and Λ be the natural map from H ⊙_{Z(Ñ)} A to H_{1}. Ñ and A naturally act on the first and the second tensor components of H_{1}, and we denote by L the von Neumann algebra generated by Ñ and A in B(H_{1}). In terms of direct integral, L can be expressed as follows (though we do not use it in the proof): Let

Ñ = ∫_{X_{N}}^{⊕} Ñ(ω)dμ_{N}(ω),

A = ∫_{X_{N}}^{⊕} A(ω)dμ_{N}(ω),

be disintegration of Ñ and A over Z(Ñ). Then L is given by

L = ∫_{X_{N}}^{⊕} Ñ(ω) ⊗ A(ω)dμ_{N}(ω).

Thanks to the Connes-Takesaki relative commutant theorem and Lemma 5.7, we have Ñ ∩N′ = Z(Ñ). We show L∩N′ = A using this and applying [69, Theorem 2.3] (or alternatively, using the above direct integral expression). To do so, we need to separate A and N by a type I von Neumann algebra. Indeed, let F be the commutant of Z(Ñ) in B(H). Then, F naturally acts on H_{1} and we have N ⊂ F and A ⊂ F′, where the commutant F′ is considered in B(H_{1}). Applying [69, Theorem 2.3] twice, we get

L ∩ N′ = (N ∨ Z(Ñ))′ ∩ (Ñ ∨ A) = (N ∨ Z(Ñ) ∨ (Ñ ∨ A)′)′
= (N ∨ Z(Ñ) ∨ (F ∩ Ñ′) ∨ (F′ ∩ A′))′
= ((F ∩ (Ñ ∨ N′)′) ∨ (F′ ∩ A′))′
= ((F ∩ Z(Ñ)′) ∨ (F′ ∩ A′))′
= (F ∨ (F′ ∩ A′))′
= Z(Ñ) ∨ A = A.

We introduce a 1-parameter automorphism group Θ on L extending both θ and ϑ. Let u(t) be the canonical implementation of θ_{t} on H. We set

U(t)Λ(ξ ⊙ a) = Λ(u(t)ξ ⊙ ϑ_{t}(a)), ξ ∈ H, a ∈ A.

Then, U(t) extends to a 1-parameter unitary group on H_{1} satisfying

U(t)xaU(t)^{∗} = θ_{t}(x)ϑ(a), x ∈ Ñ, a ∈ A.

We define Θ_{t} to be the restriction of Ad(U(t)) to L.

Next, we show that there exists a faithful normal conditional expectation ϵ̃ from L onto Ñ satisfying

ϵ̃(a) = ϵ(a), a ∈ A.

Let φ be a faithful normal state on Ñ, and ξ_{φ} ∈ H be the cyclic and separating vector of Ñ giving φ. We set Ω = Λ(ξ_{φ} ⊙1), and claim that Ω is cyclic and separating vector for L. Indeed, Ω is clearly a cyclic vector. To show that it is separating, we introduce a conjugation J on H_{1} by

JΛ(ξ ⊙ a) = Λ(JÑξ ⊙ a^{∗}), ξ ∈ H, a ∈ A

where JÑ is the modular conjugation of Ñ. It is easy to show that J extends to a conjugation such that JLJ ⊂ L′. Since JLJ is cyclic for Ω, Ω is a separating for L. We denote by ω the state of L defined by ω(x) = ⟨xΩ, Ω⟩, x ∈ L. Then, it is a routine work to show that J is the modular conjugation for ω, and the modular automorphism group {σ_{t}^{ω}}_{t∈ℝ} is given by

σ_{t}^{ω}(xa) = σ_{t}^{φ}(x)a, x ∈ Ñ, a ∈ A.

Thanks to Takesaki’s theorem [74], there exists a ω-preserving normal conditional expectation ϵ̃ from L onto Ñ. Since ω(xa) = ω(xϵ(a)) for x ∈ Ñ, a ∈ A, ϵ̃ satisfies ϵ̃(a) = ϵ(a).

We set M = L^{Θ} and E to be the restriction of ϵ̃ to M. Since ϵ̃ satisfies ϵ̃· Θ = θ · ϵ̃, E is a faithful normal conditional expectation from M onto N. Therefore, M is of type III [67]. Let ν be the restriction of Ad(λ^{ψ}(t)) to M, where ψ is a faithful normal semifinite weight on N. Then, thanks to the Landstad theorem [58, Chapter II, Section 2], L = M ⋊_{ν} ℝ and Θ is the dual action of ν. We claim that M ∩N′ = ℂ, and in particular M is a factor. Indeed, we have

M ∩ N′ ⊂ L ∩ N′ = A.

Since Θ acts on A ergodically, we get M ∩ N′ = ℂ.

Instead of showing that ν comes from the modular automorphism group directly, we show it for the dual action of Θ. We define τ_{1} = τ · ϵ̃, which is a faithful normal semifinite trace on L satisfying τ_{1}·Θ_{t} = e^{−t}τ_{1}. Indeed, since τ is a trace, σ_{t}^{τ_{1}} acts on Ñ trivially. On the other hand, since ϵ̃ is a faithful normal conditional expectation from L onto Ñ and L ∩ Ñ′ = A, the restriction of σ_{t}^{τ_{1}} to A is the same as the restriction of σ_{t}^{ω} to A [10], which is trivial as we saw before. Thus, τ_{1} is a trace with the scaling property

τ_{1} · Θ_{t} = τ · θ_{t} · ϵ̃ = e^{−t}τ_{1}.

This means that the dual action of Θ is the modular automorphism group for the dual weight τ̂_{1}. Thus, identifying L ⋊_{Θ} ℝ with M ⊗ B(L^{2}(R)) using the Takesaki duality theorem, we get

τ̂_{1} = τ̂ · (E ⊗ id),

στ̂_{1} ; t = (ν_{t} ⊗ Ad(r(t))).

Let h be the generator of r(t). Then, we have τ̂ = ψ ⊗ Tr(h·), which implies τ̂_{1} = ψ · E ⊗ Tr(h·). Thus, we get ν_{t} = σ_{t}^{ψ·E}, which implies L = M̃ and ϵ̃ = Ẽ.

Theorem 5.9 (Automatic Minimality). Let α be an action of a compact group K on a type III factor M. If the kernel of mod · α is trivial, then α is minimal. The smooth flow of weight of the fixed point algebra M^{α} is the factor flow of F^{M} by mod · α_{K}. There exists a minimal cocycle c ∈ Z^{1}(F^{M^{α}}, K) with K_{c} = K, unique up to equivalence, such that M is the skew product M^{α} ⊗_{c} L^{∞}(K), and α is the dual action of the skew product.

Proof. We use the same notation as in Lemma 5.4. Lemma A.1 implies that the smooth flow of weights of M is of the form (X_{M}, μ_{M}) = (Y × K, ν × m_{K}) with the flow and mod(α_{k}) given by

(ω, l) · t = (ω · t, lc(ω, t)), (ω, l) ∈ X_{M}, t ∈ ℝ,
(mod(α_{k})f)(ω, l) = (ω, k^{−1}l), k ∈ K, f ∈ L^{∞}(X_{M}, μ_{M}),

where c is a minimal cocycle with K_{c} = K. According to the above splitting of (X_{M}, μ_{X}), we set C = L^{∞}(K) ⊂ Z(M̃) and Z = L^{∞}(Y, ν) ⊂ Z(M̃) that is the fixed point subalgebra of Z(M̃) under α̃. Note that the factor flow (Z, θ|_{Z}) and the class of c are uniquely determined by F^{M} and mod · α. Since C is an equivariant copy of L^{∞}(K), the Landstad’s type theorem for coactions [58, Chapter II, Section 2] implies that M̃ is the crossed product Ñ ⋊ K̂ by a coaction of K, where we use the fact that Ñ is the fixed point subalgebra of M̃ under α̃. Moreover, since Ñ commutes with C, the coaction is trivial and we actually have M̃ ≅ Ñ ⊗ C, where Ñ and C are identified with Ñ ⊗ ℂ and ℂ ⊗ C in the right-hand side respectively. This implies Z = Z(Ñ), M̃ ∩ Ñ′ = Z(M̃) and M̃ = (M̃ ∩ Ñ′) ∨ Ñ. Since θ acts on Z ergodically, N is a factor. If N were semifinite, c would be a coboundary, which would imply that K is trivial and M = N. This is contradiction because M is of type III. Therefore, N is a type III factor. Now, Theorem 5.5 and Lemma 5.8 show that M is the skew product N ⊗_{c} L^{∞}(K). Since the Galois group for M ⊃ N coincides with α_{K} [63],[2] and every member α_{k} in the Galois group is specified by mod(α_{k}), we conclude that α is the dual action of the skew product.

Corollary 5.10. If M is an AFD factor, then for every compact subgroup K of Aut(F^{M}), there exists a minimal action α of K on M such that mod · α_{k} = k for all k ∈ K. α is unique up to conjugacy: namely if α^{1} also satisfies the condition, then there exists an automorphism ν ∈ Aut(M) with a trivial Connes-Takesaki module such that

α_{k} = ν · α_{k}^{1} · ν^{−1}, k ∈ K.

Proof. This is an easy consequence of Theorem 5.5, Theorem 5.9, and the fact that AFD type III factors are determined by their flows [6], [7], [8], [40], [20].

As for the existence of a homomorphic lifting in the above corollary, a more general statement is known: there always exists a homomorphic lifting α from Aut(F^{M}) to Aut(M) for every AFD type III factor M [73].

Now we characterize (not necessarily finite index) subfactors whose canonical endomorphism is decomposed into modular endomorphisms. Let M ⊃ N be an irreducible inclusion of factors with a unique normal conditional expectation from M onto N. We denote by M_{1} and Ê the basic extension and the dual operator valued weight from M_{1} to M respectively [19], [41], [35, Section 2]. We say that the inclusion M ⊃ N is discrete if the restriction of E · Ê on the relative commutant M_{1} ∩ N′ is semifinite. We say that a discrete inclusion is unimodular if E · Ê|_{M_{1}∩N′} is invariant under j(x) := J_{M}x^{∗}J_{M}, where J_{M} is the modular conjugation of M (see [35, Section 3] for conditions equivalent to these.)

Theorem 5.11. Let M ⊃ N be an inclusion of type III factors with a unique normal conditional expectation E from M onto N. Then, the following conditions are equivalent:

(1) M ⊃ N is a unimodular inclusion and the restriction of the canonical endomorphism γ : M ⟶ N to N is decomposed into modular endomorphisms.

(2) M ⊃ N is a discrete inclusion and M̃ is generated by Ñ and M̃ ∩ Ñ′.

(3) (M̃∩Ñ′, θ) is an extension of the flow (Z(Ñ), θ) with relatively discrete spectrum in the sense of Definition A.2, and M̃ is generated by Ñ and M̃ ∩ Ñ′.

(4) There exists a compact group K, a faithful ergodic action Ψ of K on a von Neumann algebra B, a minimal action α of K on a factor P such that the kernel of mod · α is trivial and

(M ⊃ N) ≅ ((P ⊗ B)^{α⊗Ψ} ⊃ P^{α} ⊗ ℂ).

In the case (4), K is determined by the cocycle c corresponding to the system of modular endomorphisms appearing in the decomposition of γ|_{N}, P is the skew product N ⊗_{c} L^{∞}(K), and α is the dual action of the skew product. B and Ψ are uniquely determined from (M̃ ∩ Ñ′ ⊃ Z(Ñ), θ) by Theorem A.4.

When one (and all) of the above conditions holds, N is AFD if and only if M is AFD.

Proof. (1) ⇒ (2). We assume that (1) holds and the irreducible decomposition of γ|_{N} is given by

[γ|_{N}] = ⨁_{ξ∈Ξ} n_{ξ}[ρ_{ξ}].

Thanks to [35, pp. 39], the multiplicity n_{ξ} of ρ_{ξ} is always finite. We set

L_{ξ} = {X ∈ M; Xx = ρ_{ξ}(x)X, ∀x ∈ N}.

Then, the dimension of L_{ξ} is n_{ξ} and N is generated by N and ⋃_{ξ∈Ξ} L_{ξ} [35, Theorem 3.3]. Let {X(ξ)_{i}}_{i=1}^{n_{ξ}} be an orthonormal basis of L_{ξ}. We choose a dominant weight ψ_{0} on N and assume that (ψ_{0}, ρ_{ξ}) is an invariant pair for every ξ ∈ Ξ as before. Then, since M ⊃ N is unimodular, [35, Theorem 3.3, Remark 3.4] imply that L_{ξ} is in the centralizer of ψ = ψ_{0} · E. By assumption, there exists an implementing system {V (ξ)_{i}}_{i=1}^{d(ξ)} ⊂ Ñ for ρ̃_{ξ}. As X(ξ)_{i} is in the centralizer of ψ, it commutes with λ^{ψ}(t), and so

V (ξ)_{i}^{∗}X(ξ)_{j} ∈ M̃ ∩ Ñ′, ξ ∈ Ξ, 1 ≤ i ≤ d(ρ_{ξ}), 1 ≤ j ≤ n_{ξ}.

Since M̃ is generated Ñ and ∪_{ξ∈Ξ}L_{ξ}, it is also generated by Ñ and M̃ ∩ Ñ′.

(2) ⇒ (3). We assume (2), and set A = M̃∩Ñ′, Z = Z(Ñ), ϵ = Ẽ|_{A}, and ϑ_{t} = θ_{t}|_{A}. Then, ϵ is a normal conditional expectation with the trace property ϵ(ab) = ϵ(ba), a, b ∈ A. Let M_{1} be the basic extension of M ⊃ N. As before, we regard M̃ as a subalgebra of M̃_{1} identifying λ^{ψ}(t) with λ^{ψ·Ê}(t). Then, M̃_{1} is the basic extension of M̃ ⊃ Ñ with respect to Ẽ [35, Lemma 2.4], [17, Theorem 2.1].

On the other hand, we consider the inclusion Z ⊂ A acting on the standard Hilbert space L^{2}(A) of A. Then, the basic extension of this inclusion is Z_{1} = Z′ ⊂ B(L^{2}(A)). Let u be the canonical implementation of ϑ, and J_{A} be the modular conjugation of A. We use the same notation as in the proof of Lemma 5.8, and identify M̃ with L. Note that Z_{1} naturally acts on H_{1} through the action on the second tensor component. In the same way as before, we get the following:

M̃_{1} = JÑ′J = JÑF JÑ ∨ J_{A}Z_{1}J_{A} = Ñ ∨ Z_{1},

where everything takes place in B(H_{1}). Thus, M̃_{1} ∩ Ñ′ = Z_{1}, and

(M̃_{1} ∩ Ñ′, θ, E · Ê) = (Z_{1}, Ad(u(t)), ϵ · ϵ̂),
((M_{1} ∩ N′)_{E·Ê}, E · Ê) = (Z_{1} ∩ u(ℝ)′, ϵ · ϵ̂),

where (M_{1}∩N′)_{E·Ê} is the fixed point subalgebra of M_{1}∩N′ under the relative modular automorphism group {σ_{t}^{E·Ê}}_{t∈ℝ} [19]. Since M ⊃ N is discrete, [35, Theorem 3.3] implies that (M_{1} ∩ N′)_{E·Ê} is a direct sum of matrix algebras and the restriction of E ·Ê to (M_{1} ∩N′)_{E·Ê} is still semifinite. Therefore, if ϵ·ϵ̂ is the center valued trace ♮ of Z_{1}, (A, ϑ) is an extension of the flow F^{N} with relatively discrete spectrum. Indeed, let μ be a faithful normal state on Z(Ñ). Then using the spatial derivative, we can show that the modular automorphism group of μ·ϵ·ϵ̂ is implemented by the modular operator Δ_{μ·ϵ}^{it} of μ · ϵ on L^{2}(A) [41]. However, since μ · ϵ is a trace and Δ_{μ·ϵ} is trivial, μ · ϵ · ϵ̂ is a trace as well. This means that ϵ · ϵ̂ = ♮(z·) holds with some density z affiliated with Z(Ñ). Let e_{Z(Ñ)} be the Jones projection for ϵ, and a ∈ A. Then, ae_{A}a^{∗} is considered as a field of scalar multiples of rank one projections in the central decomposition of Z_{1}. Since

ϵ · ϵ̂(ae_{A}a^{∗}) = ϵ(aa^{∗}) = ϵ(a^{∗}a),

we actually have z = 1.

(3) ⇒ (4). We assume (3). Thanks to Theorem A.4, there exists a compact group K, a minimal cocycle c ∈ Z^{1}(F^{N}, K) with K_{c} = K, and a faithful ergodic action Ψ of K on a von Neumann algebra B such that

(M̃ ∩ Ñ′ ⊃ Z(Ñ), θ) ≅ ((Z(Ñ) ⊗ B) ⊃ (Z(Ñ) ⊗ ℂ), Ψ^{c}),

(see Theorem A.4 for the notations). Let P be the skew product N ⊗_{c} L^{∞}(K) and α be the dual action. We set Q = (P ⊗ B)^{α⊗Ψ}. Then, thanks to ergodicity of Ψ, Q ⊃ N is an irreducible inclusion of factors. Let τ_{B} be the unique Ψ-invariant trace [30]. Then, the restriction of (id ⊗ τ_{B}) to Q gives a normal conditional expectation from Q onto N. Thanks to Lemma 5.8, to prove the statement it suffices to show that Q̃ is generated by Ñ and Q̃ ∩ Ñ′, and

(Q̃ ∩ Ñ′ ⊃ Z(Ñ), θ) ≅ (Z(Ñ) ⊗ B ⊃ Z(Ñ) ⊗ ℂ, Ψ^{c}).

Lemma 5.4 and Theorem 5.9 show that Q̃ is actually given by

Q̃ = (P̃ ⊗ B)^{α̃⊗Ψ} = Ñ ⊗ (L^{∞}(K) ⊗ B)^{Ad(l)⊗Ψ},

where l is the left regular representation. Thus, we have

Q̃ ∩ Ñ′ = L^{∞}(X_{N}, μ_{N}) ⊗ (L^{∞}(K) ⊗ B)^{Ad(l)⊗Ψ},

which shows Q̃ = Ñ ∨ (Q̃ ∩ Ñ′). Note that θ acts on B trivially and it acts on Z(P̃) = L^{∞}(X_{N}, μ_{N}) ⊗ L^{∞}(K) as in Theorem 5.9.

Now we assume that Z(P̃) ⊗ B acts on L^{2}((X_{P} , μ_{P}), L^{2}(B)), where L^{2}(B) is the standard Hilbert space of B and (X_{P} , μ_{P}) = (X_{N} × K, μ_{N} × m_{K}). Let u and v be the canonical implementations of F^{P} on L^{2}(X_{P} , μ_{P}) and Ψ on L^{2}(B) respectively. Then, u is given by

(u(t)ξ)(ω, k) = \frac{dt · μ_{P}}{dμ_{P}} (ω)^{1/2}ξ(ω · t, kc(ω, t)), ξ ∈ L^{2}(X_{P} , μ_{P}).

We introduce a unitary operator W on L^{2}(K, L^{2}(B)) by

(W η)(k) = v(k)η(k), η ∈ L^{2}(K, L^{2}(B)),

which satisfies W (l(k) ⊗ 1)W^{∗} = l(k) ⊗ v(k). Let Π be the restriction of Ad(W) on L^{∞}(K) ⊗ B, which is an automorphism of L^{∞}(K) ⊗ B. Then, the above relation implies

(L^{∞}(K) ⊗ B)^{Ad(l)⊗Ψ} = Π(ℂ ⊗ B),
(r(k) ⊗ 1)Π(1 ⊗ b)(r(k)^{∗} ⊗ 1) = Π(1 ⊗ Ψ_{k}(b)), b ∈ B.

For f ∈ L^{∞}(X_{N}, μ_{N}), b ∈ B, and ξ ∈ L^{2}((X_{N}, μ_{N}), L^{2}(K) ⊗ L^{2}(B)) we have

(u(t)(f ⊗ Π(1 ⊗ b))u(t)^{∗})ξ(ω)
= f(ω · t)(r(c(ω, t)) ⊗ 1)Π(1 ⊗ b)(r(c(ω, t))^{∗} ⊗ 1)ξ(ω)
= f(ω · t)Π(1 ⊗ Ψ_{c(ω,t)}(b))ξ(ω).

Thus, we get

(Q̃ ∩ Ñ′ ⊃ Z(Ñ), θ) ≅ (Z(Ñ) ⊗ B ⊃ Z(Ñ) ⊗ ℂ, Ψ^{c}).

(4) ⇒ (1). We assume (4) and set N = P^{α}, M = (P ⊗ B)^{α⊗Φ}. Let τ_{B} be the unique Ψ-invariant trace on B, L^{2}(B) be the GNS Hilbert space of τ_{B}, and Λ_{τ_{B}} be the natural map from B into L^{2}(B). We denote by v the canonical implementation of Ψ on L^{2}(B). For each (π, H_{π}) ∈ K̂, we denote by B_{π} the spectral subspace corresponding to π, which is the image of the map

E_{π}(x) = d(π) ∫_{K} Tr(\overline{π}(k))Ψ_{k}(x)dm_{K}(k), x ∈ B.

It is known that the multiplicity of B_{π} is finite [30] and there exists an orthonormal basis {b(π, s)_{i}} ⊂ B_{π}, 1 ≤ s ≤ n_{π}, 1 ≤ i ≤ d(π) with respect to the inner product of L^{2}(B) such that

Ψ_{k}(b(π, s)_{i}) = ∑_{j=1}^{d(π)} π_{ji}(k)b(π, s)_{j}, k ∈ K.

Since τ_{B} is given by the average of Ψ_{k} over K, the Peter-Weyl theorem implies

∑_{i=1}^{d(π)} b(π, s)_{i}^{∗}b(π, t)_{i} = ∑_{i=1}^{d(π)} b(π, t)_{i}b(π, s)_{i}^{∗} = d(π)δ_{s,t}.

Note that {b(π, s)_{i}}_{π,s,i} form an orthonormal basis of L^{2}(B). Though expansion of x ∈ B with this basis does not converge in weak topology in general, we can still show that {b(π, s)_{i}}_{π,s,i} generate B using the same argument as in [35, Lemma 3.8]. The point of the argument is that when a von Neumann subalgebra is globally invariant under the modular automorphism group of a state, the corresponding subspace in the GNS subspace determines the subalgebra. We use this principle in the rest of the proof without mentioning it.

Let E_{0} be the unique normal conditional expectation from P onto N, φ_{0} be a faithful normal state on N, and φ = φ_{0} · E_{0}. We denote by L^{2}(P), Λ_{φ}, and w the GNS Hilbert space of φ, the natural map from P into L^{2}(P), and the canonical implementation of α on L^{2}(P) respectively. We regard the subspace of w⊗v-invariant vectors in L^{2}(P) ⊗ L^{2}(B) as the GNS Hilbert space L^{2}(M) of the restriction of φ ⊗ τ_{B} to M. Let β be the Roberts action of K̂ on N whose dual action is α, and {W (π)_{i}}_{i=1}^{d(π)} ⊂ P, (π, H_{π}) ∈ K̂ be an orthonormal basis of H_{π} ⊂ P as before. Thanks to Theorem 5.9, β_{π} is a modular endomorphism for all (π, H_{π}) ∈ K̂. Note that NH_{π} is globally invariant under the modular automorphism group {σ_{t}^{φ}}_{t∈ℝ} [35, Lemma 3.8]. Let

L_{π} = {X ∈ M; Xx = β_{π}(x)X, ∀x ∈ N}.

Thanks to [35, Theorem 3.3], if M is generated by N and ⋃_{(π,H_{π})∈K̂} L_{π}, M ⊃ N is a discrete inclusion.

Now we set

X(π)_{s} = \frac{1}{\sqrt{d(π)}} ∑_{i=1}^{d(π)} W (π)_{i} ⊗ b(π, s)_{i}^{∗} ∈ L_{π}.

Then, {X(π)_{s}}_{s=1}^{n_{π}} are isometries with mutually orthogonal ranges. The definition of the crossed product by a Roberts action implies

L^{2}(P) = ⨁_{(π,H_{π})∈K̂} \overline{Λ_{φ}(H_{π}^{∗}N)}^{||·||}.

Therefore, direct computation using the orthonormal basis {b(π, s)_{i}}_{π,s,i} yields

L^{2}(M) = ⨁_{(π,H_{π})∈K̂} \overline{Λ_{φ⊗τ_{B}}(L_{π}^{∗}N)}^{||·||}.

Thus, M is generated by N and ⋃_{(π,H_{π})∈K̂} L_{π}, and so M ⊃ N is a discrete inclusion.

Let E be the restriction of id ⊗ τ_{B} to M, which is a conditional expectation from M onto N. Then, we have

d(π)E(X(π)_{s}X(π)_{t}^{∗}) = ∑_{i,j=1}^{d(π)} τ_{B}(b(π, s)_{i}^{∗}b(π, t)_{j})W (π)_{i}W (π)_{j}^{∗}

= δ_{s,t} ∑_{i=1}^{d(π)} W (π)_{i}W (π)_{i}^{∗}
= ⟨X(π)_{s}, X(π)_{t}⟩.

This shows that the inclusion is unimodular [35, Theorem 3.3].

When N is AFD, Ñ is AFD and M̃ is AFD thanks to the proof of Lemma 5.8 and Theorem A.4. Therefore, M is AFD. Since there exists a conditional expectation from M onto N, the converse also holds [7].

Remark 5.12. In the above, if M̃∩ Ñ′ is commutative, B in (4) is commutative, and there exists a unique closed subgroup L ⊂ K such that B = L^{∞}(K/L) where K action is given by the right translation. Thus,

(M ⊃ N) ≅ (P^{α|_{L}} ⊃ P^{α}).

This characterizes an inclusion coming from a (commutative) ergodic extension of the flow of weights. For related topics, see [25], [26], [33], [27], [42], [45], [71], [66], [77]. Under the condition that the kernel of mod·α is trivial, (and L does not contain non-trivial normal subgroups, which is regarded as a trivial constraint), L ⊂ K are unique up to conjugacy. However, if we drop this condition, there are essentially different description of M ⊃ N in terms of a group-subgroup subfactor [46], [36].

Another easy but interesting case is when B is a matrix algebra, that is, M ⊃ N is a Wassermann’s type subfactor. In this case, M and N have the common flow and graph change occurs (c.f. [57]).

5.3. Connes-Takesaki modules for Roberts actions. In this subsection, we show how the Connes-Takesaki modules of endomorphisms give constraints to minimal actions of compact groups, in particular semisimple compact Lie groups.

Let K be a compact group and β be a Roberts action of K̂ on a factor N such that β_{π} is irreducible and has a Connes-Takesaki module for every (π, H_{π}) ∈ K̂. We denote m_{β}(π) = mod(β_{π}) for simplicity. Then, thanks to Proposition 4.2, {m_{β}(π)}_{(π,H_{π})∈K̂} form a group, which is denoted by m_{β}(K̂). We treat m_{β}(K̂) as a discrete group, and we denote by m̂_{β} its dual group.

Theorem 5.13. Let K be a compact group and β be a Roberts action of K̂ on a factor N such that β_{π} is irreducible for every (π, H_{π}) ∈ K̂. Then,

(1) If K is connected, β_{π} has a Connes-Takesaki module for every (π, H_{π}) ∈ K̂.

(2) If β_{π} has a Connes-Takesaki module for every (π, H_{π}) ∈ K̂, then there exists a continuous isomorphism ν from m̂_{β} into the center Z(K) of K satisfying

χ(m_{β}(π))1_{π} = π(ν(χ)), χ ∈ m̂_{β}, (π, H_{π}) ∈ K̂.

Proof. (1). Suppose that there exists (π, H_{π}) ∈ K̂ such that β_{π} does not have a Connes-Takesaki module. Let N = β_{π}(M), and we consider the Loi-Hamachi-Kosaki decomposition of M ⊃ N [47], [25], [26] as follows:

M ⊃ P ⊃ Q ⊃ N,

where P and Q are determined by the conditions

Q̃ = Z(M̃ ∩ Ñ′) ∨ Ñ,
P̃ = M̃ ∩ Z(M̃ ∩ Ñ′)′.

Since β_{π} does not have a Connes-Takesaki module, either M ≠ P or Q ≠ N occurs. We may assume M ≠ P by considering β\overline{π} if necessary. Then, β_{π} · β\overline{π} contains the the canonical endomorphism γ for M ⊃ P. Note that the dual inclusion P_{1} ⊃ M satisfies the condition of Theorem 5.11 and (Z(P̃_{1} ∩ M̃′) = Z(P̃_{1}), θ) is a finite to one extension of the flow F^{M} [25], [26]. Thus, Theorem 5.11 implies that γ generates a Roberts action of the dual Ĝ of a finite group G, which is a subset of {β_{π}}_{(π,H_{π})∈K̂}. We regard Ĝ as a subset of K̂. We define a closed normal subgroup L ⊂ K by

L = ⋂_{(π,H_{π})∈Ĝ} Ker(π).

Since Ĝ is a finite set of irreducible representations closed under complex conjugate and irreducible decomposition of products of any members in Ĝ, the quotient group K/L is finite [28, Section 30]. However, this is contradiction because K is connected. Therefore, β_{π} has a Connes-Takesaki module for every (π, H_{π}) ∈ K̂.

(2). For χ ∈ m̂_{β}, we consider a family of unitary operators

{χ(m_{β}(π))1_{π}}_{(π,H_{π})∈K̂}.

Then, thanks to Theorem 4.2 and the Tannaka duality theorem [28, Section 30], there exists a unique element ν(χ) ∈ K such that χ(m_{β}(π))1_{π} = π(ν(χ)) holds. Since π(ν(χ)) is a scalar for every irreducible π, ν(χ) is in the center Z(K). ν is clearly a homomorphism from m̂_{β} into Z(K), which is injective as the set of irreducible representations separates points of K. ν is continuous because the topology of K is the same as that induced by the weak topology of the image of K by the direct sum representation of all members in K̂.

Corollary 5.14. Let α be a minimal action of a compact connected semisimple Lie group K on a type III factor M. Then,

(1) If M is of type III_{1}, so is M^{α}.

(2) If M is of type III_{λ}, 0 < λ < 1, there exists a positive integer n such that M^{α} is of type III_{λ^{n}}. Let β be the Roberts action of K̂ on M^{α} whose dual action is α, then m_{β}(K̂) ≅ m̂_{β} ≅ ℤ/nℤ.

(3) If M is of type III_{0}, so is M^{α}. (The flow of M and M^{α} could be very much different in this case.)

Proof. Since the center of a semisimple Lie group is a finite group, Proposition 5.2,

(2) implies that N := M^{α} is of type III. As before, we regard Ñ as a subalgebra of M̃ using a unique conditional expectation E. Let β be the Roberts action of K̂ on N = M^{α} whose dual action is α.

We first assume that N is of type III_{λ}, 0 < λ ≤ 1. Since every non-trivial modular endomorphism of N is a composition of an inner automorphism and a modular automorphism in this case, no non-trivial modular endomorphisms appear in {β_{π}}_{(π,H_{π})∈K̂} because a connected semisimple Lie group has no non-trivial 1 dimensional representation. Thus, the same computation as in the proof of Theorem 5.5 implies

M̃ ∩ Ñ′ = Z(M̃).

Thanks to Theorem 5.13, β_{π} has a Connes-Takesaki module for every (π, H_{π}) ∈ K̂, which implies

Z(M̃) = Z(Ñ)^{m_{β}(K̂)}.

This shows that if N is of type III_{1}, so is M. When N is of type III_{λ}, 0 < λ < 1, we have Aut(F^{N}) ≅ 𝕋. Since Z(K) is a finite group, m̂_{β} is finite and its dual group m_{β}(K̂) is a finite subgroup of 𝕋. Thus, m_{β}(K̂) is a finite cyclic group and so is m̂_{β}.

Now we assume that N is of type III_{0}. Then, in the same way as above we can show

Z(M̃) ⊃ Z(Ñ)^{m_{β}(K̂)}.

Since m_{β}(K̂) is a finite group, (Z(Ñ)^{m_{β}(K̂)}, θ) is a non-transitive ergodic flow and so is F^{M}. Therefore, M is of type III_{0}.

We give a simple example of a minimal action of a compact group with non-trivial m_{β}(K̂). Let λ be a positive number with 0 < λ < 1, and set T = −2π/ log λ. We define a state of the 3 by 3 matrix algebra M(3, ℂ) by

φ_{0}(x) = Tr(Diag(\frac{1}{1 + 2λ}, \frac{λ}{1 + 2λ}, \frac{λ}{1 + 2λ})x),

where Diag means a diagonal matrix. We set

(M, φ) = ⨂_{n=1}^{∞}(M(3, ℂ), φ_{0}),

which is a Powers factor of type III_{λ}. For g ∈ SU(2), we define

α_{g} = ⨂_{n=1}^{∞}(Ad(Diag(1, g))).

Then, α is a minimal action of SU(2). It is easy to show σ_{T/2}^{φ} = α_{−1}, which implies that P = M^{α_{−1}} is of type III_{λ^{2}}. Note that the restriction of α to P induces a minimal action of SO(3), whose center is trivial. Thus, Corollary 5.14 implies that M^{α} and P has the common flow and M^{α} is of type III_{λ^{2}}. Corollary 5.14 again implies that the endomorphism in the Roberts action corresponding to the spin 1/2 representation has a non-trivial Connes-Takesaki module.

Appendix A. Miscellaneous results in ergodic theory

In this appendix, we collect some results from ergodic theory, (or its simple noncommutative generalization), used in the main body of the present notes. We provide proofs here because we could not find appropriate references, though some (or all) of them are probably well-known to specialists.

For a locally compact group G, we say that (X, μ) is a G-space if X is a standard Borel space, μ is a probability measure on X, and G acts on (X, μ) as a non-singular Borel transformation group. We say that a G-action is faithful if the corresponding action on L^{∞}(X, μ) is faithful. We say that two G-spaces (X_{1}, μ_{1}) and (X_{2}, μ_{2}) are isomorphic if the corresponding G-actions on the von Neumann algebras (or equivalently, measure algebras) L^{∞}(X_{1}, μ_{1}) and L^{∞}(X_{2}, μ_{2}) are conjugate. For g ∈ G and a Borel subset E ⊂ X, we use the notation g · μ(E) := μ(E · g).

A.1. Compact automorphism groups of ergodic transformation groups.

Lemma A.1. Let G be a locally compact group and (X, μ) be an ergodic G-space. We assume that a compact group K faithfully acts on (X, μ) commuting with G. We use the notation with G acting on X from right and K acting on X from left. Let (Y, ν) be the factor space of (X, μ) by K. Then, there exists a minimal K-valued cocycle c : Y × G ⟶ K with K_{c} = K such that (X, μ) is isomorphic to the skew product (Y ×_{c} K, ν × m_{K}), that is, under this isomorphism, the factor map from X onto Y corresponds to the projection onto the first component, and the K × G-action corresponds to the following action:

k · (ω, l) = (ω, kl), ω ∈ Y, k, l ∈ K,
(ω, l) · g = (ω · g, lc(ω, g)), g ∈ G.

c is unique up to equivalence.

Proof. Considering the Gelfand spectrum of an appropriate weakly dense separable C^{∗}-subalgebra of L^{∞}(X, μ), we may and do assume that X is a compact metric space, on which G × K acts continuously, and that μ is invariant under K. Since K is compact, every K-orbit is closed and there exists a Borel subset Y ⊂ X that meets each K-orbit exactly once (see, for example, [3, Section 3.4]). Let K be the set of all closed subgroups of K. Then, K has a Polish space structure such that K continuously acts on K by k · H = kHk^{−1}, H ∈ K, k ∈ K [16]. We denote by [H] the class of H in the quotient space of K/K. For each ω ∈ X, let K_{ω} ⊂ K be the stabilizer subgroup of ω. Then, thanks to [4, Chapter II, Proposition 2.3], the map

X ∋ ω ↦ [K_{ω}] ∈ K/K

is a Borel map satisfying [K_{ω·g}] = [K_{ω}] for g ∈ G. Thanks to the ergodicity of the G-action, there exist H ∈ K and K × G-invariant Borel null set E ⊂ X such that for every ω ∈ X_{0} := X \ E, we have [K_{ω}] = [H]. We set Y_{0} = Y ∩ X_{0}.

Let N(H) be the normalizer subgroup of H in K. Choosing a Borel cross section from N(H)\K to K, we have a Borel map φ : Y_{0} ⟶ K such that φ(ω) · H = K_{ω}. We set

Y_{1} = {φ(ω)^{−1} · ω ∈ X_{0}; ω ∈ Y_{0}}.

Then, Y_{1} is a Borel subset of X_{0} that meets each K-orbit exactly once and K_{ω} = H for all ω ∈ Y_{1}. Choosing a Borel cross section ψ : H\K ⟶ K, we set

Φ(ω, kH) = ψ(kH) · ω, ω ∈ Y_{1}, kH ∈ H\K.

Thanks to [3, pp. 72, Corollary], Φ is a Borel isomorphism from Y_{1} × H\K onto X_{0}. Since μ is K-invariant, μ·Φ is of the form ν ×m_{H\K}, where m_{H\K} is the (normalized) Haar measure of H\K. We introduce a K × G-action on Y_{1} × H\K from the action on X_{0} through Φ.

When we regard (Y_{1}, ν) as the factor space of X_{0} by the K-action, we denote the action of g ∈ G on Y_{1} by ω ∘ g, ω ∈ Y_{1}. Then, for each ω ∈ Y_{1} and g ∈ G, there exists a unique c_{(ω,g)}H ∈ H\K such that ω · g = c_{(ω,g)} · (ω ∘ g). Using a Borel cross section from H\K to H again, we can take c(ω, g) ∈ K to be a Borel map from Y_{1} × G to K. Let π : K ⟶ H\K be the quotient map. Commutativity of the K-action and G-action implies that for each fixed g ∈ G, c takes its values in N(H) almost everywhere. Moreover, π · c is an N(H)/H-valued cocycle and

(ω, π(k)) · g = (ω ∘ g, π(kc(ω, g))), ω ∈ Y_{1}, g ∈ G, k ∈ K.

This implies that for every subset U ⊂ K, Y_{1} × π(U · N(H)) is G-invariant. Thus, thanks to the ergodicity of the G-action, we conclude that whenever U is an open subset of K,

m_{K}(U · N(H)) = m_{H\K}(π(U · N(H))) = 1,

which implies N(H) = K, that is, H is a normal subgroup of K. Since the K-action is faithful, we get H = {e}.

Uniqueness of c up to equivalence is obvious.

A.2. Non-commutative extensions with relatively discrete spectrum. In [80], Zimmer introduced the notion of ergodic extensions with relatively discrete spectrum. The following is a slight generalization of this notion to the non-commutative case.

Definition A.2. Let G be locally compact group and (X, μ) be an ergodic G-space. Let A be a von Neumann algebra with an ergodic G-action Φ such that the center of A equivariantly includes Z = L^{∞}(X, μ), that is

Φ_{g}(f)(ω) = f(ω · g), f ∈ Z, ω ∈ X, g ∈ G.

Let u be the canonical implementation of Φ on the standard Hilbert space for A. We say that A is an extension of Z with relatively discrete spectrum if the von Neumann algebra u(G)′∩Z′ is decomposed into a direct sum of type I factors, and for each minimal projection p ∈ u(G)′ ∩Z′, ♮(p) is bounded, where ♮ is the (non-normalized) center valued trace on Z′, (in other words, the restriction of ♮ to u(G)′ ∩ Z′ is semifinite).

Note that the above definition coincides with Zimmer’s one [80, Definition 4.2] when A is commutative (although the two definitions appear different).

Lemma A.3. Let G be a locally compact group, K be a compact group, (X, μ) be a G-space, and c : X ×G ⟶ K be a cocycle. If Ψ is an action of K on a von Neumann algebra B standardly acting on a Hilbert space H, then there exists an action Ψ^{c} of G on L^{∞}(X, μ) ⊗ B such that for all g ∈ G, f ∈ L^{∞}(X, μ), x ∈ B, ξ ∈ L^{2}((X, μ), H),

(Ψ_{g}^{c}(f ⊗ x)ξ)(ω) = f(ω · g)Ψ_{c(ω,g)}(x)ξ(ω)

holds for almost every ω ∈ X.

Proof. Let v be the canonical implementation of Ψ on H. We introduce a unitary u(g), g ∈ G on L^{2}((X, μ), H) by

(u(g)ξ)(ω) = \frac{dg · μ}{dμ}(ω)^{1/2}v(c(ω, g))ξ(ω · g), ξ ∈ L^{2}((X, μ), H), ω ∈ X.

Then, u is a representation of G, which is a Borel map from G to the unitary group of B(L^{2}((X, μ), H)) with respect to the Borel structure coming from the weak (and strong ) topology. Since any unitary Borel representation of a locally compact group is continuous, (or more strongly, any Borel homomorphism between Polish groups is continuous [4, Chapter I, Propositon 3.3]), so is u. Ψ_{g}^{c} is defined by the restriction of Ad(u(g)) to L^{∞}(X, μ) ⊗ B.

The proof of (1) of the following theorem is a straightforward generalization of Zimmer’s argument in [80, Theorem 4.3] except for the proof of uniqueness, which does not exist there. (Zimmer’s comment about uniqueness in [80, pp.401] does not seem to be relevant.)

Theorem A.4. Let G be a locally compact group, (X, μ) be an ergodic G-space, and Φ be an action of G on a von Neumann algebra A. We assume that A is an extension of Z := L^{∞}(X, μ) with relatively discrete spectrum. Then, (1) There exist a compact group K, a faithful ergodic action Ψ of K on a von

Neumann algebra B, and a minimal cocycle c : X × G ⟶ K with K_{c} = K such that

(Z ⊂ A, Φ) ≅ (Z ⊗ ℂ ⊂ Z ⊗ B, Ψ^{c}),

that is, there exists an isomorphism from A onto Z ⊗ B that intertwines Φ and Ψ^{c}, and is identity on Z.

(2) A is always an injective von Neumann algebra, and there exists a unique faithful normal conditional expectation ϵ from A onto Z satisfying Φ_{g} · ϵ = ϵ · Φ_{g}, g ∈ G and ϵ(ab) = ϵ(ba), a, b ∈ A.

(3) (K, c, Ψ, B) satisfying the condition of (1) are unique in the following sense: if (K^{1}, c^{1}, Ψ^{1}, B^{1}) satisfy the same condition, then there exist a group isomorphism ν from K onto K_{1} and an isomorphism σ from B onto B^{1} such that ν · c and c^{1} are equivalent and Ψ_{ν(k)}^{1} · σ = σ · Ψ_{k}, k ∈ K.

Proof. (1). By assumption, we have a direct sum decomposition

Z′ ∩ u(G)′ = ⨁_{λ∈Λ} M_{λ},

where M_{λ} is a type I factor. We take a system of matrix units {e_{ij}^{λ}}_{i,j=1}^{n_{λ}} of M_{λ} with n_{λ} either a natural number or possibly infinite such that {e_{ii}^{λ}}_{i=1}^{n_{λ}} form a partition of unity consisting of minimal projections of M_{λ}. Since Ad(u(g)), g ∈ G normalizes Z, it does Z′ as well, and we set Φ′_{g} to be the restriction of Ad(u(g)) to Z′. Then, we have Φ_{g} · ♮ = ♮ · Φ′_{g}, and in particular,

Φ_{g}(♮(e_{11}^{λ})) = ♮(e(λ)_{11}), ∀g ∈ G.

Thus, by ergodicity ♮(e_{11}^{λ}) is a constant, say d(λ) ∈ ℕ.

In what follows, when we claim that “some statement involving g ∈ G and ω ∈ X holds for almost all ω ∈ X”, we always mean that for each fixed g ∈ G, there exists a conull set E_{g} ⊂ X such that the statement holds for all ω ∈ E_{g}. The annoying fact that E_{g} varies according to g ∈ G is usually taken care of by Mackey’s argument in [54]. We consider the disintegration of H over Z

H = ∫_{X}^{⊕} H(ω)dμ(ω).

Then, ♮(e_{11}^{λ}) = d(λ) means that the dimension of e_{11}^{λ}(ω)H(ω) is d(λ) for almost all ω ∈ X. We take ξ_{a}^{λ} = (ξ_{a}^{λ}(ω)) ∈ H, a = 1, 2, ⋯, d(λ) such that for almost all ω ∈ X {ξ_{a}^{λ}(ω)}_{a=1}^{d(λ)} is an orthonormal basis of e_{11}^{λ}(ω)H(ω). Since u(g) is a unitary satisfying (u(g)fu(g)^{∗})(ω) = f(ω · g) for f ∈ Z, for ξ, η ∈ H and f ∈ Z we have, on one hand,

⟨u(g)fξ, u(g)η⟩ = ∫_{X} f(ω · g)⟨(u(g)ξ)(ω), (u(g)η)(ω)⟩dμ(ω),

and on the other hand,

⟨u(g)fξ, u(g)η⟩ = ⟨fξ, η⟩ = ∫_{X} f(ω)⟨ξ(ω), η(ω)⟩dμ(ω)
= ∫_{X} f(ω · g)\frac{dg · μ}{dμ}(ω)⟨ξ(ω · g), η(ω · g)⟩dμ(ω).

This implies,

⟨(u(g)ξ)(ω), (u(g)η)(ω)⟩ = \frac{dg · μ}{dμ}(ω)⟨ξ(ω · g), η(ω · g)⟩,

for almost all ω. Thus,

\{\frac{dg · μ}{dμ}(ω)^{−1/2}(u(g)ξ_{a}^{λ})(ω)\}_{i=1}^{d(λ)}

is an orthonormal basis of e_{11}^{λ}(ω)H(ω) for almost all ω. We set

c^{λ}(ω, g)_{ab} = \frac{dg · μ}{dμ}(ω)^{−1/2}⟨(u(g)ξ_{a}^{λ})(ω), ξ_{b}^{λ}(ω · g)⟩.

Then, c^{λ}(ω, g) = (c^{λ}(ω, g)_{ab}) is an element of Z^{1}(G, U(d(λ))). We define a compact group K and a cocycle c by

K = ∏_{λ∈Λ} U(d(λ)),

c = (c^{λ})_{λ∈Λ} ∈ Z^{1}(G, K).

Changing the system {ξ_{a}^{λ}}_{a=1}^{d(λ)} if necessary, we may and do assume that c is minimal. We denote by K the minimal subgroup of c. Note that c^{λ} is minimal for all λ ∈ Λ and that if λ_{1} ≠ λ_{2}, c_{λ_{1}} and c_{λ_{2}} are not equivalent (otherwise, Z′ ∩ u(G)′ would be bigger).

Let H_{λ}^{1} and H_{λ}^{2} be Hilbert spaces with basis {f_{a}^{λ}}_{a=1}^{d(λ)} and {e_{i}^{λ}}_{i=1}^{n_{λ}} respectively. We set

H_{0} = ⨁_{λ∈Λ} H_{λ}^{1} ⊗ H_{λ}^{2}.

Since {(e_{i1}^{λ}ξ_{a}^{λ})(ω)}_{λ,i,a} is an orthonormal basis of H(ω) for almost all ω, we identify (e_{i1}^{λ}ξ_{a}^{λ})(ω) with f_{a}^{λ} ⊗e_{i}^{λ}, and H with L^{2}(X, μ)⊗H_{0}. There exists a natural embedding of K = ∏_{λ∈Λ} U(H_{λ}^{1}) in the group U(H_{0}) of all unitaries H_{0} as those unitaries acting “only on H_{λ}^{1}, λ ∈ Λ”. Through this embedding, we also regard K as a subgroup of U(H_{0}). Then, we have

(u(g)ξ)(ω) = (\frac{dg · μ}{dμ}(ω))^{1/2}c(ω, g)ξ(ω · g), ξ ∈ L^{2}((x, μ), H_{0}), (A.1)

for almost all ω ∈ X. Let

A = ∫_{X}^{⊕} A(ω)dμ(ω)

be the disintegration of A over Z = L^{∞}(X, μ)⊗ℂ1_{H_{0}}. Then, the map X ∋ ω ↦ A(ω) is measurable with respect to the Effros Borel structure of the set of von Neumann algebras on H_{0} [59, Chapter 4], [75, Chapter 4, Section 8]. For x ∈ A, (A.1) implies

α_{g}(x)(ω) = c(ω, g)x(ω · g)c(ω, g)^{−1}, (A.2)

for almost all ω. Since A has countable generators (we assume that Hilbert spaces are separable throughout this paper), this implies

c(ω, g)A(ω · g)c(ω, g)^{−1} = A(ω), (A.3)

for almost all ω.

Using the Effros Borel structure in the same way as in the proof of [80, Theorem 4.3] and passing to an equivalent cocycle if necessary (and changing A(ω) and c(ω, g) on a null set), we may and do assume that there exists a von Neumann algebra B ⊂ B(H_{0}) such that A(ω) = B and c(ω, g) normalizes B for all ω ∈ X. Since c is a minimal cocycle with K_{c} = K, K normalizes B, and so we define Ψ to be the restriction of Ad(k), k ∈ K to B. Then, (A.2) implies Φ = Ψ^{c}. Since Φ is ergodic, so is Ψ.

(2). Thanks to [30], B is injective and so is A.

Let ϵ be a normal conditional expectation from A onto Z. Then, since H is the standard Hilbert space of A, there exists ξ = (ξ(ω)) ∈ L^{2}((X, μ), H_{0}) whose vector state is μ · ϵ, where μ means the state of Z corresponding to the measure μ. Thus, for all x ∈ A and f ∈ Z we have

∫_{X} f(ω)ϵ(x)(ω)dμ(ω) = ⟨ϵ(fx)ξ, ξ⟩. (A.4)

We introduce a measurable field of normal states {φ_{ω}} of B by

φ_{ω}(b) = ⟨bξ(ω), ξ(ω)⟩, b ∈ B.

Then, (A.4) implies ϵ(x)(ω) = φ_{ω}(x(ω)) for almost all ω. Moreover, if ϵ commutes with Φ_{g}, we get

φ_{ω·g} = φ_{ω} · Ψ_{c(ω,g)},

for almost all ω. Let 𝔖(B) be the set of all normal states of B, which is a Polish space in the norm topology. Since Ψ is continuous in u-topology, K acts on 𝔖(B) as a continuous transformation group. Applying the same argument as above to 𝔖(B) and passing to an equivalent cocycle (and changing φ_{ω} and c(ω, g) on a null set), we may assume that there exists φ ∈ 𝔖(B) such that φ_{ω} = φ for all ω and φ·Ψ_{c(ω,g)} = φ holds. Thus, φ is a Ψ-invariant state, which is unique because of ergodicity of Ψ. Thus, ϵ = id ⊗ φ is uniquely determined. ϵ has the trace property as φ is a trace thanks to [30].

(3). We assume that (A, Φ) are also given by A = Z ⊗ B^{1}, Φ = (Ψ^{1})^{c^{1}}. Let τ be the unique Ψ^{1}-invariant trace, L^{2}(B^{1}) be the GNS Hilbert space of τ, and v(k) be the canonical implementation of Ψ_{k}^{1}, k ∈ K^{1}. Then, we have H = L^{2}(X, μ) ⊗ L^{2}(B^{1}) and u is given by

u(g)ξ(ω) = \frac{dg · μ}{dμ}(ω)^{1/2}v(c^{1}(ω, g))ξ(ω · g), ξ ∈ L^{2}((X, μ), L^{2}(B^{1})).

Since K^{1} is a compact group, we have the irreducible decomposition

(L^{2}(B^{1}), v(k)) = ( ⨁_{(π,H_{π})∈K̂^{1}} H_{π} ⊗ L_{π}, ⨁_{(π,H_{π})∈K̂^{1}} π(k) ⊗ 1_{L_{π}}),

where L_{π} is the multiplicity space, which may be zero. Since the G-action on (X, μ) is ergodic and c^{1} is a minimal cocycle with K_{c^{1}} = K we have

Z′ ∩ u(G)′ = ⨁_{(π,H_{π})∈K̂^{1}} ℂ ⊗ ℂ ⊗ B(L_{π}).

Thus, we identify Λ with the subset of K̂^{1} consisting of the irreducibles appearing in (v, L^{2}(B^{1})), and identify H_{λ}^{2} with L_{π} when λ ∈ Λ is identified with (π, H_{π}) ∈ K̂^{1}. In what follows, we abuse notation and use only Λ, instead of K̂^{1}, if there is no possibility of confusion. We set

K_{1} = ∏_{λ∈Λ} U(H_{λ}),

and regard it as a subgroup of the unitary group U(H) as before. Since Ψ^{1} is faithful, we identify K_{1} with v(K_{1}) ⊂ K_{1}.

Let w(ω) : H_{0} ⟶ L^{2}(B^{1}) be the measurable field of unitaries describing the identity map on H with respect to the two distinct splitting H = L^{2}(X, μ) ⊗ H_{0} and H = L^{2}(X, μ) ⊗ L^{2}(B^{1}). w(ω) is decomposed as

w(ω) = ⨁_{λ∈Λ} w_{λ}(ω) ⊗ 1_{H_{λ}^{2}},

where w_{λ}(ω) : H_{λ}^{1} ⟶ H_{λ} is a measurable field of unitaries. Then, we have

B^{1} = w(ω)Bw(ω)^{∗},
c^{1}(ω, g) = w(ω)c(ω, g)w(ω · g)^{∗},

for almost all ω ∈ X. Thanks to uniqueness of the minimal subgroup up to conjugacy, there exists a unitary w_{λ}^{0} from H_{λ}^{1} to H_{λ} such that if we set

ν_{0} = (∏_{λ∈Λ} Ad(w_{λ}^{0})) : K ⟶ K_{1},

then ν_{0}(K) = K_{1}. We denote by ν_{1} the restriction of ν_{0} to K. Let w^{1}(ω) = w_{λ}(ω)w_{λ}^{0}^{∗}, and

w^{0} = ⨁_{λ∈Λ} w_{λ}^{0} ⊗ 1_{H_{λ}^{2}},

w^{1}(ω) = ⨁_{λ∈Λ} w_{λ}^{1}(ω) ⊗ 1_{H_{λ}^{2}},

Then, we have

B^{1} = w^{1}(ω)w^{0}Bw^{0}^{∗}w^{1}(ω)^{∗},
c^{1}(ω, g) = w^{1}(ω)ν^{1} · c(ω, g)w^{1}(ω · g)^{∗},

for almost all ω ∈ X. Since c^{1} and ν_{1} · c are mutually equivalent minimal K_{1}-valued cocycles with (K_{1})_{c^{1}} = (K_{1})_{ν·c} = K_{1}, [80, Theorem 6.1] implies that there exist κ ∈ K_{1} normalizing K_{1} and a Borel map a : X ⟶ K_{1} such that w^{1}(ω) = a(ω)κ for almost all ω ∈ X. Let ν(k) = κν_{1}(k)κ^{−1}, then we get

B^{1} = κw^{0}Bw^{0}^{∗}κ^{∗},
c^{1}(ω, g) = a(ω)ν · c(ω, g)a(ω · g)^{∗},

for almost all ω. Let σ be the restriction of Ad(κw^{0}) to B. Then ν and σ have the desired property.

A.3. Existence of minimal cocycles on ergodic flows. The following is an easy generalization of [81, Theroem 2] to the non-singular case, which asserts that there are plenty of non-trivial modular endomorphisms for any type III_{0} factors.

Proposition A.5. Let (X, μ) be an ergodic non-transitive ℝ-space. Then, for every compact group K, there exists a minimal cocycle c : X × ℝ ⟶ K with K_{c} = K.

Proof. As in [81, Theorem 2], it suffices to show the statement for a ℤ-action instead of the ℝ-action thanks to Ambrose-Kakutani’s theorem [1]. Let S be a given nonsingular and non-transitive ergodic transformation on (X, μ). We show that there exists a minimal cocycle c : X × ℤ ⟶ K with K_{c} = K. We take a non-transitive ergodic transformation T on (Y, ν) such that ν is T -invariant. Thanks to [81, Theorem 1], there exists a minimal cocycle c′ : Y × ℤ ⟶ K such that K_{c′} = K. Let R be the ℤ^{2}-action on X × Y given by R_{(n_{1},n_{2})} = S^{n_{1}} × T^{n_{2}}. We define a cocycle c′′ on (X × Y) × ℤ^{2} by

c′′((ω_{1}, ω_{2}), (n_{1}, n_{2})) = c′(ω_{2}, n_{2}),

which is a minimal cocycle with K_{c′′} = K. Since (X, μ, S) is orbit equivalent to (X × Y, μ × ν, R) [40], we get the result.

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Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

  E-mail address: izumi@kusm.kyoto-u.ac.jp