Full Factors and Co-amenable Inclusions

Abstract

We show that if M is a full factor and $N \subset M$ is a co-amenable subfactor with expectation, then N is also full. This answers a question of Popa from 1986. We also generalize a theorem of Tomatsu by showing that if M is a full factor and $\sigma :G \curvearrowright M$ is an outer action of a compact group G, then $\sigma$ is automatically minimal and $M^G$ is a full factor which has w-spectral gap in M. Finally, in the appendix, we give a proof of the fact that several natural notions of co-amenability for an inclusion $N\subset M$ of von Neumann algebras are equivalent, thus closing the cycle of implications given in Anantharaman-Delaroche’s paper in 1995.

Appendix A. Bimodule Characterization of Co-amenability

Let M be a von Neumann algebra. We will denote by ${\mathrm{L}}^2(M)$ the standard Hilbert space for M, by $J_M$ the modular conjugation, by ${\mathrm{L}}^2(M)^+$ the natural positive cone, and by $\pi _M :M \odot M^{\mathrm {op}}\rightarrow \mathbf {B}({\mathrm{L}}^2(M))$ the natural binormal $*$-representation. We define the norm

$$\begin{aligned} M \odot M^{\mathrm {op}}\ni T \mapsto \Vert T \Vert _{\pi _M}=\Vert \pi _M(T) \Vert _{\mathbf {B}({\mathrm{L}}^2(M))}. \end{aligned}$$

For $x \in M$, we will use the notation $\overline{x}=(x^*)^{\mathrm {op}}=(x^{\mathrm {op}})^* \in M^{\mathrm {op}}$.

Haagerup has made a deep study of the norm

$$\begin{aligned} M^{\oplus n} \ni (x_1, \ldots , x_n) \mapsto \Vert \sum _i x_i \otimes \overline{x_i} \Vert _{\pi _M} \end{aligned}$$

and obtained many remarkable results in [14], but we need only the following (Theorem 2.1 in [14]). This was proved earlier in the semifinite case by Pisier (Theorem 2.1 in [31]). Alternative proofs are found in [21] and [41].

Theorem A.1

(Pisier–Haagerup). For any von Neumann algebra M and any $x_1,\ldots ,x_n\in M$,

$$\begin{aligned} \Vert \sum _i x_i \otimes \overline{x_i} \Vert _{\pi _M}^{1/2}=\Vert (x_1,\ldots ,x_n)\Vert _{1/2}, \end{aligned}$$

where the latter norm is the complex interpolation of equivalent norms $\Vert \,\cdot \,\Vert _{0}$ and $\Vert \,\cdot \,\Vert _{1}$ on $M^{\oplus n}$ that are given by

$$\begin{aligned} \Vert (a_1,\ldots ,a_n)\Vert _{0}:=\Vert \sum _i a_ia_i^*\Vert ^{1/2} \text{ and } \Vert (a_1,\ldots ,a_n)\Vert _{1}:=\Vert \sum _i a_i^*a_i\Vert ^{1/2}. \end{aligned}$$

In particular, for any (not necessarily normal) unital completely positive map $\theta$ from M into a von Neumann algebra N and any $x_1,\ldots ,x_n\in M$,

$$\begin{aligned} \Vert \sum _i \theta (x_i) \otimes \overline{\theta (x_i)} \Vert _{\pi _N} \le \Vert \sum _i x_i \otimes \overline{x_i} \Vert _{\pi _M}. \end{aligned}$$

Note that the second half of Theorem A.1 is a trivial consequence of the interpolation theorem, because $\theta \otimes \mathrm{id}_n:(M^{\oplus n},\Vert \,\cdot \,\Vert _{t}) \rightarrow (N^{\oplus n},\Vert \,\cdot \,\Vert _{t})$ is contractive at $t=0,1$. The purpose of this appendix is to give a proof of the following corollary to Theorem A.1.

Corollary A.2

(cf. Corollary 3.8 in [14]). For von Neumann algebras $N\subset M$, the following are equivalent.

• (i) There is a conditional expectation of M onto N.

• (ii) For every $x_1,\ldots ,x_n\in N$,

  $$\begin{aligned} \Vert \sum _i x_i \otimes \overline{x_i} \Vert _{\pi _N}=\Vert \sum _i x_i \otimes \overline{x_i} \Vert _{\pi _M}. \end{aligned}$$

• (iii) ${}_N{\mathrm{L}}^2(N)_N\prec {}_N{\mathrm{L}}^2(M)_N$. Namely, for every $x_i,y_i\in N$,

  $$\begin{aligned} \Vert \sum _i x_i \otimes \overline{y_i} \Vert _{\pi _N} \le \Vert \sum _i x_i \otimes \overline{y_i} \Vert _{\pi _M}. \end{aligned}$$

The following reformulation of Corollary A.2 closes the circle of implications of Propositions 2.5 in [1].

Corollary A.3

(cf. Proposition 3.6 in [1]). Let ${}_M\mathcal {H}_N$ be an M-N correspondence and put $N_1=\mathbf {B}_{N^{\mathrm {op}}}(\mathcal {H})$, the commutant of the right N-action. Then the following are equivalent.

• (i) ${}_M\mathcal {H}_N$ is left-amenable, i.e. ${}_M {\mathrm{L}}^2(M)_M\prec {}_M {\mathrm{L}}^2(N_1)_M$.

• (ii) There is a conditional expectation of $N_1$ onto M.

We recall the notion of the selfpolar forms ( [10, 40]), which plays a central role in [14] as well as in the proof of Corollary A.2. Associated with any normal state $\varphi$ on M, there are a canonical unit vector $\xi _\varphi$ in ${\mathrm{L}}^2(M)^+$ and the selfpolar form $s_\varphi :M\times M\rightarrow \mathbb {C}$ given by

$$\begin{aligned} s_\varphi (x,y)=\mathopen {\langle }\pi _M(x \otimes \overline{y})\xi _\varphi ,\xi _\varphi \mathclose {\rangle }_{{\mathrm{L}}^2(M)}. \end{aligned}$$

Proposition A.4

(Theorems 1.1 and 2.1 in [40]). The selfpolar form $s_\varphi$ satisfies the following properties.

• (i) $s_\varphi$ is a positive semi-definite sesqui-linear form on $M\times M$,

• (ii) $s_\varphi \ge 0$ on $M_+\times M_+$,

• (iii) $s_\varphi (x,1)=\varphi (x)$ for $x\in M$, and

• (iv) $s_{\varphi }$ is selfpolar.

Moreover, if s is any sesqui-linear form on M which satisfies (i)–(iii) above (with $s_\varphi$ exchanged by s), then $s(x,x)\le s_\varphi (x,x)$ for every $x\in M$.

We do not introduce the selfpolar property (iv), since we will not use it (in an explicit way). We use the following well-known variant of the Hahn–Banach theorem.

Lemma A.5

Let C be a convex cone of a real vector space V, $p:V\rightarrow \mathbf {R}$ a sublinear function, and $q:C\rightarrow \mathbf {R}$ a superlinear function. This means that $p(x+y)\le p(x)+p(y)$ and $p(\lambda x)=\lambda p(x)$ for every $x,y\in V$ and $\lambda \in \mathbf {R}_{\ge 0}$; and that $-q$ is sublinear. If $q\le p$ on C, then there is a linear functional $\psi$ on V such that $\psi \le p$ on V and $q\le \psi$ on C.

Proof

Observe that $r(x):=\inf \{ p(x+y)-q(y) : y\in C\}$ is a sublinear function on V such that $-p(-x) \le r(x) \le p(x)$ for $x\in V$ (in particular r takes values in $\mathbf {R}$). By the Hahn–Banach theorem, there is a linear functional $\psi$ on V such that $\psi \le r$. One has $-\psi (x)\le r(-x)\le -q(x)$ for $x\in C$. $\quad \square$

Proof of Corollary A.2

The equivalence $(\mathrm{i})\Leftrightarrow (\mathrm{ii})$ is proved in Corollary 3.8 in [14]. That $(\mathrm{i})\Rightarrow (\mathrm{ii})$ follows from Theorem A.1. That $(\mathrm{iii})\Rightarrow (\mathrm{i})$ is proved in Proposition 2.5 in [1]. Thus it is left to show $(\mathrm{ii})\Rightarrow (\mathrm{iii})$. We closely follow the proof of Corollary 3.8 (Theorem 3.7) in [14]. Take any normal state $\varphi$ on N and consider the convex cone $C:=\{\sum _i x_i\otimes \overline{x_i} : x_i \in N \}\subset M\odot M^{\mathrm {op}}$ and the seminorm $p(\,\cdot \,)=\Vert \pi _M(\,\cdot \,)\Vert _{\mathbf {B}(({\mathrm{L}}^2(M))}$ on $M\odot M^{\mathrm {op}}$. Since

$$\begin{aligned} \sum _i s_{\varphi }(x_i,x_i) =\mathopen {\langle }\pi _N(\sum _i x_i\otimes \overline{x_i})\xi _\varphi ,\xi _\varphi \mathclose {\rangle } \le p(\sum _i x_i\otimes \overline{x_i}) \end{aligned}$$

by condition (ii), Lemma A.5 yields $\psi \in \mathbf {B}({\mathrm{L}}^2(M))^*$ of norm 1 such that $s_{\varphi }(x,x)\le \mathfrak {R}\psi (\pi _M(x \otimes \overline{x}))$ for $x\in N$. Observe that $\Vert \psi \Vert \le 1$ and $1\le \mathfrak {R}\psi (1)$ imply that $\psi$ is a state. The state $\psi ^*$ on $\mathbf {B}({\mathrm{L}}^2(M))$ defined by $\psi ^*(T)=\psi (J_M T^* J_M)$ satisfies $\psi ^*(\pi _M(x \otimes \overline{y}))=\overline{\psi (\pi _M(y \otimes \overline{x}))}$ for every $x,y\in M$. Thus by replacing $\psi$ with the state $\frac{1}{2}(\psi +\psi ^*)$, we may assume that $s:M\times M\ni (x,y)\mapsto \psi (\pi _M(x \otimes \overline{y}))$ is a sesqui-linear form such that $s_{\varphi }(x,x) \le s(x,x)$ for every $x\in N$. Since $s_{\varphi }(1+\lambda x,1+\lambda x) \le s(1+\lambda x,1+\lambda x)$ for all $\lambda \in \mathbf {R}$ and a fixed $x\in N$, one sees $\varphi (x)=s(x,1)$ for $x\in N$. It follows from Proposition A.4 that $s_{\varphi }(x,x)=s(x,x)$ for every $x\in N$. Hence by polarization identity, $s_{\varphi }=s$ and so for every $T=\sum _i x_i \otimes \overline{y_i} \in N \odot N^{\mathrm {op}}$ we have

$$\begin{aligned} \langle \pi _N(T) \xi _\varphi ,\xi _\varphi \rangle = \sum _i s_\varphi (x_i,y_i)=\sum _i s(x_i,y_i) =\psi (\pi _M(T)). \end{aligned}$$

Since $\{ \xi _\varphi \}_\varphi$ is a $\pi _N$-cyclic family, one obtains

$$\begin{aligned} \Vert \pi _N(T) \Vert _{\mathbf {B}({\mathrm{L}}^2(N))} \le \Vert \pi _M(T) \Vert _{\mathbf {B}({\mathrm{L}}^2(M))} \end{aligned}$$

which is precisely condition (iii). $\quad \square$