Proof. Denote $\mathcal {Q}=\mathscr N_{p{\mathcal M} p}(\mathcal P)"$ . Following [Reference Chifan, Ioana and KidaCIK13, Section 3], consider the $\ast $ -embedding $\Delta : {\mathcal M} \rightarrow {\mathcal M} \bar \otimes {\mathcal L}(\Sigma )$ given by $\Delta (au_g)=(au_g) \otimes v_{\varepsilon (g)}$ for all $a\in {\mathcal D}$ and $g\in \Gamma $ , where $(v_g)_{g\in \Gamma }$ are the canonical group unitaries in ${\mathcal L}(\Sigma )$ . Then, using [Reference Popa and VaesPV12, Theorem 1.4], one of the following must hold:

1. a) $\Delta (\mathcal P)\prec _{{\mathcal M}\bar \otimes {\mathcal L}(\Sigma )} {\mathcal M}\otimes 1$ , or

2. b) $\Delta (\mathcal {Q})$ is amenable relative to ${\mathcal M} \otimes 1$ inside ${\mathcal M}\bar \otimes {\mathcal L}(\Sigma )$ .

Using [Reference Chifan, Ioana and KidaCIK13, Proposition 3.4], one can see that case a) implies 1. Moreover, using Proposition 2.12, case b) yields 2.

Proof. Let $\varepsilon :\Gamma \to B$ be the quotient homomorphism given by $\Gamma \in \mathcal W\mathcal R (A, B \curvearrowright I)$ , and let $\{u_g\}_{g\in \Gamma }$ and $\{v_h\}_{h\in B}$ be the canonical generating unitaries of ${\mathcal L}(\Gamma )$ and ${\mathcal L}(B)$ , respectively. Denote by $\Delta :{\mathcal M}\to {\mathcal M}\bar \otimes \mathcal L(B)$ the $*$ -homomorphism given by $\Delta (au_g)=au_g\otimes v_{\varepsilon (g)}$ , for all $a\in {\mathcal D},g\in \Gamma $ . From [Reference Krogager and VaesKV15, Lemma 5.2], we derive that either $\Delta (\mathcal A)\prec _{{\mathcal M}\bar \otimes {\mathcal L}(B)} {\mathcal M}\otimes 1$ or $\Delta (\mathcal B)$ is amenable relative to ${\mathcal M}\bar \otimes 1$ . The first possibility implies by [Reference Chifan, Ioana and KidaCIK13, Proposition 3.4] that $\mathcal A\prec _{{\mathcal M}} {\mathcal D}\rtimes A^{I}$ , while the second one implies by Proposition 2.12 that $\mathcal B$ is amenable relative to ${\mathcal D}\rtimes A^{I}$ inside ${\mathcal M}$ . Since A is amenable, the conclusion follows from [Reference Ozawa and PopaOP07, Proposition 2.4].

Proof. Let $\mathcal A \subset p{\mathcal L}(\Gamma )p$ be a diffuse von Neumann subalgebra, and assume by contradiction that $\mathcal B:=\mathcal A'\cap p{\mathcal L}(\Gamma )p$ is nonamenable. Thus, there exists a nonzero projection ${z\in \mathcal Z(\mathcal B)}$ , such that $ q \mathcal B q$ is nonamenable for every projection $q \in \mathcal B z$ . By applying Proposition 4.1, we deduce that $\mathcal A z\prec ^s_{{\mathcal L}(\Gamma )} {\mathcal L}(A^{I})$ , and hence, by using [Reference Chifan, Ioana, Osin and SunCIOS21, Corollary 4.7] (for $m=1$ ) and also [Reference Chifan, Ioana, Osin and SunCIOS21, Lemma 4.11], we get that $\mathcal B z$ is amenable, contradiction.

Proof. Denote $\Gamma =\Gamma _1\times \dots \times \Gamma _n$ and $\mathcal P=\mathcal P_1\vee \dots \vee \mathcal P_{m}$ . Following the augmentation technique from [Reference Chifan, Diaz-Arias and DrimbeCD-AD20, Section 3], we consider a Bernoulli action with Abelian base $\Lambda \curvearrowright {\mathcal D}^\Lambda $ , and let $\tilde {\mathcal M}={\mathcal D}^\Lambda \rtimes \Lambda $ . Let $\Delta :\tilde {{\mathcal M}}\to \tilde {{\mathcal M}}\bar \otimes {\mathcal M}$ be the $*$ -homomorphism given by $\Delta (dv_h)=dv_h\otimes v_h$ , for all $d\in {\mathcal D}^\Lambda , h\in \Lambda $ . The assumption implies that $\mathcal P\prec ^{s'}_{\tilde {\mathcal M}} {\mathcal D}^\Lambda \rtimes \Lambda _0$ , and hence, by applying [Reference DrimbeDr19b, Lemma 2.3], we get that $\Delta (\mathcal P)\prec _{\tilde {\mathcal M}\bar \otimes {\mathcal M}}^{s'} \tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Lambda _0)$ . Next, from [Reference DrimbeDr19b, Lemma 2.4(2)], we derive that $\Delta (\mathcal P)\prec _{\tilde {\mathcal M}\bar \otimes {\mathcal M}} \tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Gamma )$ .

Thus, there exist projections $r\in \mathcal P, q\in \tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Gamma )$ , a nonzero partial isometry ${v\in q(\tilde {{\mathcal M}}\bar \otimes {\mathcal M})\Delta (r)}$ and a $*$ -homomorphism $\theta : \Delta (r\mathcal P r)\to q (\tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Gamma )) q$ , such that $\theta (x)v=vx$ , for any $x\in \Delta (rPr)$ , and the support projection of $E_{\tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Gamma )}(vv^*)$ equals q. Using [Reference Chifan, de Santiago and SucpikarnonCdSS17, Lemma 4.5], we can assume that $r\in \mathcal P_1$ . Denote $\mathcal N=\tilde {{\mathcal M}} \bar \otimes {\mathcal L}(\Gamma )$ and $\mathcal {Q}_i=\Delta (r\mathcal P_i r)$ for any $1\leq i\leq m$ . From [Reference Ioana, Popa and VaesIPV10, Lemma 10.2], it follows that $\Delta (\mathcal P_i)$ is strongly nonamenable relative to $\tilde {{\mathcal M}}\otimes 1$ inside $\tilde {{\mathcal M}}\bar \otimes {\mathcal M}$ , for any $1\leq i\leq m$ . Therefore, $\mathcal {Q}_i$ is nonamenable relative to $\tilde {{\mathcal M}}\otimes 1$ inside $\mathcal N$ for any $1\leq i\leq m$ . In particular, we derive from [Reference Popa and VaesPV11, Proposition 2.7] that there exists $1\leq j\leq n$ , such that $\mathcal {Q}_m$ is nonamenable relative to $\tilde {{\mathcal M}}\otimes {\mathcal L}(\Gamma _{\widehat {j}})$ inside $\mathcal N$ . Without loss of generality, we may assume that $j=n$ . By applying Proposition 4.1, it follows that $\bigvee _{i=1}^{m-1} Q_i\prec _{\mathcal N} \tilde {{\mathcal M}}\otimes \mathcal L(\Gamma _{\widehat {n}}\times A_{n}),$ where $A_n<\Gamma _{n}$ is an amenable subgroup. Thus, $ \Delta (\bigvee _{i=1}^{m-1} \mathcal P_i)\prec _{\tilde {{\mathcal M}}\bar \otimes {\mathcal M}} \tilde {{\mathcal M}}\bar \otimes {\mathcal L}(\Gamma _{\widehat {n}}\times A_{n}). $

Assume, by contradiction, that $m>n$ . By repeating the previous argument finitely many times, we obtain that there exists an amenable subgroup $A<\Gamma $ with the property that $\Delta (\bigvee _{i=1}^{m-n} \mathcal P_i)\prec _{\tilde {{\mathcal M}}\bar \otimes {\mathcal M}}\tilde {{\mathcal M}}\bar \otimes {\mathcal L}(A)$ . By applying [Reference Ioana, Popa and VaesIPV10, Lemma 10.2], it follows that $\mathcal P_1$ is amenable, contradiction.

Proof. Throughout the proof, we denote by $\Gamma _S=\times _{i\in S}\Gamma _i$ the subproduct supported on a subset $S\subset \{1,\dots , n\}$ . We also denote by $\widehat S$ the complement of S inside $\{1,\dots , n\}$ . Denote ${\mathcal M}={\mathcal L}(\Gamma )$ . Since $\mathcal P_n$ has no amenable direct summand, it follows that $\mathcal P_n z$ is nonamenable for any nonzero projection $z\in (\bigvee _{i=0}^n \mathcal P_i)'\cap pMp$ . By [Reference Popa and VaesPV11, Proposition 2.7], we get that there exists $1\leq j\leq n$ , such that $\mathcal P_n z$ is nonamenable relative to ${\mathcal L}(\Gamma _{\widehat j})$ inside ${\mathcal M}$ . Without loss of generalization, assume $j=n$ . Using Proposition 4.1, we get that $(\bigvee _{i=0}^{n-1} \mathcal P_i)z\prec _{{\mathcal M}} {\mathcal L}(\Gamma _{\widehat n})\bar \otimes {\mathcal L}(A_n^{(I_n)})$ . Thus, there exist projections $r\in \bigvee _{i=0}^{n-1} \mathcal P_i, q\in {\mathcal L}(\Gamma _{\widehat n})\bar \otimes \mathcal L(A_n^{(I_n)})$ with $rz\neq 0$ , a nonzero partial isometry $v\in q{\mathcal M} r$ and a $*$ -homomorphism $\theta : r(\bigvee _{i=0}^{n-1} \mathcal P_i)rz \to q({\mathcal L}(\Gamma _{\widehat n})\bar \otimes \mathcal L(A_n^{(I_n)}))q$ , such that $\theta (x)v=vx$ and the support projection of $E_{\mathcal L(\Gamma _{\widehat n})\bar \otimes \mathcal L(A_n^{(I_n)})}(vv^*)$ equals q. Since $\mathcal P_{n-1}$ is a II $_1$ factor, by using [Reference Chifan, de Santiago and SucpikarnonCdSS17, Lemma 4.5], we can assume that $r\in \mathcal P_{n-1}$ . Denote $\mathcal N=q({\mathcal L}(\Gamma _{\widehat n})\bar \otimes {\mathcal L}(A_n^{(I_n)}))q$ and $\mathcal {Q}_i=\theta (r\mathcal P_irz)$ for any $0\leq i\leq n-1$ . Since $\mathcal {Q}_{n-1}$ is nonamenable, we use [Reference Popa and VaesPV11, Proposition 2.7] and derive that exists $1\leq k\leq n-1$ , such that $\mathcal {Q}_{n-1}$ is nonamenable relative to $\mathcal L( \Gamma _{\widehat {k,n}})\bar \otimes {\mathcal L}(A_k^{(I_k)}\times A_n^{(I_n)})$ inside $\mathcal N$ . Without loss of generalization, we assume $k=n-1$ . By applying Proposition 4.1, we derive that $ \bigvee _{i=0}^{n-2} \mathcal {Q}_i\prec _{\mathcal N} {\mathcal L}( \Gamma _{\widehat {n-1,n}})\bar \otimes {\mathcal L}(A_{n-1}^{(I_{n-1})}\times A_n^{(I_n)}). $ Using that $\mathcal {Q}_i v=v r\mathcal P_i rz$ , we derive that $(\bigvee _{i=0}^{n-2} \mathcal P_i)z\prec _{{\mathcal M}} {\mathcal L}(\Gamma _{\widehat {n-1,n}})\bar \otimes {\mathcal L}(A_{n-1}^{(I_{n-1})}\times A_n^{(I_n)})$ . By using the previous argument finitely many times, we obtain $\mathcal P_0z\prec _{{\mathcal M}}{\mathcal L}(\times _{i=1}^n A_i^{(I_i)})$ .

Assume, by contradiction, that $\mathcal P_0z\nprec _{{\mathcal M}} \mathbb C1$ . The previous paragraph implies that there exists a nonempty subset $F\subset \{1,\dots ,n\}$ , such that

(4.1) $$ \begin{align} \mathcal P_0z\prec_{{\mathcal M}}{\mathcal L}(\times_{i\in F} A_i^{(I_i)}) \text{ and } \mathcal P_0z\nprec_{{\mathcal M}}{\mathcal L}(\times_{i\in F\setminus\{j\}} A_i^{(I_i)}) \text{ for any }j\in F. \end{align} $$

Therefore, there exist projections $p_0\in P_0, s\in {\mathcal L}(\times _{i\in F} A_i^{(I_i)})$ with $p_0z\neq 0$ , a nonzero partial isometry $w\in s{\mathcal M} p_0$ , a $*$ -homomorphism $\Psi : p_0P_0p_0z\to s{\mathcal L}(\times _{i\in F} A_i^{(I_i)})s$ , such that $\Psi (x)w=wx$ , for any $x\in p_0P_0p_0z$ , and the support projection of $E_{\mathcal L(\times _{i\in F} A_i^{(I_i)})}(ww^*)$ equals s. By letting $T=\Psi (p_0P_0z)$ and $A=\times _{i\in F} A_i^{(I_i)}$ , one can check that (4.1) implies that

$$ \begin{align*}T\subset s{\mathcal L}(A)s \text{ and } T\nprec_{{\mathcal L}(A)}{\mathcal L}(\times_{i\in F\setminus\{j\}} A_i^{(I_i)}) \text{ for any }j\in F. \end{align*} $$

By applying [Reference Chifan, Ioana, Osin and SunCIOS21, Corollary 4.7 and Lemma 4.11], we have $T'\cap s {\mathcal L}(\Gamma _{F}) s$ is amenable. Note that the $*$ -isomorphism $\mathrm {Ad}(w): w^*w {\mathcal M} w^*w\to ww^* {\mathcal M} ww^*$ sends $w^*w (p_0P_0p_0z'\cap p_0z{\mathcal M} p_0z) w^*w$ onto $ww^* (T'\cap s{\mathcal M} s) ww^*$ . Hence,

$$ \begin{align*} w ( p_0P_0p_0z'\cap p_0z{\mathcal M} p_0z ) w^*=ww^* ((T'\cap s {\mathcal L}(\Gamma_{F}) s)\bar\otimes {\mathcal L}(\Gamma_{\widehat F})s ) ww^*. \end{align*} $$

Since $\bigvee _{i=1}^n \mathcal P_i\subset \mathcal P_0'\cap p{\mathcal M} p$ , it follows that $\bigvee _{i=1}^n \mathcal P_i\prec _{{\mathcal M}} (T'\cap s {\mathcal L}(\Gamma _{F}) s)\bar \otimes \mathcal L(\Gamma _{\widehat F})s $ . Since $\widehat F$ has at most $n-1$ elements and $T'\cap s \mathcal L(\Gamma _{F}) s$ is amenable, we can apply the previous arguments finitely many times and derive that there exists an amenable subalgebra $S\subset {\mathcal M} $ , such that $\mathcal P_n\prec _{{\mathcal M}} S$ . This shows that $\mathcal P_n$ has an amenable direct summand, contradiction.

Finally, we conclude that $\mathcal P_0z\prec _{{\mathcal M}} \mathbb C1$ , for all nonzero projections $z\in (\bigvee _{i=0}^n \mathcal P_i)'\cap p{\mathcal M} p$ . Using [Reference Drimbe, Hoff and IoanaDHI16, Lemma 2.4], we conclude that $\mathcal P_0\prec ^s_{{\mathcal M}} \mathbb C1$ , which implies that $\mathcal P_0$ is completely atomic.

Proof. Let $\mathcal {Q}={\mathcal L}(\Omega )$ as $\mathcal A\prec \mathcal {Q}$ , then using [Reference Chifan, Kida and PantCKP14, Proposition 2.4] and its proof, one can find nonzero projections $a \in \mathcal A$ , $q \in \mathcal {Q}$ , a partial isometry $v\in q {\mathcal M} a$ , a subalgebra ${\mathcal D} \subseteq q\mathcal {Q} q$ and a $\ast $ -isomorphism $\phi : a\mathcal A a \rightarrow {\mathcal D} $ , such that

1. 1. ${\mathcal D} \vee ({\mathcal D}'\cap q\mathcal {Q} q) \subseteq q\mathcal {Q} q$ has finite index, and

2. 2. $\phi (x)v = vx$ for all $x \in a\mathcal A a$ .

The intertwining relation 2. implies that $vv^* \in {\mathcal D}' \cap q{\mathcal M} q$ and $v^*v \in (a\mathcal A a)' \cap a{\mathcal M} a = a \otimes \mathcal B.$ Hence, there is a projection $b \in \mathcal B$ satisfying $v ^*v = a \otimes b$ . Choosing $u \in \mathscr {U}({\mathcal M} )$ , such that $v = u(a \otimes b) $ then relation 2. entails that

(5.1) $$ \begin{align}{\mathcal D} vv^* = v\mathcal A v^* = u(a\mathcal A a \otimes b)u^*.\end{align} $$

Using this relation and passing to the relative commutants, we obtain $vv^* ({\mathcal D}' \cap q{\mathcal M} q)vv^* = u(a \otimes b\mathcal B b)u ^*$ . Thus, one can find scalars $s_1,s_2> 0$ satisfying

(5.2) $$ \begin{align}({\mathcal D}' \cap q{\mathcal M} q)z = u(a \otimes b\mathcal B b) ^{s_1} u ^* \cong \mathcal B ^{s_2}.\end{align} $$

Here, z denotes the central support projection of $vv^*$ in ${\mathcal D}'\cap q{\mathcal M} q$ . Now notice that

(5.3) $$ \begin{align} {\mathcal D}' \cap q {\mathcal M} q \supseteq (q\mathcal{Q} q)' \cap q{\mathcal M} q = (\mathcal{Q}' \cap {\mathcal M} )q \supseteq {\mathcal L}(C_\Lambda(\Omega))q,\end{align} $$

where ${\mathcal L}(C_\Lambda (\Omega ))$ has no amenable direct summand since $C_\Lambda (\Omega )$ is a nonamenable group. Moreover, we also have ${\mathcal D}' \cap q{\mathcal M} q \supseteq {\mathcal D}' \cap q \mathcal {Q} q$ . Thus, $(\mathcal {Q} ' \cap {\mathcal M} )z$ and $({\mathcal D}' \cap q\mathcal {Q} q)z$ are commuting subalgebras of $({\mathcal D}' \cap q {\mathcal M} q)z$ where $(\mathcal {Q}' \cap {\mathcal M} )z$ has no amenable direct summand. Since $\mathcal B^{s_2}$ is solid, then $({\mathcal D}' \cap q\mathcal {Q} q)z$ must be purely atomic. Thus, cutting by a central projection $0\neq r \in {\mathcal D}' \cap q\mathcal {Q} q$ , we may assume that ${\mathcal D} \subseteq q\mathcal {Q} q$ is a finite index inclusion of algebras. Since ${\mathcal D}$ is a factor, shrinking r if necessary, we can actually assume that ${\mathcal D} \subseteq q\mathcal {Q} q$ is an irreducible inclusion of finite index II $_1$ factors. Moreover, one can check that if one replaces v by the partial isometry of the polar decomposition of $r v \neq 0$ , then all relations (5.1), (5.2) and (5.3) are still satisfied. In addition, we can assume without any loss of generality that the support projection satisfies $s(E_{\mathcal {Q}}(vv^*)) = q$ .

Using [Reference JonesJo81, Lemma 3.1], one can find a projection $e\in q\mathcal {Q} q$ and a subfactor $\mathcal R \subseteq {\mathcal D} \subseteq q \mathcal {Q} q$ , such that $e \mathcal {Q} e =\mathcal R e$ and the index $[{\mathcal D}:\mathcal R]=[q \mathcal {Q} q: {\mathcal D}]$ . Now, notice the restriction $\phi ^{-1}: \mathcal R \rightarrow a\mathcal A a$ is an injective $\ast $ -homomorphism, such that $\mathcal T:= \phi ^{-1}(\mathcal R)\subseteq a\mathcal A a$ is finite index and

(5.4) $$ \begin{align} \phi^{-1}(y)v^*=v^*y \text{ for all } y\in \mathcal R. \end{align} $$

Let $\phi ': \mathcal R e \rightarrow \mathcal R$ be the $\ast $ -homomorphism given by $\phi '(y e)=y$ , for any $y\in \mathcal R$ . Since e has full support in $\langle {\mathcal D}, e\rangle =q\mathcal {Q} q$ , we have $ev\neq 0$ . Letting $w_0$ be a partial isometry, such that $w_0 |v^*e |=v^*e$ , the (5.4) gives that $\theta = \phi ^{-1} \circ \phi ': e \mathcal {Q} e \rightarrow a\mathcal A a$ is an injective $\ast $ -isomorphism satisfying $\theta (e \mathcal {Q} e)= \mathcal T$ and

(5.5) $$ \begin{align} \theta(y) w_0^*=w_0^* y \text{ for all } y\in e \mathcal{Q} e. \end{align} $$

Notice that $w_0^*w_0\in \mathcal T'\cap (a \mathcal A a \bar \otimes \mathcal B)=(\mathcal T'\cap a \mathcal A a) \bar \otimes \mathcal B $ . Since we have $\mathcal T'\cap (a\mathcal A a\bar \otimes \mathcal B )\supseteq \mathscr Z(\mathcal T'\cap a \mathcal A a) \bar \otimes \mathcal B$ and these von Neumann algebras have the same center then proceeding as in the proof of [Reference Ozawa and PopaOP03, Proposition 12], one can see that $w_0^*w_0$ is equivalent in ${\mathcal T'\cap (a \mathcal A a \bar \otimes \mathcal B)}$ with projection in $\mathscr Z(\mathcal T'\cap a \mathcal A a) \bar \otimes \mathcal B$ . Thus, one can assume without any loss of generality that $w_0^*w_0\in \mathscr Z(\mathcal T'\cap a \mathcal A a) \bar \otimes \mathcal B$ . As $[a\mathcal A a:\mathcal T]<\infty $ , then $\mathcal T'\cap a\mathcal A a$ is finite-dimensional. Thus, replacing $w_0$ by $w=w_0r_0$ for a minimal projection $r_0\in \mathscr Z(\mathcal T'\cap a \mathcal A a)$ with $r_0 w_0^*|v^*e|\neq 0$ , we see all previous relations are satisfied, including (5.5). Moreover, we can assume that $w^*w= z_1\otimes z_2$ for some $z_1\in \mathscr Z(\mathcal T'\cap a \mathcal A a)$ and $z_2\in B$ . Hence, (5.5) implies

(5.6) $$ \begin{align} w^* \mathcal{Q} w= \theta(e \mathcal{Q} e) w^*w= \mathcal T z_1 \otimes z_2. \end{align} $$

Since $\mathcal T \subseteq a \mathcal A a$ is a finite index inclusion of II $_1$ factors, using the local index formula [Reference JonesJo81, Lemma 2.2.1], we have that $\mathcal T z_1\subseteq z_1 \mathcal A z_1$ is a finite index inclusion of II $_1$ factors as well. In addition, we have

(5.7) $$ \begin{align} (w^* \mathcal{Q} w)'\cap (z_1\otimes z_2) {\mathcal M} (z_1\otimes z_2)= ((\mathcal T z_1)' \cap z_1 \mathcal A z_1) \bar\otimes z_2 \mathcal B z_2. \end{align} $$

Altogether, the previous relations (5.6) and (5.7) imply that

(5.8) $$ \begin{align} \begin{aligned} \mathcal T z_1 \bar\otimes z_2 \mathcal B z_2& \subset \mathcal T z_1 \vee (\mathcal T z_1'\cap z_1 \mathcal A a)\bar\otimes z_2 \mathcal B z_2\\ &= w^* \mathcal{Q} w\vee (w^* \mathcal{Q} w)'\cap (z_1\otimes z_2) {\mathcal M} (z_1\otimes z_2)\\ &\subseteq z_1 \mathcal A z_1\bar\otimes z_2 \mathcal B z_2 = (z_1\otimes z_2) {\mathcal M} (z_1\otimes z_2). \end{aligned} \end{align} $$

Since $\mathcal T z_1\subseteq z_1 \mathcal A z_1$ is a finite index inclusion of II $_1$ factors, then so is $\mathcal T z_1 \bar \otimes z_2 \mathcal B z_2\subseteq z_1 \mathcal A z_1\bar \otimes z_2 \mathcal B z_2$ . Let $f= ww^*$ , and note that $f=re$ for some projection $r\in \mathcal {Q}'\cap {\mathcal M}$ . Now notice (5.8) implies that

(5.9) $$ \begin{align} e\mathcal{Q} e r \vee r (\mathcal{Q} '\cap {\mathcal M})r e = e\mathcal{Q} e r \vee ((e\mathcal{Q} er) '\cap er{\mathcal M} er)\subseteq f {\mathcal M} f \end{align} $$

is a finite index inclusion of von Neumann algebras. Since ${\mathcal M}$ is a factor, the finite index condition implies that the center $\mathscr Z(r(\mathcal {Q}'\cap {\mathcal M} )r)$ is completely atomic (see [Reference Chifan, de Santiago and SucpikarnonCdSS17, Proposition 2.1(3)]). Thus, compressing r more if necessary, we further obtain that $e\mathcal {Q} e r \vee r (\mathcal {Q} '\cap {\mathcal M})r e \subseteq f {\mathcal M} f$ is a finite index inclusion of II $_1$ factors, as desired.

Proof. Without any loss of generality, we can assume that $t=1$ , as the other cases do not hide any difficulties. Throughout the proof, we denote by $\Gamma _S=\times _{i\in S}\Gamma _i$ the subproduct supported on a subset $S\subset \{1,\dots , n\}$ . We also denote by $\widehat S$ the complement of S inside $\{1,\dots , n\}$ .

Let $\Delta : {\mathcal M}\rightarrow {\mathcal M}\bar \otimes {\mathcal M}$ be the $\ast $ -embedding given by $\Delta (v_h)=v_h\otimes v_h$ for all $h\in \Lambda $ . Let $1\leq i,j\leq n $ , and observe that $\Delta ({\mathcal L}(\Gamma _{\hat i})), \Delta (\mathcal L(\Gamma _i))\subset {\mathcal M} \bar \otimes \mathcal L(\Gamma _{\hat j})\bar \otimes \mathcal L(\Gamma _j)$ are commuting subalgebras. Using Proposition 4.1 (see also [Reference Chifan, Ioana, Osin and SunCIOS21, Theorem 6.15]), we have either $\Delta ({\mathcal L}(\Gamma _{\hat i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes {\mathcal L}(\Gamma _{\hat j})\bar \otimes L(A^{(I_j)}_j)$ or $\Delta ({\mathcal L}(\Gamma _{i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes {\mathcal L}(\Gamma _{\hat j})\bar \otimes {\mathcal L}(A^{(I_j)}_j)$ . Since ${\mathcal L}(A^{(I_j)}_j)$ is Abelian and $\Delta ({\mathcal L}(\Gamma _{\hat i}))$ , $\Delta ({\mathcal L}(\Gamma _{i}))$ have property (T), the prior intertwining relations further imply that either $\Delta ({\mathcal L}(\Gamma _{\hat i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes {\mathcal L}(\Gamma _{\hat j})$ or $\Delta (\mathcal L(\Gamma _{i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes \mathcal L(\Gamma _{\hat j})$ . Moreover, using [Reference Drimbe, Hoff and IoanaDHI16, Lemma 2.4(3)], we have either $\Delta (\mathcal L(\Gamma _{\hat i}))\prec ^s_{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes \mathcal L(\Gamma _{\hat j})$ or $\Delta (\mathcal L(\Gamma _{i}))\prec ^s_{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes \mathcal L(\Gamma _{\hat j})$ . If the former would hold for all $1\leq j\leq n$ , then by using [Reference Drimbe, Hoff and IoanaDHI16, Lemma 2.8(2)], we would get $\Delta ({\mathcal L}(\Gamma _{i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} \bigcap ^n_{j=1} ({\mathcal M}\bar \otimes {\mathcal L}(\Gamma _{\hat j})) ={\mathcal M}\otimes 1$ . This contradicts [Reference Ioana, Popa and VaesIPV10, Lemma 7.2]. Hence, for every $1\leq i\leq n$ , there is $1\leq j\leq n$ so that $\Delta (\mathcal L(\Gamma _{\hat i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes \mathcal L(\Gamma _{\hat j})$ . Furthermore, using [Reference DrimbeDr20, Theorem 4.3], we actually have that $\Delta (\mathcal L(\Gamma _{\hat i}))\prec _{{\mathcal M}\bar \otimes {\mathcal M}} {\mathcal M}\bar \otimes \mathcal L(\Gamma _{\hat i})$ for all $1\leq i\leq n$ . Now, since $\mathcal L(\Gamma _i)$ has property (T), Theorem 6.1 shows that there is a subgroup $\Sigma <\Lambda $ , such that

(5.10) $$ \begin{align} {\mathcal L}(\Gamma_{\hat i})\prec_{\mathcal M} {\mathcal L}(\Sigma)\text{ and }{\mathcal L}(\Gamma_{i})\prec_{\mathcal M} {\mathcal L}(C_\Lambda(\Sigma)). \end{align} $$

Since ${\mathcal L}(\Gamma _i)$ is solid, (5.10) allows to apply Proposition 5.1 and find a projection ${0\neq p= qr}$ with $q\in {\mathcal L}(\Sigma ), r \in {\mathcal L}(\Sigma )'\cap {\mathcal M}$ , such that $p({\mathcal L}(\Sigma )\vee ({\mathcal L}(\Sigma )'\cap {\mathcal M}))p \subseteq p{\mathcal M} p$ is a finite index inclusion of II $_1$ factors. In particular, $p{\mathcal L}(\Sigma \mathrm {vC}_\Lambda (\Sigma ))p \subseteq p{\mathcal M} p$ is also a finite index inclusion of von Neumann algebras, and thus, $[\Lambda : \Sigma \mathrm {vC}_\Lambda (\Sigma )]<\infty $ . Since $\Lambda $ is icc property (T), then so is $\Sigma \mathrm {vC}_\Lambda (\Sigma )$ . Now, observe that one can find an increasing sequence of groups $\cdots \leqslant \Omega _n \leqslant \Omega _{n+1}\leqslant \cdots \leqslant \mathrm {vC}_\Lambda (\Sigma )$ normalized by $\Sigma $ with $\cup _n \Omega _n=\mathrm {vC}_\Lambda (\Sigma )$ whose centralizers form a descending sequence $\Sigma \geqslant C_\Sigma (\Omega _1)\geqslant \cdots \geqslant C_\Sigma (\Omega _n )\geqslant C_\Sigma (\Omega _{n+1}) \cdots $ of finite index subgroups. Therefore, $\Sigma \Omega _n \nearrow \Sigma \mathrm {vC}_\Lambda (\Sigma )$ , and using property (T), there is n so that ${\Sigma \Omega _n = \Sigma \mathrm {vC}_\Lambda (\Sigma )}$ . Since $C_\Sigma (\Omega _n) \Omega _n \leqslant \Sigma \Omega _n$ has finite index, we conclude that $C_\Lambda (\Omega _n) \Omega _n\leq \Lambda $ has finite index as well. Denote $\Sigma _0=C_\Lambda (\Omega _n)$ , and note that $C_\Lambda (\Sigma )\leq C_\Lambda (\Sigma _0)$ . From (5.10), it follows that

(5.11) $$ \begin{align} {\mathcal L}(\Gamma_{\hat i})\prec_{\mathcal M} {\mathcal L}(\Sigma_0)\text{ and }{\mathcal L}(\Gamma_{i})\prec_{\mathcal M} {\mathcal L}(C_\Lambda(\Sigma_0)). \end{align} $$

By passing to relative commutants in (5.11), we get that ${\mathcal L}(\Sigma _0)\prec _{\mathcal M} \mathcal L(\Gamma _{\hat i})$ and $\mathcal L(C_{\Lambda }(\Sigma _0))\prec _{\mathcal M} \mathcal L(\Gamma _i)$ . Since $[\Lambda :\Sigma _0 C_\Lambda (\Sigma _0)]<\infty $ , we derive that $\mathcal L(\Sigma C_\Lambda (\Sigma ))'\cap {\mathcal M}=\mathbb C 1$ . Therefore, using [Reference Drimbe, Hoff and IoanaDHI16, Lemma 2.4(3)], we see that

(5.12) $$ \begin{align} {\mathcal L}(\Sigma_0)\prec^s_{\mathcal M} {\mathcal L}(\Gamma_{\hat i}) \text{ and } {\mathcal L}(C_{\Lambda}(\Sigma_0))\prec^s_{\mathcal M} {\mathcal L}(\Gamma_i). \end{align} $$

By using (5.11), (5.12) together with [Reference Drimbe, Hoff and IoanaDHI16, Theorem 6.1] and a standard inductive argument, we obtain the desired conclusion.

Proof. This follows directly from Theorem 5.2 and [Reference Chifan, Ioana, Osin and SunCIOS21, Theorem 9.1].

Proof. 1. Following [Reference Ioana, Popa and VaesIPV10, Reference IoanaIo10], consider the $\ast $ -embedding $\Delta : {\mathcal M} \rightarrow {\mathcal M} \bar \otimes {\mathcal M}$ given by $\Delta (v_h)=v_h\otimes v_h$ for all $h\in \Lambda $ . Fix $\mathscr {C} \in \mathrm {cliq} (\mathscr G)$ . From assumptions, we have $\Delta ({\mathcal L}(\Gamma _{\mathscr {C}}))\subseteq {\mathcal M} \bar \otimes {\mathcal M} = \mathcal L(\Gamma \times \Gamma )$ is a property (T) von Neumann algebra. Thus, applying Theorem 2.8, one can find cliques $\mathscr {C}_1,\mathscr {C}_2\in \mathrm {cliq}(\mathscr {G})$ , such that $\Delta (\mathcal L(\Gamma _{\mathscr {C}}))\prec _{{\mathcal M} \bar \otimes {\mathcal M}} \mathcal L(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2})$ . This implies that one find projections $q\in \Delta (\mathcal L(\Gamma _{\mathscr {C}})) p\in \mathcal L(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2})$ , partial isometry ${w\in {\mathcal M}\bar \otimes {\mathcal M} }$ and a $\ast $ -isomorphism onto its image $\theta : q\Delta (\mathcal L(\Gamma _{\mathscr {C}})) q\rightarrow \mathcal R:= \theta (q\Delta (\mathcal L(\Gamma _{\mathscr {C}})) q)\subseteq p \mathcal L(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2}) p$ , such that $\theta (x)w=wx$ for all $x\in q \Delta (\mathcal L(\Gamma _{\mathscr {C}})) q$ . Notice that $ww^*\in \mathcal R'\cap q({\mathcal M} \bar \otimes {\mathcal M})q $ and $w^*w\in (\Delta ({\mathcal L}(\Gamma _{\mathscr {C}}))'\cap {\mathcal M}\bar \otimes {\mathcal M} )q$ . Moreover, we can assume without loss of generality that the support of $E_{\mathcal L(\Gamma _{\mathscr {C}_1} \times \Gamma _{\mathscr {C}_2})}(ww^*)$ equals p.

Since $\Gamma _t, t\in \mathscr V$ , are icc, we can assume without loss of generality that $q\in \Delta ({\mathcal L}(\Gamma _{v_0}))$ for some $v_0\in \mathscr {C}$ . For every $\mathscr {D} \subseteq \mathscr {C}$ , we denote $\mathcal R_{\mathscr {D}}= \theta (q\Delta (\mathcal L(\Gamma _{\mathscr {D}}))q)$ . Thus, $\mathcal R= \vee _{t\in \mathscr {C}} \mathcal R_v$ where $\mathcal R_t= \theta (q\Delta (\mathcal L(\Gamma _t))q)$ are mutually commuting nonamenable II $_1$ factors.

Now, fix $\emptyset \neq \mathscr {D} \subseteq \mathscr {C}$ , and notice that $\mathcal R_{\mathscr {D}} \vee \mathcal R_{\mathrm {lk}(\mathscr {D})} = \mathcal R\subset {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2})=:\tilde {\mathcal N}$ are commuting nonamenable factors. Since $\mathcal R_{\mathscr {D}}$ and $\mathcal R_{\mathrm {lk}(\mathscr {D})}$ are commuting property (T) algebras, then [Reference Chifan, Ioana, Osin and SunCIOS21, Theorem 6.15] implies that for every $t \in \mathscr {C}_2$ , either $\mathcal R_{\mathscr {D}}\prec _{\tilde {\mathcal N}} {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}}\times A^{(B_t)}_t ) $ or $\mathcal R_{\mathrm {lk}(\mathscr {D})}\prec _{\tilde {\mathcal N}} {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}}\times A^{(B_t)}_t ) $ . However, since $R_{\mathscr {D}}, \mathcal R_{\mathrm { lk}(\mathscr {D})}$ have property (T) and $A_t^{(B_t)}$ is amenable, we further conclude that either $\mathcal R_{\mathscr {D}}\prec _{\tilde {\mathcal N}} {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}} )$ or $\mathcal R_{\mathrm {lk}(\mathscr {D})}\prec _{\tilde {\mathcal N}} {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}} )$ , and using factoriality, we actually have either a) $\mathcal R_{\mathscr {D}}\prec ^s {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}} )$ or b) $\mathcal R_{\mathrm {lk}(\mathscr {D})}\prec ^s {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}} )$ . Assume by contradiction that b) holds for all $t\in \mathscr {C}_2$ . By [Reference Drimbe, Hoff and IoanaDHI16, Lemma 2.6], this would imply that $\mathcal R_{\mathrm {lk}(\mathscr {D})}\prec \cap _{t\in \mathscr {C}_2} {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}} )= {\mathcal L}(\Gamma _{\mathscr {C}_1})\otimes 1$ . Using [Reference Ioana, Popa and VaesIPV10, Proposition 7.2], this entails that $\mathcal R_{\mathrm {lk}(\mathscr {D})}$ is atomic, a contradiction. Hence, there is $t\in \mathscr {C}_2$ , such that $\mathcal R_{\mathscr {D}}\prec ^s {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{v\}} )$ . Combining this with the first part, we get again that $\Delta ({\mathcal L}(\Gamma _{\mathscr {D}}))\prec {\mathcal L}(\Gamma _{\mathscr {C}_1}\times \Gamma _{\mathscr {C}_2 \setminus \{t\}}) )$ . The conclusion of the first part follows from Theorem 6.1 by letting $\mathscr {D}=\mathscr {C}\setminus \{v\}$ .

2. The conclusion follows from Proposition 4.5.

Proof. Since ${\mathcal L}(\Gamma _{\hat v})\prec _{\mathcal M} {\mathcal L}(\Lambda _{\hat v})$ , one can find projections $a\in {\mathcal L}(\Gamma _{\hat v})$ , $f\in {\mathcal L}(\Lambda _{\hat v})$ , a nonzero partial isometry $v\in f{\mathcal M} a$ and a $\ast $ -isomorphism onto its image $\phi :a \mathcal L(\Gamma _{\hat v})a \rightarrow \mathcal B:=\phi (a\mathcal L(\Gamma _{\hat v})a) \subseteq f\mathcal L(\Lambda _{\hat v})f $ , such that

(6.1) $$ \begin{align} \phi(x)v=vx \text{ for all } x\in a{\mathcal L}(\Gamma_{\hat v})a. \end{align} $$

Notice that $vv^*\in \mathcal B'\cap f{\mathcal M} f$ and $v^*v\in a{\mathcal L}(\Gamma _{\hat v})a'\cap a{\mathcal M} a$ . Since the virtual centralizer satisfies $\mathrm {vC}_\Gamma (\Gamma _{\hat v})= C_\Gamma (\Gamma _{\hat v}) = \Gamma _v$ , then ${\mathcal L}(\Gamma _{\hat v})'\cap {\mathcal M} = {\mathcal L}(\Gamma _v)$ . Thus, we can write ${v^*v=aa_0}$ for a projection $a_0\in {\mathcal L}(\Gamma _v)$ . Equation (6.1) implies that $\mathcal B vv^* = v {\mathcal L}(\Gamma _{\hat v})v^* = u_1 {\mathcal L}(\Gamma _{\hat v}) v^*v u^*_1$ , where $u_1\in {\mathcal M}$ is a unitary extending v. Taking relative commutants, we get $vv^*(\mathcal B' \cap f{\mathcal M} f)vv^*= u_1 v^*v( a{\mathcal L}(\Gamma _{\hat v})a'\cap a{\mathcal M} a )v^*v u^*_1= u_1 v^*v\mathcal L(\Gamma _{v} ) v^*v u^*_1$ . Hence, $vv^*(\mathcal B \vee \mathcal B'\cap f{\mathcal M} f)vv^*=\mathcal B vv^*\vee vv^*(\mathcal B' \cap f{\mathcal M} f)vv^*= u_1 v^*v\mathcal L(\Gamma _{\mathscr {C}}) v^*vu^*_1$ . Therefore, since $\mathcal L(\Gamma _{\mathscr {C}})$ is a factor, one can find a new unitary $u_2 \in \mathscr {U}({\mathcal M})$ , such that

(6.2) $$ \begin{align} (\mathcal B \vee \mathcal B'\cap f{\mathcal M} f) z_2 \subseteq u_2 {\mathcal L}(\Gamma_{\mathscr{C}}) u^*_2, \end{align} $$

here, $z_2$ is the central support of $vv^*$ in $\mathcal B \vee \mathcal B'\cap f{\mathcal M} f$ . In particular, we have $z_2\in \mathcal Z(\mathcal B'\cap f{\mathcal M} f )$ and $vv^*\leq z_2 \leq f$ .

Now, observe that (6.2) implies that

(6.3) $$ \begin{align} {\mathcal L}(C_\Lambda(\Lambda_{\hat v}))z_2\subseteq (f{\mathcal L}(\Lambda_{\hat v} )f'\cap f {\mathcal M} f )z_2\subseteq (\mathcal B'\cap f {\mathcal M} f)z_2\subseteq u_2 {\mathcal L}(\Gamma_{\mathscr{C}})u_2^*. \end{align} $$

Next, we will show the following containment

(6.4) $$ \begin{align}z_2({\mathcal L}(C_\Lambda(\Lambda_{\hat v}))f'\cap f{\mathcal M} f) z_2\subseteq u_2\mathcal L(\Gamma_{\mathscr{C}})u_2^*.\end{align} $$

Since we have ${\mathcal M}= u_2{\mathcal L}(\Gamma _{\mathscr V\setminus \{v\}} \ast _{\Gamma _{\hat v}} \Gamma _{\mathscr {C}} ) u^*_2=u_2{\mathcal L}(\Gamma _{\mathscr V\setminus \{v\}})u_2^* \ast _{u_2{\mathcal L}(\Gamma _{\hat v})u_2^*} u_2\mathcal L (\Gamma _{\mathscr {C}} ) u^*_2$ , then by [Reference Ioana, Peterson and PopaIPP05, Theorem 1.2.1] and (6.3) to get (6.4), it suffices to show that

(6.5) $$ \begin{align}\mathcal L(C_\Lambda(\Lambda_{\hat v}))z_2\nprec_{u_2{\mathcal L} (\Gamma_{\mathscr{C}} ) u^*_2} u_2{\mathcal L}(\Gamma_{\hat v})u_2^*.\end{align} $$

Since $\Gamma _v$ is icc, it follows that $vv^*(\mathcal B' \kern1.3pt{\cap}\kern1.3pt f{\mathcal M} f)vv^*{=} u_1 v^*v{\mathcal L}(\Gamma _{v} ) v^*v u^*_1{=} u_1 a_0(\mathcal L(\Gamma _{v} ) )a_0 a u^*_1$ is a factor. Since $z_2$ is the central support of $vv^*$ in $\mathcal B' \cap f{\mathcal M} f$ , then $(\mathcal B' \cap f{\mathcal M} f) z_2$ is a factor as well.

Assume that (6.5) does not hold. Thus, one can find projections $t\in {\mathcal L}(C_\Lambda (\Lambda _{\hat v})), r\in u_2 \mathcal L(\Gamma _{\hat v})u_2^*$ , a partial isometry $w\in ru_2{\mathcal L}(\Gamma _{\mathscr {C}})u_2^* tz_2$ and a $\ast $ -isomorphism onto its image $\psi : t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2\rightarrow ru_2\mathcal L(\Gamma _{\hat v})u_2^* r$ , such that

(6.6) $$ \begin{align} \psi(x)w=wx \text{ for all } x\in t{\mathcal L}(C_\Lambda(\Lambda_{\hat v}))tz_2. \end{align} $$

Notice that $t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2\subseteq t(\mathcal B'\cap f {\mathcal M} f)tz_2\subseteq u_2 {\mathcal L}(\Gamma _{\mathscr {C}})u_2^*$ , and observe one can pick t small enough so that $tz_2$ is subequivalent to $v^*v$ inside $(\mathcal B'\cap f {\mathcal M} f)z_2$ . Using this, one can find a unitary $u_3\in {\mathcal M}$ , such that $t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2\subseteq u_3 v^*v \mathcal L(\Gamma _v)v^*v u_3^*$ . Hence, using this in combination with relation (6.6), we see that for all unitaries $x\in t\mathcal L(C_\Lambda (\Lambda _{\hat v}))tz_2$ , we have that $\psi (x)$ is a unitary in $r u_2 \mathcal L(\Gamma _{\mathscr {C}}) u_2^* r$ , and therefore,

(6.7) $$ \begin{align} \begin{aligned} \|E_{u_2 {\mathcal L}(\Gamma_{\hat v}) u_2^*}(ww^*)\|_2&=\|E_{u_2 {\mathcal L}(\Gamma_{\hat v}) u_2^*}(\psi(x)ww^*)\|_2= \| E_{u_2{\mathcal L}(\Gamma_{\hat v})u_2^*}(wxw^*)\|_2\\ &=\| E_{u_2{\mathcal L}(\Gamma_{\hat v})u_2^*}(wE_{u_3 {\mathcal L}(\Gamma_v)u_3^*}(x)w^*)\|_2\\ &=\| E_{{\mathcal L}(\Gamma_{\hat v})}(u_2^*w u_3E_{{\mathcal L}(\Gamma_v)}(u_3^*x u_3)u_3^*w^* u_2)\|_2. \end{aligned} \end{align} $$

Since $w\neq 0$ , then (6.7) and basic approximations of $u_2^*w u_3$ and $u_3$ imply the existence of a finite subset $F\subset \Gamma $ and a constant $C>0$ , such that for all unitaries $x\in t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2$ , we have

(6.8) $$ \begin{align} \sum_{g,h,k,l\in F }| E_{{\mathcal L}(\Gamma_{\hat v})}(u_gE_{{\mathcal L}(\Gamma_v)}(u_hx u_k) u_l)\|_2\geq C. \end{align} $$

However, we see that for all $x\in t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2$ , we have

$$ \begin{align*}\begin{aligned}\sum_{g,h,k,l\in F }| E_{{\mathcal L}(\Gamma_{\hat v})}(u_gE_{{\mathcal L}(\Gamma_v)}(u_hx u_k) u_l)\|_2&= \sum_{g,h,k,l\in F }| P_{ g^{-1}\Gamma_{\hat v}l^{-1}\cap \Gamma_v}(u_hx u_k)\|_2 \\&= \sum_{g,h,k,l\in F, gl^{-1}\in \Gamma_v }| P_{ (g^{-1}\Gamma_{\hat v}g^{-1}\cap \Gamma_v)gl^{-1}}(u_hx u_k)\|_2\\&= \sum_{g,h,k,l\in F, gl^{-1}\in \Gamma_v }| E_{ {\mathcal L}(g^{-1}\Gamma_{\hat v}g^{-1}\cap \Gamma_v)}(u_hx u_{klg^{-1}})\|_2\\ &=\sum_{g,h,k,l\in F, gl^{-1}\in \Gamma_v }| \tau(x u_{klg^{-1}h})|.\end{aligned}\end{align*} $$

Here, we denoted by $P_S:\ell ^2(\Gamma )\to \ell ^2(\Gamma )$ the orthogonal projection onto the $\|\cdot \|_2$ -closure of the $\mathrm {span}\{u_g|g\in S\}$ for any subset $S\subset \Gamma $ . As F is finite and $t{\mathcal L}(C_\Lambda (\Lambda _{\hat v}))tz_2$ is diffuse, there is a sequence of unitaries $x_n \in t\mathcal L(C_\Lambda (\Lambda _{\hat v}))tz_2$ , such that $\sum _{g,h,k,l\in F, gl^{-1}\in \Gamma _v }| \tau (x_n u_{klg^{-1}h})|\rightarrow 0$ as $n\rightarrow \infty $ . This, however, contradicts (6.8). Hence, (6.5) must hold.

Thus, (6.4) implies that $z_2 ({\mathcal L}(C_\Lambda (\Lambda _{\hat v}))f\vee {\mathcal L}(C_\Lambda (\Lambda _{\hat v}))f'\cap f{\mathcal M} f) z_2\subseteq u_2\mathcal L(\Gamma _{\mathscr {C}})u_2^*$ . Again, since $\mathcal L(\Gamma _{\mathscr {C}})$ is a factor, there is $u\in \mathscr {U}({\mathcal M})$ so that

(6.9) $$ \begin{align} ({\mathcal L}(C_\Lambda(\Lambda_{\hat v}))f\vee {\mathcal L}(C_\Lambda(\Lambda_{\hat v}))f'\cap f{\mathcal M} f) z\subseteq u\mathcal L(\Gamma_{\mathscr{C}})u^*,\end{align} $$

where z is the central support of $z_2$ in $(\mathcal L(C_\Lambda (\Lambda _{\hat v}))f\vee \mathcal L(C_\Lambda (\Lambda _{\hat v}))f'\cap f{\mathcal M} f)$ . In particular, we have $vv^*\leq z_2 \leq z \leq f$ . Now since $f\mathcal L(\Lambda _{\hat v} )f\subseteq \mathcal L(C_\Lambda (\Lambda _{\hat v}))f'\cap f{\mathcal M} f$ , then by (6.9), we get $(f{\mathcal L}(\Lambda _{\hat v} )f\vee {\mathcal L}(C_\Lambda (\Lambda _{\hat v}))f)z\subseteq u{\mathcal L}(\Gamma _{\mathscr {C}})u^*$ , and hence

(6.10) $$ \begin{align} u^*({\mathcal L}(C_\Lambda(\Lambda_{\hat v}))f\vee f {\mathcal L}(\Lambda_{\hat v} )f) z u\subseteq {\mathcal L}(\Gamma_{\mathscr{C}}).\end{align} $$

In particular, (6.10) implies that $u^*f{\mathcal L}(\Lambda _{\hat v})fzu\subseteq {\mathcal L}(\Gamma _{\mathscr {C}})$ . Since $vv^*\leq z\in f{\mathcal L}(C_\Lambda (\Lambda _{\hat v})) f'\cap f{\mathcal M} f$ and $\mathcal B$ is a factor, then the map $\phi ': a\mathcal L(\Gamma _{\hat v})a \rightarrow u^* \mathcal B z u \subseteq u^*f \mathcal L(\Lambda _{\hat v})fz u$ given by $\phi '(x)=u^* \phi (x)z u$ still defines a $\ast $ -isomorphism that satisfies $\phi '(x) w=wx$ , for any $x\in a\mathcal L(\Gamma _{\hat v})a$ , where $w= u^*zv$ is a nonzero partial isometry. Hence, $\mathcal L(\Gamma _{\hat v}) \prec _{{\mathcal M}} u^*f\mathcal L(\Lambda _{\hat v})fzu$ . Thus, by Corollary 2.17, it follows that $\mathcal L(\Gamma _{\hat v})\prec _{\mathcal L(\Gamma _{\mathscr {C}})} u^*f\mathcal L(\Lambda _{\hat v})fzu$ .

To this end, using [Reference Chifan, Kida and PantCKP14, Proposition 2.4] and its proof, we can find nonzero ${p \in \mathscr P({\mathcal L}(\Gamma _{\hat v}))}$ , $r=u^*ezu \in u^*f{\mathcal L}(\Lambda _{\hat v})fzu$ with $e\in \mathscr P(f{\mathcal L}(\Lambda _{\hat v})f)$ , a von Neumann subalgebra $\mathcal C \subseteq u^*e\mathcal L(\Lambda _{\hat v})ezu $ and a $\ast $ -isomorphism $\theta : p\mathcal L(\Gamma _{\hat v})p \rightarrow \mathcal C$ , such that:

1. a) the inclusion $\mathcal C \vee (\mathcal C' \cap u^*e{\mathcal L}(\Lambda _{\hat v})ezu ) \subseteq u^*e{\mathcal L}(\Lambda _{\hat v})ezu $ has finite index;

2. b) there is a nonzero partial isometry $y \in {\mathcal L}(\Gamma _{\mathscr {C}})$ , such that $\theta (x) y =y x$ for all ${x\in p{\mathcal L}(\Gamma _{\hat v})p}$ , where $y^*y\in p{\mathcal L}(\Gamma _{\hat v})p'\cap p{\mathcal M} p$ and $yy^*\in \mathcal C '\cap r{\mathcal M} r$ .

Note that $r\in {\mathcal L}(\Gamma _{\mathscr {C}})$ and $\mathcal C$ , $\mathcal C' \cap u^* r{\mathcal L}(\Lambda _{\hat v})r z u$ and $u^*\mathcal L(C_\Lambda (\Lambda _{\hat v}))r z u$ are commuting von Neumann subalgebras of $r {\mathcal L}(\Gamma _{\mathscr {C}})r$ . Since $\mathcal C$ is isomorphic to a corner of $\mathcal L(\Gamma _{\hat v})$ and $u^*\mathcal L(C_\Lambda (\Lambda _{\hat v}))ezu$ has no amenable direct summand, then the assumption implies that $\mathcal C' \cap u^* e\mathcal L(\Lambda _{\hat v})ezu$ is purely atomic. Thus, one can find a nonzero projection $q \in \mathcal Z(\mathcal C' \cap u^* e{\mathcal L}(\Lambda _{\hat v})e zu )$ , such that after compressing the containment in a) by q and replacing $\mathcal C$ by $\mathcal C q$ , y by $qy$ and $\theta (x)$ by $\theta (x) q$ in b), we can assume in addition that $\mathcal C \subseteq u^*e{\mathcal L}(\Lambda _{\hat v})ezu $ is a finite index inclusion of nonamenable II $_1$ factors. By [Reference Pimsner and PopaPP86, Proposition 1.3], it follows that $\mathcal C \subseteq u^*e{\mathcal L}(\Lambda _{\hat v})ezu $ admits a finite Pimsner-Popa basis, which implies that there exist $x_1,\dots ,x_m\in u^*e{\mathcal L}(\Lambda _{\hat v})ezu$ , such that $u^*e{\mathcal L}(\Lambda _{\hat v})ezu=\sum _{i=1}^m x_i \mathcal C$ . Note also that $u^*e{\mathcal L}(\Lambda _{\hat v})ezu \subset r{\mathcal M} r$ since $r=u^*ezu$ . Hence,

(6.11) $$ \begin{align} \mathcal {QN}_{ r{\mathcal M} r}(\mathcal C ) "=\mathcal {QN}_{ r{\mathcal M} r}( u^*e{\mathcal L}(\Lambda_{\hat v})ezu )". \end{align} $$

Also, the intertwining relation in b) shows that $\mathcal C yy^*= y p{\mathcal L}(\Gamma _{\hat v})p y^*= l p{\mathcal L}(\Gamma _{\hat v})p y^*y l$ where $yy^*\in \mathcal C'\cap r {\mathcal M} r$ and $l\in {\mathcal L}(\Gamma _{\mathscr {C}})$ is a unitary extending y, meaning $y=ly^*y$ . Therefore, using the quasi-normalizer formulas for group von Neumann algebras and for compressions, Lemmas 2.15 and 2.13 (and Remark 2.14), respectively, we deduce that

(6.12) $$ \begin{align}\begin{aligned} ly^*y {\mathcal L}(\Gamma_{\mathscr{C}}) y^*yl^* & = ly^*y {\mathcal L}(\mathrm{QN}_\Gamma (\Gamma_{\hat v})) y^*yl^* \overset{L.2.15}{=} l y^*y \mathcal{QN}_{{\mathcal M}}({\mathcal L}(\Gamma_{\hat v}))"y^*yl^*\\ & \overset{L.2.13}{=}\mathcal {QN}_{l y^*y {\mathcal M} y^*yl}(l p{\mathcal L}(\Gamma_{\hat v})p y^*y l)"= \mathcal {QN}_{yy^*{\mathcal M} yy^*}(\mathcal C yy^*)" \\ &\overset{R.2.14}{=}yy^*\mathcal {QN}_{ r{\mathcal M} r}(\mathcal C ) "yy^*\overset{(6.11)}{=}yy^*\mathcal {QN}_{ r{\mathcal M} r}( u^* e{\mathcal L}(\Lambda_{\hat v})ezu )" yy^*\\ &\overset{L.2.13}{=} yy^* u^*z e\mathcal {QN}_{ {\mathcal L}(\Lambda)}({\mathcal L}(\Lambda_{\hat v}) )"ez u yy^*\overset{L.2.15}{=}yy^* u^* z e{\mathcal L} (\mathrm{QN}_{\Lambda}(\Lambda_{\hat v}) )e zu yy^*. \end{aligned}\end{align} $$

Denote $\Upsilon = \mathrm {QN}_\Lambda (\Lambda _{\hat v})$ and $\Sigma = \langle \mathrm {QN}^{(1)}_\Lambda (\Upsilon ))\rangle <\Lambda $ . As $\mathrm {QN}^{(1)}_\Gamma (\Gamma _{\mathscr {C}})= \Gamma _{\mathscr {C}}$ , then formula (6.12) together with the corresponding formulas for one-sided quasi-normalizers, Lemmas 2.13 and 2.15, show that

(6.13) $$ \begin{align}yy^* u^* ze{\mathcal L} (\Upsilon)e zu yy^* =yy^* u^* z e{\mathcal L} (\Sigma)ez u yy^*= ly^*y {\mathcal L}(\Gamma_{\mathscr{C}}) y^*yl^*.\end{align} $$

In particular, by [Reference Chifan and IoanaCI17, Lemma 2.2], we have $[\Sigma :\Upsilon ]<\infty $ , and one can also check that $\mathrm {QN}^{(1)}_\Lambda (\Sigma )=\Sigma $ .

Notice the above relations also show that $yy^*= u^* d u$ for some projection $d\in ze{\mathcal L} (\Sigma )ez$ . Thus, relation (6.13) entails that $ u^* d{\mathcal L} (\Sigma ) d u = ly^*y {\mathcal L}(\Gamma _{\mathscr {C}}) y^*yl^*$ , and letting $w_0:=ul \in \mathscr {U}({\mathcal M})$ and $t=y^*y$ , we conclude that $w^*_0 d {\mathcal L}(\Sigma )d w_0= t {\mathcal L}(\Gamma _{\mathscr {C}}) t$ . Moreover, if we replace $w^*_0 \Sigma w_0$ by $\Sigma $ and we use $w^*_0 d w_0= t$ , we have that $t {\mathcal L}(\Sigma )t = t {\mathcal L}(\Gamma _{\mathscr {C}})t$ . As $\mathcal L(\Gamma _{\mathscr {C}})$ is a factor, one can find a unitary $w_1 \in {\mathcal M}$ so that if c denotes the central support of $t \in {\mathcal L}(\Sigma )$ , we have that ${\mathcal L}(\Sigma ) c \subseteq w_1\mathcal L(\Gamma _{\mathscr {C}})w_1^*$ . This implies that there exists a projection $h \in {\mathcal L}(\Gamma _{\mathscr {C}})$ , such that $t= w_1 hw_1^*$ . Moreover, since ${\mathcal L}(\Gamma _{\mathscr {C}})$ is a factor, there is a unitary $w_2 \in \mathcal L(\Gamma _{\mathscr {C}})$ so that $t = w_2 hw^*_2$ . Altogether, these relations show that $wt=tw$ where $w :=w_1 w_2^*$ . Also, we have that ${\mathcal L}(\Sigma )c\subseteq w {\mathcal L}(\Gamma _{\mathscr {C}})w^*$ . Multiplying on both sides by t, we get $t {\mathcal L}(\Gamma _{\mathscr {C}})t=t\mathcal L(\Sigma )t\subseteq t w\mathcal L(\Gamma _{\mathscr {C}})w^* t$ , and hence $tw^*t \mathcal L(\Gamma _{\mathscr {C}}) t\subseteq t \mathcal L(\Gamma _{\mathscr {C}})t w^* t$ . In particular, using Corollary 2.17, we get $w^*t =tw^*t\in \mathcal {QN}^{(1)}_{t{\mathcal M} t}(t{\mathcal L}(\Gamma _{\mathscr {C}})t)=t {\mathcal L}(\Gamma _{\mathscr {C}})t$ , and hence $wt \in t {\mathcal L}(\Gamma _{\mathscr {C}})t$ . Altogether, these relations imply that $t {\mathcal L}(\Sigma )t= t{\mathcal L}(\Gamma _{\mathscr {C}})t=w t \mathcal L(\Gamma _{\mathscr {C}})t w^*$ . Since $\mathcal L(\Sigma ) c\subseteq w \mathcal L(\Gamma _{\mathscr {C}})w^*$ , we apply the ‘moreover’ part in [Reference Chifan and IoanaCI17, Lemma 2.6] and derive that $\mathcal L(\Sigma ) c= c \mathcal L(\Gamma _{\mathscr {C}})c$ .

In conclusion, we showed there is a subgroup $\Lambda _{\hat v} \mathrm {C}_\Lambda (\Lambda _{\hat v})<\mathrm {QN}_\Lambda (\Lambda _{\hat v})<\Sigma <\Lambda $ , such that $[\Sigma :\mathrm {QN}_\Lambda (\Lambda _{\hat v})]<\infty $ , and $\mathrm {QN}^{(1)}_\Lambda (\Sigma )=\Sigma $ . Moreover, there are $r\in \mathscr P(\mathscr Z({\mathcal L}(\Sigma )))$ , $v_0\in \mathscr {U}({\mathcal M})$ and $n\in \mathscr P(\mathcal L(\Gamma _{\mathscr {C}}))$ so that

(6.14) $$ \begin{align} v_0{\mathcal L}(\Sigma) rv^*_0=c{\mathcal L}(\Gamma_{\mathscr{C}})c.\end{align} $$

Finally, we show that $[\Sigma : \Lambda _{\hat v} \mathrm {vC}_\Lambda (\Lambda _{\hat v})]<\infty $ . Let $u^*qzu=s\in u^* e {\mathcal L}(\Lambda _{\hat v})e z u$ be a projection, and let $\mathcal C_1\subseteq \mathcal C \subseteq u^* e {\mathcal L}(\Lambda _{\hat v})e z u$ be a subfactor, such that $u^* e {\mathcal L}(\Lambda _{\hat v})e z u=\langle \mathcal C, s \rangle $ is the basic construction for $\mathcal C_1\subseteq \mathcal C$ . Notice that ${\mathcal D}_1 := \theta ^{-1}(\mathcal C_1)\subseteq p\mathcal L(\Gamma _{\hat v})p$ is a finite index subfactor. Let $\Theta : {\mathcal D}_1 \rightarrow \mathcal C_1s= u^*q \mathcal L(\Lambda _{\hat v})qzu$ be the isomorphism given by $\Theta (x)=\theta (x)s$ for all $x\in {\mathcal D}_1$ . Using b), we have that $\Theta (x) sy=sy x$ for all $x\in {\mathcal D}_1$ . Since s is a Jones’s projection for $\mathcal C_1\subseteq \mathcal C$ , one can check that $sy\neq 0$ . Letting w be the polar decomposition of $sy$ , we further have that $\Theta (x) w=w x$ for all $x\in {\mathcal D}_1$ . By Corollary 2.17, we have $s\in {\mathcal L}(\Gamma _{\mathscr {C}})$ , $ww^*\in u^*q\mathcal L(\Lambda _{\hat v})qzu'\cap s{\mathcal M} s \subset s\mathcal L(\Gamma _{\mathscr {C}})s$ and $w^*w \in p{\mathcal D}_1 p' \cap p{\mathcal M} p\subset p{\mathcal D}_1 p' \cap pL(\Gamma _{\mathscr {C}}) p$ . Letting $w_o$ be a unitary extending w and using the previous intertwining relation, we see that $u^* q{\mathcal L}(\Lambda _{\hat v})qzu ww^*= w_o {\mathcal D}_1 w^*_o$ . By taking relative commutants, we obtain that $ ww^*(u^* q\mathcal L(\Lambda _{\hat v})qzu'\cap s {\mathcal M} s)ww^*=w_o {\mathcal D}_1'\cap p\mathcal L(\Gamma _{\mathscr {C}})p w_o^* $ , and hence,

(6.15) $$ \begin{align}\begin{aligned} &w_o ({\mathcal D}_1\vee \mathcal L(\Gamma_v)p )w_o^*\subset w_o ({\mathcal D}_1\vee {\mathcal D}_1'\cap p\mathcal L(\Gamma_{\mathscr{C}})p )w_o^*=\\ &=ww^*(u^* q{\mathcal L}(\Lambda_{\hat v})qzu \vee u^* qL(\Lambda_{\hat v})qzu'\cap s {\mathcal M} s)ww^* \\ &=ww^*u^* qz ({\mathcal L}(\Lambda_{\hat v}) \vee {\mathcal L}(\Lambda_{\hat v})'\cap {\mathcal M} ) qzu ww^*\subseteq ww^* {\mathcal L}(\Gamma_{\mathscr{C}})ww^*.\end{aligned}\end{align} $$

Since $\Gamma _{\mathscr {C}}$ is malnormal in $\Gamma $ , the containment $w_o ({\mathcal D}_1\vee {\mathcal L}(\Gamma _v)p )w_o^*\subseteq ww^* {\mathcal L}(\Gamma _{\mathscr {C}})ww^*$ implies that $w=w_o p\in {\mathcal L}(\Gamma _{\mathscr {C}})$ , and hence $ww^* {\mathcal L}(\Gamma _{\mathscr {C}})ww^*= w{\mathcal L}(\Gamma _{\mathscr {C}})w^*= w_o p{\mathcal L}(\Gamma _{\mathscr {C}})pw_o^*$ . Therefore, since ${\mathcal D}_1\vee {\mathcal L}(\Gamma _v)p\subseteq p\mathcal L(\Gamma _{\mathscr {C}})p$ has finite index, then so does $w_o ({\mathcal D}_1\vee \mathcal L(\Gamma _v)p )w_o^*\subset ww^*\mathcal L(\Gamma _{\mathscr {C}}) ww^*$ . In particular, all inclusions in (6.15) are of finite index. Hence, we have ${\mathcal L}(\Gamma _{\mathscr {C}})\prec _{\mathcal M} {\mathcal L}(\Lambda _{\hat v})\vee \mathcal L(\Lambda _{\hat v})'\cap {\mathcal M}$ and since $\mathcal L(\Lambda _{\hat v})\vee \mathcal L(\Lambda _{\hat v})'\cap {\mathcal M}\subseteq \mathcal L(\Lambda _{\hat v} vC_\Lambda (\Lambda _{\hat v}))$ , we get $\mathcal L(\Gamma _{\mathscr {C}})\prec _{\mathcal M} \mathcal L(\Lambda _{\hat v} vC_\Lambda (\Lambda _{\hat v}))$ . Using (6.14), this further implies that $\mathcal L(\Sigma )\prec _{\mathcal M} \mathcal L(\Lambda _{\hat v} \mathrm { vC}_\Lambda (\Lambda _{\hat v}))$ . Moreover, since $\Lambda _{\hat v}\mathrm {vC}_\Lambda (\Lambda _{\hat v})<\Sigma $ and $\mathrm {QN}^{(1)}(\Sigma )=\Sigma $ , we actually have that ${\mathcal L}(\Sigma )\prec _{{\mathcal L}(\Sigma )} {\mathcal L}(\Lambda _{\hat v} \mathrm {vC}_\Lambda (\Lambda _{\hat v}))$ . Using [Reference Drimbe, Hoff and IoanaDHI16, Lemma 2.5], this implies that $[\Sigma : \Lambda _{\hat v} \mathrm {vC}_\Lambda (\Lambda _{\hat v})]<\infty $ , as claimed.