Assumption 2.4. Let $\omega $ be a state on $\mathfrak {A}$ such that:

1. (SA1) $\omega $ has central support (that is, $\xi _\omega $ is separating for $\mathcal {R}_\omega $ );

2. (SA2) the lifted action $\widetilde {\alpha }:G\to \mathrm {Aut}(\mathcal {R}_\omega )$ exists and commutes with the modular group $\sigma _\omega ^t$ of the standard form $(\mathcal {R}_\omega , \mathcal {H}_\omega , J_\omega , \mathcal {P}_\omega )$ .

Proof. We verify the two parts of Assumption 2.4 separately.

Central support. Let $\mathcal {H}_\psi := \overline {\pi _\omega (\mathfrak {A}^H)\xi _\omega } \subset \mathcal {H}_\omega $ . The restriction of $\pi _\omega $ to $\mathfrak {A}^H$ leaves $\mathcal {H}_\psi $ invariant because $\mathfrak {A}^H$ commutes with itself. The vector $\xi _\omega $ is cyclic for $\pi _\omega (\mathfrak {A}^H)$ on $\mathcal {H}_\psi $ by definition. For $a\in \mathfrak {A}^H$ ,

$$ \begin{align*} \langle \pi_\omega(a)\xi_\omega, \xi_\omega\rangle = \omega(a) = \psi(a), \end{align*} $$

so $(\mathcal {H}_\psi , \pi _\omega |_{\mathfrak {A}^H}, \xi _\omega )$ is the GNS triple for $\psi $ . The vector $\xi _\omega $ is separating for ${\mathcal {R}_\psi := \pi _\omega (\mathfrak {A}^H)"}$ because it is separating for $\mathcal {R}_\omega $ and $\mathcal {R}_\psi \subset \mathcal {R}_\omega $ . Hence, $\psi $ has central support.

Commutation with the modular group. Let $\sigma _\psi ^t$ denote the modular group of the standard form $(\mathcal {R}_\psi , \mathcal {H}_\psi , J_\psi , \mathcal {P}_\psi )$ . We need to show that the lifted K-action on $\mathcal {R}_\psi $ commutes with $\sigma _\psi ^t$ .

We use the following theorem of Takesaki [Reference Takesaki9, Theorem IX.4.2]: if $\mathcal {N}\subset \mathcal {M}$ is a von Neumann subalgebra with a faithful normal conditional expectation $E:\mathcal {M}\to \mathcal {N}$ that commutes with the modular group $\sigma ^{\mathcal {M}}$ , then $\sigma ^{\mathcal {N}}_t = \sigma ^{\mathcal {M}}_t|_{\mathcal {N}}$ .

In our setting, $\mathcal {M} = \mathcal {R}_\omega $ , $\mathcal {N} = \mathcal {R}_\omega ^H$ and $E = \widetilde {E}_H$ (the extension of $E_H$ to $\mathcal {R}_\omega $ ). By [Reference Griseta4, Lemma 2.3], $\widetilde {E}_H$ exists and is faithful and normal. By [Reference Griseta4, Theorem 2.4], $\widetilde {E}_H$ commutes with $\sigma _\omega ^t$ . Since $\mathfrak {A}^H$ is globally invariant under $\sigma _\omega ^t$ (as follows from the commutation of $E_H$ with $\sigma _\omega ^t$ ), Takesaki’s theorem applies and gives $\sigma _{\psi }^t = \sigma _\omega ^t|_{\mathcal {R}_\omega ^H}$ .

Now, for any $g\in G$ and $a\in \mathfrak {A}^H$ , the lifted action $\widetilde {\beta }_{[g]}$ on $\mathcal {R}_\psi $ is given by ${\widetilde {\beta }_{[g]}(\pi _\omega (a)) = \pi _\omega (\alpha _g(a))}$ . Since $\pi _\omega (\alpha _g(a)) \in \mathcal {R}_\omega ^H$ (because H is normal), we have, using the commutation of $\widetilde {\alpha }_g$ with $\sigma _\omega ^t$ ,

$$ \begin{align*} \begin{aligned} \sigma_\psi^t(\widetilde{\beta}_{[g]}(\pi_\omega(a))) &= \sigma_\omega^t(\pi_\omega(\alpha_g(a))) = \pi_\omega(\sigma_\omega^t(\alpha_g(a))) \\ &= \pi_\omega(\alpha_g(\sigma_\omega^t(a))) = \widetilde{\beta}_{[g]}(\sigma_\psi^t(\pi_\omega(a))). \end{aligned} \end{align*} $$

Thus, the lifted K-action commutes with the modular group. The existence of the lifted action on $\mathcal {R}_\omega ^H$ follows from [Reference Griseta4, Lemma 2.2] applied to the induced action.

Proof. The second assertion is immediate because $E_H(a)=a$ for $a\in \mathfrak {A}^H$ .

For the first assertion, by Theorem 2.2 applied to the group H, we have a unitary equivalence $ U: \mathcal {H}_\omega \to \mathcal {H}_{\omega \circ E_H} $ such that

$$ \begin{align*} U\pi_\omega(a)U^* = \pi_{\omega\circ E_H}(E_H(a)) \end{align*} $$

for all $a\in \mathfrak {A}$ . This unitary is constructed explicitly in the proof of [Reference Griseta4, Theorem 2.4]. The proof also shows that the modular groups are conjugated:

$$ \begin{align*} \sigma_{\omega\circ E_H}^t = U\sigma_\omega^t U^*. \end{align*} $$

The central support condition is preserved under unitary equivalence: the cyclic vector $\xi _{\omega \circ E_H} = U\xi _\omega $ is separating for $\mathcal {R}_{\omega \circ E_H} = U\mathcal {R}_\omega U^*$ because $\xi _\omega $ is separating for $\mathcal {R}_\omega $ . The lifted action of G on $\mathcal {R}_{\omega \circ E_H}$ is given by $\widetilde {\alpha }_g^{(\omega \circ E_H)} = U\widetilde {\alpha }_g U^*$ , which commutes with $\sigma _{\omega \circ E_H}^t$ because $\widetilde {\alpha }_g$ commutes with $\sigma _\omega ^t$ . Thus, $\omega \circ E_H$ satisfies Assumption 2.4.

Proof. Since $\omega $ is H-invariant, the unitaries $U_h$ implementing $\widetilde {\alpha }_h$ on $\mathcal {H}_\omega $ satisfy $U_h\xi _\omega = \xi _\omega $ .

(i) Let $\mathcal {H}_\omega ^H := \{\eta \in \mathcal {H}_\omega : U_h\eta = \eta \text { for all }h\in H\}$ . The projection onto $\mathcal {H}_\omega ^H$ is $P = \int _H U_h\,dh$ . Since P commutes with $\pi _\omega (\mathfrak {A}^H)$ , we have $P\pi _\omega (a)\xi _\omega = \pi _\omega (a)\xi _\omega $ for $a\in \mathfrak {A}^H$ , so ${\mathcal {H}_\psi \subset \mathcal {H}_\omega ^H}$ . Conversely, if $\eta \in \mathcal {H}_\omega ^H$ , by cyclicity of $\xi _\omega $ , there exist $a_n\in \mathfrak {A}$ such that $\pi _\omega (a_n)\xi _\omega \to \eta $ . Then, $P\pi _\omega (a_n)\xi _\omega = \pi _\omega (E_H(a_n))\xi _\omega \in \mathcal {H}_\psi $ converges to $P\eta = \eta $ , so $\eta \in \mathcal {H}_\psi $ . Thus, $\mathcal {H}_\psi = \mathcal {H}_\omega ^H$ .

(ii) For $a\in \mathfrak {A}^H$ , $\pi _\omega (a)$ commutes with all $U_h$ , so it leaves $\mathcal {H}_\psi $ invariant. The cyclic vector $\xi _\omega $ is cyclic for $\pi _\omega (\mathfrak {A}^H)$ on $\mathcal {H}_\psi $ by the definition of $\mathcal {H}_\psi $ . The positivity and normalisation of $\psi $ give $\langle \pi _\omega (a)\xi _\omega ,\pi _\omega (b)\xi _\omega \rangle = \omega (a^*b)$ for $a,b\in \mathfrak {A}^H$ , establishing the GNS representation.

(iii) The equality $\pi _\omega (\mathfrak {A}^H)" = \mathcal {R}_\omega ^H$ follows from the fact that $E_H$ extends to a faithful normal conditional expectation from $\mathcal {R}_\omega $ onto $\mathcal {R}_\omega ^H$ (see [Reference Griseta4, Lemma 2.3]), and $\pi _\omega (\mathfrak {A}^H)$ is $\sigma $ -weakly dense in $\mathcal {R}_\omega ^H$ . Since $\xi _\omega $ is separating for $\mathcal {R}_\omega $ , it is separating for the subalgebra $\mathcal {R}_\omega ^H$ .

Proof. We view $\mathcal {H}_\psi $ as the GNS Hilbert space of $\psi $ , which is a left $\mathcal {R}_\omega ^H$ -module. The von Neumann algebra $\mathcal {R}_\omega $ is a right $\mathcal {R}_\omega ^H$ -module via right multiplication. The standard construction of the relative tensor product of correspondences (see, for example, [Reference Connes3, Section 2] or [Reference Takesaki9, Appendix B]) defines the Hilbert space $\mathcal {R}_\omega \otimes _{\mathcal {R}_\omega ^H} \mathcal {H}_\psi $ as the completion of $\mathcal {R}_\omega \odot \mathcal {H}_\psi $ modulo the relation

$$ \begin{align*} xn \otimes \xi = x \otimes n\xi \quad (x\in\mathcal{R}_\omega,\ n\in\mathcal{R}_\omega^H,\ \xi\in\mathcal{H}_\psi), \end{align*} $$

with inner product

$$ \begin{align*} \langle x\otimes\xi,\ y\otimes\eta\rangle := \langle \xi,\ E_H(x^*y)\eta\rangle. \end{align*} $$

The standard construction shows that this sesquilinear form is positive semidefinite and descends to the balanced tensor product. Since $E_H$ is faithful and $\psi $ is faithful on $\mathcal {R}_\omega ^H$ , the form is in fact positive definite.

The map $\Phi : \mathcal {R}_\omega \otimes _{\mathcal {R}_\omega ^H} \mathcal {H}_\psi \to \mathcal {H}_\omega $ defined by $\Phi (x\otimes \xi ) = x\xi $ is well defined and isometric:

$$ \begin{align*} \|\Phi(x\otimes\xi)\|^2 = \langle x\xi,x\xi\rangle = \langle \xi,x^*x\xi\rangle = \langle \xi,E_H(x^*x)\xi\rangle = \|x\otimes\xi\|^2. \end{align*} $$

Here, the third equality uses the fact that $\xi \in \mathcal {H}_\psi $ and the conditional expectation acts as the projection onto the fixed-point subspace on $\mathcal {H}_\psi $ ; this is a standard property of the GNS representation of a conditional expectation (see [Reference Griseta4, Lemma 2.3] or [Reference Takesaki9, Theorem IX.4.2]).

Surjectivity follows because the image of $\Phi $ contains $\mathcal {H}_\psi $ (take $x=1$ ) and is invariant under left multiplication by $\mathcal {R}_\omega $ ; hence, contains $\mathcal {R}_\omega \mathcal {H}_\psi = \mathcal {H}_\omega $ . Thus, $\Phi $ is a unitary.

Proof. Since $H\subset G$ , G-quasi-invariance immediately implies H-quasi-invariance.

By Theorem 2.2 applied to the group G, we have $\pi _\omega \cong \pi _{\omega \circ E_G}$ . By Lemma 3.2, ${\omega \circ E_G}$ satisfies Assumption 2.4. Now, $\omega \circ E_G = (\omega \circ E_H)\circ E_K$ because ${E_G = E_K\circ E_H}$ . By Lemma 3.2, $\omega \circ E_H$ also satisfies Assumption 2.4.

Let $\widetilde {\omega } := \omega \circ E_H$ . Then, $\widetilde {\omega }$ is H-invariant and $\widetilde {\omega }|_{\mathfrak {A}^H} = \psi $ . Moreover, $\widetilde {\omega }$ is G-quasi-invariant because $\pi _{\widetilde {\omega }}\cong \pi _\omega $ by Lemma 3.2 and unitary equivalence preserves quasi-invariance.

By Lemma 3.3, the GNS representation of $\psi $ is on the subspace $\mathcal {H}_\psi = \overline {\pi _{\widetilde {\omega }}(\mathfrak {A}^H)\xi _\omega }$ of H-fixed vectors, with cyclic vector $\xi _\omega $ . The G-quasi-invariance of $\widetilde {\omega }$ provides unitaries $U_g$ on $\mathcal {H}_{\widetilde {\omega }}$ such that $U_g\pi _{\widetilde {\omega }}(a)U_g^* = \pi _{\widetilde {\omega }}(\alpha _g(a))$ for all $a\in \mathfrak {A}$ . For $a\in \mathfrak {A}^H$ , $\alpha _g(a) = \beta _{[g]}(a)$ . Since H is normal, the subspace $\mathcal {H}_\psi $ of H-fixed vectors is invariant under $U_g$ : for $h\in H$ and $\eta \in \mathcal {H}_\psi $ ,

$$ \begin{align*} U_h U_g\eta = U_g U_{g^{-1}hg}\eta = U_g\eta \end{align*} $$

because $g^{-1}hg\in H$ fixes $\eta $ . Thus, the restrictions of the $U_g$ to $\mathcal {H}_\psi $ provide unitaries implementing the K-action on $\pi _{\widetilde {\omega }}(\mathfrak {A}^H)$ , showing that $\psi $ is K-quasi-invariant.

Proof. By H-quasi-invariance and Theorem 2.2, $\pi _\omega \cong \pi _{\omega \circ E_H}$ . Let $\widetilde {\omega }:=\omega \circ E_H$ . Then $\widetilde {\omega }$ satisfies Assumption 2.4 by Lemma 3.2, is H-invariant and $\widetilde {\omega }|_{\mathfrak {A}^H} = \psi $ . It suffices to prove that $\widetilde {\omega }$ is G-quasi-invariant, since $\pi _\omega \cong \pi _{\widetilde {\omega }}$ .

Let $\varphi := \widetilde {\omega }\circ E_G = \psi \circ E_K\circ E_H$ . By Lemma 3.4, the GNS spaces satisfy

$$ \begin{align*} \mathcal{H}_{\widetilde{\omega}} \cong \mathcal{R}_{\widetilde{\omega}} \otimes_{\mathcal{R}_{\widetilde{\omega}}^H} \mathcal{H}_\psi, \end{align*} $$

and similarly,

$$ \begin{align*} \mathcal{H}_\varphi \cong \mathcal{R}_{\widetilde{\omega}} \otimes_{\mathcal{R}_{\widetilde{\omega}}^H} \mathcal{H}_{\psi\circ E_K}. \end{align*} $$

Since $\psi $ is K-quasi-invariant, Theorem 2.2 applied to the group K acting on $\mathfrak {A}^H$ gives a unitary equivalence

$$ \begin{align*} V: \mathcal{H}_\psi \to \mathcal{H}_{\psi\circ E_K} \end{align*} $$

such that $V\pi _\psi (b) = \pi _{\psi \circ E_K}(b)V$ for all $b\in \mathfrak {A}^H$ . Since $\pi _\psi (\mathfrak {A}^H)$ generates $\mathcal {R}_{\widetilde {\omega }}^H$ , the unitary V extends uniquely to an $\mathcal {R}_{\widetilde {\omega }}^H$ – $\mathcal {R}_{\widetilde {\omega }}^H$ bimodule unitary. Indeed, the intertwining relation for a $\sigma $ -weakly dense $*$ -subalgebra implies the same relation for the von Neumann closures by continuity of the representations in the $\sigma $ -weak topology. Identifying the two representations of $\mathcal {R}_{\widetilde {\omega }}^H$ shows that V intertwines the left actions of $\mathcal {R}_{\widetilde {\omega }}^H$ .

Define a unitary $W: \mathcal {R}_{\widetilde {\omega }} \otimes _{\mathcal {R}_{\widetilde {\omega }}^H} \mathcal {H}_\psi \to \mathcal {R}_{\widetilde {\omega }} \otimes _{\mathcal {R}_{\widetilde {\omega }}^H} \mathcal {H}_{\psi \circ E_K}$ by $W(x\otimes \xi ) = x\otimes V\xi $ . This is well defined because V intertwines the $\mathcal {R}_{\widetilde {\omega }}^H$ -actions. The induced map gives a unitary equivalence

$$ \begin{align*} \mathcal{H}_{\widetilde{\omega}} \cong \mathcal{H}_\varphi. \end{align*} $$

For $a\in \mathfrak {A}$ , the representation of a on the balanced tensor product is left multiplication by $\pi _{\widetilde {\omega }}(a)$ . The unitary W intertwines left multiplication by $\pi _{\widetilde {\omega }}(a)$ with left multiplication by $\pi _\varphi (a)$ because

$$ \begin{align*} W(\pi_{\widetilde{\omega}}(a)\otimes\xi) = \pi_{\widetilde{\omega}}(a)\otimes V\xi = \pi_\varphi(a)(1\otimes V\xi). \end{align*} $$

Thus, $\pi _{\widetilde {\omega }} \cong \pi _\varphi $ . By Lemma 2.3, since $\widetilde {\omega }$ satisfies Assumption 2.4, $\pi _{\widetilde {\omega }}\cong \pi _{\widetilde {\omega }\circ E_G}$ implies that $\widetilde {\omega }$ is G-quasi-invariant. Therefore, $\omega $ is G-quasi-invariant.

Proof of Theorem 4.1.

(i) $\Rightarrow $ (ii) by Proposition 4.2; (ii) $\Rightarrow $ (i) by Proposition 4.3.

Proof. This follows by induction on n using Theorem 4.1. The base case $n=1$ is exactly the theorem. For the induction step, apply Theorem 4.1 to the subgroup $G_1$ to reduce G-quasi-invariance to $G_1$ -quasi-invariance and $G/G_1$ -quasi-invariance of $\omega |_{\mathfrak {A}^{G_1}}$ . Then, apply the induction hypothesis to the $G_1$ -series.

Proof. We prove the corollary by descending induction on i from n down to $0$ . For $i=n$ , $G_n=\{e\}$ , so $\mathcal {R}_\omega ^{G_n} = \mathcal {R}_\omega $ , which is a factor by hypothesis.

Assume that $\mathcal {R}_\omega ^{G_{i+1}}$ is a factor and that the action of $K_i=G_i/G_{i+1}$ on $\mathcal {R}_\omega ^{G_{i+1}}$ is minimal. By Corollary 5.1, the restriction $\omega _i := \omega |_{\mathfrak {A}^{G_{i+1}}}$ is $K_i$ -quasi-invariant. By Lemma 3.3, the GNS representation of $\omega _i$ has von Neumann algebra $\mathcal {R}_{\omega _i} = \mathcal {R}_\omega ^{G_{i+1}}$ , which is a factor by the induction hypothesis. The lifted action of $K_i$ on $\mathcal {R}_{\omega _i}$ commutes with the modular group by Lemma 3.1. Since the action is minimal by assumption, [Reference Griseta4, Proposition 3.5] applies directly (the proposition requires exactly the conditions we have: a minimal action of a compact group on a factor with modular compatibility) and implies that the fixed-point algebra $(\mathcal {R}_{\omega _i})^{K_i} = \mathcal {R}_\omega ^{G_i}$ is a factor. Thus, by induction, all $\mathcal {R}_\omega ^{G_i}$ are factors.