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# Strictly outer actions of locally compact groups: Beyond the full factor case

Basile Morando

► To cite this version:

Basile Morando. Strictly outer actions of locally compact groups: Beyond the full factor case. Journal of Functional Analysis, 2025, 289 (2), pp.110897. 10.1016/j.jfa.2025.110897. hal-04997358

HAL Id: hal-04997358

https://hal.science/hal-04997358v1

Submitted on 19 Mar 2025

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# Strictly outer actions of locally compact groups: beyond the full factor case

Basile Morando

March 19, 2025

## Abstract

We show that, given a continuous action $\alpha$ of a locally compact group $G$ on a factor $M$, the relative commutant $M' \cap (M \rtimes_{\alpha} G)$ is contained in $M \rtimes_{\alpha} H$ where $H$ is the subgroup of elements acting without spectral gap. As a corollary, we answer a question of Marrakchi and Vaes by showing that if $M$ is semifinite and $\alpha_g$ is not approximately inner for all $g \neq 1$, then $M' \cap (M \rtimes_{\alpha} G) = \mathbb{C}$.

# 1 Introduction and results

Given a continuous action $\alpha : G \curvearrowright M$ of a locally compact group $G$ on a von Neumann algebra $M$, we can build the *crossed product* $M \rtimes_{\alpha} G$: this algebra is generated by a copy of $M$ and a unitary representation of $G$ such that, given $u_g$ in the representation of $G$ and $x$ in the copy of $M$, we have $u_g x u_g^* = \alpha_g(x)$. Understanding how properties of the action reflect into properties of the algebra $M \rtimes_{\alpha} G$ and of the inclusion $M \subset M \rtimes_{\alpha} G$ is a fundamental question in the theory of non-commutative dynamical systems. In this article, we focus on the following question:

**Problem 1.** Find a criterion on $\alpha : G \curvearrowright M$ a continuous action of a locally compact group on a factor $M$, ensuring that $M' \cap (M \rtimes_{\alpha} G)$ is trivial. Such an action is said to be strictly outer, and this property trivially implies factoriality of $M \rtimes_{\alpha} G$.

When the group $G$ is discrete, many properties of the crossed product can be deduced from the *Fourier decomposition*: every element of $M \rtimes_{\alpha} G$ is uniquely written as a sum $\sum_{\gamma \in G} x_{\gamma} u_{\gamma}$ (where $x_{\gamma}$ and $u_{\gamma}$ respectively refers to elements of the copy of $M$ and $G$ inside $M \rtimes_{\alpha} G$). It allows in particular to answer problem 1.

* When $M = \mathbb{C}$, $\mathbb{C} \rtimes G$ is just the group of von Neumann algebra $L(G)$. This algebra is a factor if and only if $G$ has infinite non-trivial conjugacy classes.

* When $M$ is a factor, the relative commutant $M' \cap (M \rtimes_{\alpha} G)$ is trivial if and only if $\alpha$ is outer: for all $\gamma \neq 1$ in $G$, there is no unitary $u \in M$ such that $\alpha_{\gamma} = \text{Ad}(u)$.

* More generally, consider the subgroup $H = \{\gamma \in G \mid \alpha_{\gamma}$ is inner\}$. Then $M' \cap (M \rtimes_{\alpha} G) \subset M \rtimes_{\alpha} H$. When $M$ is a type $\Pi_1$ factor, it implies that $M \rtimes_{\alpha} G$ is a factor if and only if the subgroup $H$ of inner-acting elements of $G$ is $G$-icc (for every $h \in H \setminus \{1\}$, the set $\{\gamma h \gamma^{-1} \mid \gamma \in G\}$ is infinite).

In the non-discrete case, these questions become much more subtle:

* Even in the simplest case where $M = \mathbb{C}$, we do not have any criterion for factoriality of $\mathbb{C} \rtimes G = L(G)$.

* When $M$ is a factor, it is still true that $M' \cap (M \rtimes_{\alpha} G) = \mathbb{C}$ implies that $\alpha$ is outer. However, the converse is no longer true. For example, if $M$ is a factor of type $\Pi_0$ with trivial $T$-invariant, any action $\sigma_t^\varphi : \mathbb{R}^{+*} \curvearrowleft M$ by modular automorphisms is outer, but $M \rtimes_{\sigma_t^\varphi} \mathbb{R}^{+*}$ is not even a factor. As explained in [12, Example 8], there also exist outer actions $\alpha : \mathbb{S}^1 \curvearrowleft R$ of $\mathbb{S}^1$ on the hyperfinite $\Pi_1$ factor such that $R \rtimes_{\alpha} \mathbb{S}^1$ is not a factor either.
• Consider $(M, \tau)$ a $\Pi_\infty$ factor, $\tau$ being a semifinite faithful trace on $M$, and $\alpha : \mathbb{R}^{+*} \curvearrowleft M$ a continuous action. Assume that $\alpha$ scales the trace, meaning that for all $s \in \mathbb{R}^{+*}$ and $x \in M$, $\tau(\alpha_s(x)) = s^{-1}\tau(x)$. It trivially implies that $\alpha$ is outer, but the fact that $M' \cap (M \times_\alpha \mathbb{R}^{+*}) = \mathbb{C}$ is a deep result of modular theory, the *Connes-Takesaki relative commutant theorem* [8, Theorem 5.1] (see also [14, XII.1.7 and XII.1.1]).

In [12], Marrakchi and Vaes were able to completely solve Problem 1 for full factors (in the $\Pi_1$ setting, factors that do not have property $\Gamma$).

**Theorem 1.1 ([12, Theorem A]).** Let $\alpha : G \curvearrowleft M$ be a continuous action of a locally compact group on a full factor with separable predual. Assume that $\alpha$ is outer. Then $M' \cap (M \times_\alpha G) = \mathbb{C}$.

Recall that a factor $M$ with separable predual is full if and only if the subgroup $\text{Inn}(M)$ of inner automorphisms is closed in the Polish group $\text{Aut}(M)$ [5]. Therefore, any outer automorphism $\theta$ of a full factor has formally stronger properties:

• $\theta$ is not in the closure $\overline{\text{Inn}}(M)$ of $\text{Inn}(M)$ in the topological group $\text{Aut}(M)$ (it is not approximately inner).

• The bimodule $L^2(\theta)$ does not weakly contain the standard bimodule $L^2(\text{id})$ by [6, Theorem 3.1] and [11, Proposition 7.2]. If $M$ is a $\Pi_1$ factor, this is equivalent to the following property: there exists $\varepsilon > 0$ and $a_1, ..., a_n \in M$ such that for all $b \in M$,

$$ \sum_i \|\theta(a_i)b - ba_i\|_2^2 \geq \varepsilon \|b\|_2^2. $$

In the non-full setting, the properties above define distinct refinements of outerness. We therefore need to introduce corresponding terminologies: for $\theta \in \text{Aut}(M)$ we say that

<table><thead><tr><td>Terminology</td><td>Definition</td></tr></thead><tbody><tr><td>$\theta$ is inner</td><td>$\theta \in \text{Inn}(M)$</td></tr><tr><td>$\theta$ is outer</td><td>$\theta \notin \text{Inn}(M)$</td></tr><tr><td>$\theta$ is topologically (or approximately) inner</td><td>$\theta \in \overline{\text{Inn}}(M)$</td></tr><tr><td>$\theta$ is topologically outer</td><td>$\theta \notin \overline{\text{Inn}}(M)$</td></tr><tr><td>$\theta$ is spectrally (or weakly) inner</td><td>$L^2(\text{id}) \prec L^2(\theta)$</td></tr><tr><td>$\theta$ is spectrally outer (or $\theta$ acts with spectral gap)</td><td>$L^2(\text{id}) \not\prec L^2(\theta)$</td></tr></tbody></table>

We recall the known implications between the various notions of outerness introduced above. The missing implications are known not to hold.

<table><tr><td>$M$ factor</td><td>$\theta$ spectrally outer $\Rightarrow \theta$ topologically outer $\Rightarrow \theta$ outer</td></tr><tr><td>$M \Pi_1$ factor</td><td>$\theta$ spectrally outer $\overset{[6]}{\iff} \theta$ topologically outer $\Rightarrow \theta$ outer</td></tr><tr><td>$M$ full factor</td><td>$\theta$ spectrally outer $\overset{[6, 11]}{\iff} \theta$ topologically outer $\iff \theta$ outer</td></tr></table>

When $\alpha : G \curvearrowleft M$ is a continuous action on a factor, we say that $\alpha$ is inner (resp. outer, topologically inner, topologically outer, spectrally inner, spectrally outer) if for all $g \in G \setminus \{1\}$, $\alpha_g$ is inner (resp. outer, topologically inner, topologically outer, spectrally inner, spectrally outer).

As pointed out before, Theorem 1.1 does not hold for non-full factors. Nevertheless, Marrakchi and Vaes ask in [12] for the following natural generalization:

**Question 1 ([12, Question 9]).** Let $\alpha : G \curvearrowleft M$ be a continuous action of a locally compact group on a factor. Assume that for all $g \in G \setminus \{1\}$, $\alpha_g \notin \overline{\text{Inn}}(M)$ (i.e. $\alpha$ is topologically outer). Do we have $M' \cap (M \times_\alpha G) = \mathbb{C}$?

In this paper, we solve this question at least when $M$ is a semifinite factor. This is a consequence of the following main theorem.
**Theorem A.** Let $\alpha : G \curvearrowleft M$ be an action of a locally compact group on a $\sigma$-finite factor. Let $H = \{g \in G | L^2(\text{id}) \prec L^2(\alpha_g)\} < G$ be the subgroup of elements acting without spectral gap. Then

$$M' \cap (M \times_{\alpha} G) \subset M \times_{\alpha} H.$$

This theorem does not answer Question 1 when $M$ is a type III factor because not every topologically outer action on a type III factor is spectrally outer. However, this holds true when $M$ is of type $\Pi_1$, which implies the following corollary:

**Corollary B.** Let $\alpha : G \curvearrowleft M$ be a continuous action of a locally compact group on a $\sigma$-finite type II factor. If $\alpha$ is topologically outer (i.e. for all $g \in G \setminus \{1\}$, $\alpha_g \notin \overline{\text{Inn}}(M)$), then $M' \cap (M \times_{\alpha} G) = \mathbb{C}$.

If $\alpha$ is a trace scaling action on a $\Pi_\infty$ factor, it is topologically outer. Therefore, Corollary B generalizes Connes-Takesaki's relative commutant theorem. However, our proof of this corollary uses Connes-Takesaki's result, and therefore we do not get a new proof of this deep result.

To establish Theorem A the key step is the following uniform spectral gap for group actions result:

**Theorem C.** Let $\alpha : G \curvearrowleft M$ be a continuous action of a locally compact group on a $\sigma$-finite factor. Let $g \in G$ such that $L^2(\text{id}) \prec L^2(\alpha_g)$. Then there exists an open neighborhood $\mathcal{O}$ of $g$ in $G$ such that $L^2(\text{id}) \not\prec \bigoplus_{h \in \mathcal{O}} L^2(\alpha_h)$.

Specializing this result in the $\Pi_1$ factors case makes it clearer: saying that $g$ is such that $L^2(\text{id}) \prec L^2(\alpha_g)$ is exactly saying that there exists $\varepsilon > 0$ and $a_1, ..., a_n \in M$ such that

$$\forall \xi \in L^2 M, \sum_i \|\alpha_g(a_i)\xi - \xi a_i\|_2^2 \ge \varepsilon \|\xi\|_2^2.$$

By Connes' result, this is equivalent to $\alpha_g \notin \overline{\text{Inn}}(M)$. Therefore, by continuity of the action, there is an open neighborhood $\mathcal{O}$ of $g$ in $G$ such that for all $h \in \mathcal{O}$, $\alpha_h$ possesses a similar spectral gap property, but a priori with $\varepsilon$ and $a_1, ..., a_n$ depending on the group element. Theorem C exactly says that it is possible to choose a neighborhood $\mathcal{O}$ such that the spectral gap is uniform over $\mathcal{O}$: there exists $\varepsilon > 0$, $a_1, ..., a_n \in M$ such that for all $h$ in $\mathcal{O}$,

$$\forall \xi \in L^2 M, \sum_i \|\alpha_h(a_i)\xi - \xi a_i\|_2^2 \ge \varepsilon \|\xi\|_2^2.$$

Whereas in the full case, this result (see [12, lemma 1]) is a direct consequence of a uniform spectral gap result at the automorphism group level by Jones [10], it is not clear at all that such a property (or a satisfying weakening of it) remains true for non-full factors. We obtain Theorem C using different methods: we rely on the locally compact group structure of $G$ to apply smoothing procedures to well-chosen states.

# 2 Preliminaries

**Topologies on von Neumann algebras.** Let $M$ be a von Neumann algebra, $M_*$ its predual. By weak topology on $M$ we mean the weak-* topology relative to $M_*$. The strong topology (resp. the *-strong topology) is the topology induced by the seminorms $\|x\|_\varphi \doteq \varphi(x^*x)^{1/2}$ (resp. $\|x\|_\varphi^\sharp \doteq \varphi(x^*x)^{1/2} + \varphi(xx^*)^{1/2}$) for $\varphi \in M_*^+$.

**Automorphisms group.** The automorphism group Aut($M$) acts on $M_*$ by $\theta(\varphi) = \varphi \circ \theta$ for $\theta \in \text{Aut}(M)$ and $\varphi \in M_*$. Following [9], it is endowed with the u-topology which is the topology of pointwise norm convergence on $M_*$: a net $(\theta_i)_{i \in I}$ converges to $\theta$ if and only if for all $\varphi \in M_*$, $\|\theta_i(\varphi) - \theta(\varphi)\| \to 0$. With this topology Aut($M$) is a topological group that is Polish when $M_*$ is separable. The map

$$\text{Ad} : \mathcal{U}(M) \to \text{Aut}(M)$$
$$u \mapsto [x \mapsto u x u^*]$$

is continuous, and its image is the subgroup of inner automorphisms denoted by $\mathrm{Inn}(M)$. When $M$ is a factor, we say that it is full if the map $\mathrm{Ad}$ is open on its range [5]. When $M_*$ is separable, $M$ is full if and only if $\mathrm{Inn}(M) = \mathrm{Inn}(M)$, where $\mathrm{Inn}(M)$ is the closure of $\mathrm{Inn}(M)$ in the $u$-topology. The topological group of classes of outer automorphisms is defined by $\mathrm{Out}(M) = \mathrm{Aut}(M)/\mathrm{Inn}(M)$, and the quotient map is denoted by $p: \mathrm{Aut}(M) \to \mathrm{Out}(M)$. When $\mathrm{Inn}(M)$ is not closed, we will consider $\mathrm{Out}(M) = \mathrm{Aut}(M)/\mathrm{Inn}(M)$ and the corresponding quotient map $p: \mathrm{Aut}(M) \to \mathrm{Out}(M)$.

**Crossed products.** We refer to [14, Chapter X, §1] for proofs and details. Let $G$ be a locally compact group. We denote by $\mu$ one left Haar measure on it, by $\lambda: G \to L^2(G, \mu)$ (resp. $\rho: G \to L^2(G, \mu)$) its left (resp. right) regular representation and by $L(G)$ (resp. $R(G)$) the von Neumann algebra $\lambda(G)''$ (resp. $\rho(G)''$). A continuous action $\alpha: G \ltimes M$ of $G$ on a von Neumann algebra $M$ is a continuous group homomorphism $\alpha: G \to \mathrm{Aut}(M)$. The crossed product associated to the action $\alpha: G \ltimes M$ is the von Neumann algebra generated by a copy of $M$ and a family of unitaries $\{u_g \mid g \in G\}$ verifying $u_g x u_g^* = \alpha_g(x)$ for $g \in G$ and $x \in M$ that admits a representation $\alpha: M \ltimes_{\alpha} G \to M \bar{\otimes} B(L^2(G))$ such that $\alpha(u_g) = 1 \otimes \lambda_g$ and $\alpha(x) = [s \mapsto \alpha_{s^{-1}}(x)] \in M \bar{\otimes} L^\infty(G) \subset M \bar{\otimes} B(L^2(G))$.

**Smoothing procedure for von Neumann algebra valued functions.** Let $\alpha: G \ltimes M$ be a continuous action on a von Neumann algebra. Given $x \in M$, the map $g \mapsto \alpha_g(x)$ is weakly continuous, but generally not norm continuous. Consider $M_\alpha = \{x \in M \mid g \mapsto \alpha_g(x) \text{ is norm continuous}\}$: $M_\alpha$ is an $\alpha$ invariant $C^*$-subalgebra of $M$, and it turns out that $M_\alpha$ is $*$-strongly dense in $M$. This is a consequence of the following useful smoothing procedure.

First, recall that we can integrate some von Neumann algebra valued functions, even if they are not Bochner measurable. Let $f: (X, \mu) \to M$ be a map from a measure space to a von Neumann algebra with $x \mapsto \|f(x)\|$ integrable. It is weakly measurable if for all $\omega \in M_*$, $\omega \circ f: X \to \mathbb{C}$ is measurable. For such a function, there exists a unique element $\int_X f d\mu \in M$ such that for all $\omega \in M_*$, $\omega(\int_X f d\mu) = \int_X (\omega \circ f) d\mu$. For all $x \in M$, $h \mapsto \alpha_h(x)$ is weakly integrable and therefore, for any Borel $X \subset G$ of non-zero finite Haar measure, we can define the smoothing map $\alpha_X$ by

$$\alpha_X: M \to M \\ x \mapsto \frac{1}{\mu(X)} \int_X \alpha_h(x) d\mu(h).$$

For all $X \subset G$, $0 < \mu(X) < \infty$ and for all $x \in M$, $\alpha_X(x) \in M_\alpha$. Furthermore, if $\mathcal{B}$ is a neighborhood basis of the identity in $G$, $\lim_{U \in \mathcal{B}} \alpha_U(x) = x$ in weak topology which implies the $*$-strong density of $M_\alpha$ in $M$. Note that for any $x \in M_\alpha$, $h \mapsto \alpha_h(x)$ is norm-continuous. Therefore, it is Bochner measurable which implies that for all $\omega \in M^*$ and $x \in M_\alpha$, $\omega(\alpha_X(x)) = \frac{1}{\mu(X)} \int_X \omega(\alpha_h(x)) d\mu(h)$ (this does not hold for a generic element in $M$).

**Comparisons of bimodules.** We refer to [7] and [1] for proofs and details. Given $M$ and $N$ two von Neumann algebras, an $M-N$-bimodule $(\mathcal{H}, \pi_\mathcal{H})$ (sometimes abbreviated to $\pi_\mathcal{H}$ or to $\mathcal{H}$) is a binormal $*$-representation of the algebra $M \odot N^{\mathrm{op}}$. We denote by $\lambda_\mathcal{H}$ and $\rho_\mathcal{H}$ the restriction of $\pi_\mathcal{H}$ to $M$ and $N^{\mathrm{op}}$, and for $\xi \in \mathcal{H}$, $a \in M$ and $b \in N$ we use the abbreviations: $a \cdot \xi \doteq \lambda_\mathcal{H}(a)\xi$ and $\xi \cdot b \doteq \rho_\mathcal{H}(b)\xi$ when there is no risk of confusion. There are several ways of comparing two $M-N$-bimodules $(\mathcal{H}, \pi_\mathcal{H})$ and $(\mathcal{K}, \pi_\mathcal{K})$:

* $(\mathcal{H}, \pi_\mathcal{H})$ is contained in $(\mathcal{K}, \pi_\mathcal{K})$ if there exists an isometry $V: \mathcal{H} \to \mathcal{K}$ intertwining the representations $\pi_\mathcal{H}$ and $\pi_\mathcal{K}$. If we can choose $V$ as a unitary, we say that the two bimodules are unitarily equivalent, and we write $(\mathcal{H}, \pi_\mathcal{H}) \simeq (\mathcal{H}, \pi_\mathcal{K})$. This is in fact equivalent to $(\mathcal{H}, \pi_\mathcal{H}) \subset (\mathcal{K}, \pi_\mathcal{K})$ and $(\mathcal{K}, \pi_\mathcal{K}) \subset (\mathcal{H}, \pi_\mathcal{H})$. We denote by $\mathrm{Hom}((\mathcal{H}, \pi_\mathcal{H}), (\mathcal{K}, \pi_\mathcal{K}))$ the set of linear maps between $\mathcal{H}$ and $\mathcal{K}$ that intertwine the two representations, and we put $\mathrm{End}(\mathcal{H}, \pi_\mathcal{H}) \doteq \mathrm{Hom}((\mathcal{H}, \pi_\mathcal{H}), (\mathcal{H}, \pi_\mathcal{H}))$: it is exactly the von Neumann algebra $\pi_\mathcal{H}(M \odot N^{\mathrm{op}})' \subset B(\mathcal{H})$.

* $(\mathcal{H}, \pi_\mathcal{H})$ is quasi-contained in $(\mathcal{K}, \pi_\mathcal{K})$ if there exists a normal $*$-homomorphism $\Phi: \pi_\mathcal{K}(M \odot N^{\mathrm{op}})'' \to \pi_\mathcal{H}(M \odot N^{\mathrm{op}})''$ such that the following diagram commutes.
<figure><img src="image_3.png" /></figure>

Equivalently, $(\mathcal{H}, \pi_{\mathcal{H}})$ is quasi-contained in $(\mathcal{K}, \pi_{\mathcal{K}})$ if there exists a central projection $z \in \mathcal{Z}(\pi_{\mathcal{K}}(M \odot N^{\text{op}}))''$ such that $\pi_{\mathcal{H}}(M \odot N^{\text{op}})''$ is isomorphic to $z[\pi_{\mathcal{K}}(M \odot N^{\text{op}})]''$. If $(\mathcal{H}, \pi_{\mathcal{H}})$ is quasi-contained in $(\mathcal{K}, \pi_{\mathcal{K}})$ and $(\mathcal{K}, \pi_{\mathcal{K}})$ is quasi-contained in $(\mathcal{H}, \pi_{\mathcal{H}})$, we say that $(\mathcal{H}, \pi_{\mathcal{H}})$ and $(\mathcal{K}, \pi_{\mathcal{K}})$ are quasi-equivalent. When $p$ is a projection in $\pi_{\mathcal{H}}(M \odot N^{\text{op}})'$, $p\mathcal{H}$ is naturally endowed with an $M$-$N$-bimodule structure. If $p$ and $q$ are two such projections, $(p\mathcal{H}, \pi_{p\mathcal{H}})$ is quasi-contained in $(q\mathcal{H}, \pi_{q\mathcal{H}})$ if and only if $z(p) \le z(q)$, $z(p)$ being the central support of $p$ (check [3, Chapter 6] for details).

• $(\mathcal{H}, \pi_{\mathcal{H}})$ is weakly contained in $(\mathcal{K}, \pi_{\mathcal{K}})$, written $(\mathcal{H}, \pi_{\mathcal{H}}) \prec (\mathcal{K}, \pi_{\mathcal{K}})$ if there exists a $*$-homomorphism $\Phi : \overline{\pi_{\mathcal{K}}(M \odot N^{\text{op}})}^{\|\cdot\|} \to \overline{\pi_{\mathcal{H}}(M \odot N^{\text{op}})}^{\|\cdot\|}$ such that the following diagram commutes.

<figure><img src="image_4.png" /></figure>

Equivalently, $(\mathcal{H}, \pi_{\mathcal{H}}) \prec (\mathcal{K}, \pi_{\mathcal{K}})$ if for all $x \in M \odot N^{\text{op}}$ (equivalently, for all $x \in M \otimes_{\max} N^{\text{op}}$), $\|\pi_{\mathcal{H}}(x)\| \le \|\pi_{\mathcal{K}}(x)\|$. If $(\mathcal{H}, \pi_{\mathcal{H}}) \prec (\mathcal{K}, \pi_{\mathcal{K}})$ and $(\mathcal{K}, \pi_{\mathcal{K}}) \prec (\mathcal{H}, \pi_{\mathcal{H}})$, $(\mathcal{H}, \pi_{\mathcal{H}})$ and $(\mathcal{K}, \pi_{\mathcal{K}})$ are weakly equivalent, and we write $(\mathcal{H}, \pi_{\mathcal{H}}) \sim (\mathcal{K}, \pi_{\mathcal{K}})$.

As indicated by the terminology, containment of bimodules implies quasi-containment which in turn implies weak containment. We will need the following formulation of weak containment in terms of states.

**Proposition 2.1.** Let $(\mathcal{H}, \pi_{\mathcal{H}})$ and $(\mathcal{K}, \pi_{\mathcal{K}})$ be $M$-$N$-bimodules. Suppose that $\mathcal{H}$ admits a cyclic vector for $\pi_{\mathcal{H}}$. The following properties are equivalent:

1. $(\mathcal{H}, \pi_{\mathcal{H}}) \prec (\mathcal{K}, \pi_{\mathcal{K}})$

2. For all unit vector $\xi \in \mathcal{H}$ there exists a state $\omega \in B(\mathcal{K})^*$ such that $\omega(\pi_{\mathcal{K}}(a \otimes b^{\text{op}})) = \langle \pi_{\mathcal{H}}(a \otimes b^{\text{op}})\xi, \xi \rangle$ for all $a \in M, b \in N$.

3. There exists a $\pi_{\mathcal{H}}$-cyclic unit vector $\xi_0 \in \mathcal{H}$ and a state $\omega \in B(\mathcal{K})^*$ such that $\omega(\pi_{\mathcal{K}}(a \otimes b^{\text{op}})) = \langle \pi_{\mathcal{H}}(a \otimes b^{\text{op}})\xi_0, \xi_0 \rangle$ for all $a \in M, b \in N$.

*Proof.*

1) 2) Let $\Phi : \overline{\pi_{\mathcal{K}}(M \odot N^{\text{op}})}^{\|\cdot\|} \to \overline{\pi_{\mathcal{H}}(M \odot N^{\text{op}})}^{\|\cdot\|}$ be the $*$-homomorphism given by the weak containment, and $\xi \in \mathcal{H}$ a unit vector. Extending $x \mapsto \langle \Phi(x)\xi, \xi \rangle$ to $B(\mathcal{K})$ by Hahn-Banach produces a state with the required properties.

3) 1) Let $x \in M \otimes_{\max} N^{\text{op}}$.

$$
\begin{align*}
\|\pi_{\mathcal{H}}(x)\| &= \|\pi_{\mathcal{H}}(|x|)\| = \sup_{\|\xi\|=1} \langle \pi_{\mathcal{H}}(|x|)\xi, \xi \rangle = \sup_{\substack{y \in M \otimes_{\max} N^{\text{op}} \\ \|\pi_{\mathcal{H}}(y)\xi_0\| \le 1}} \langle \pi_{\mathcal{H}}(|x|)\pi_{\mathcal{H}}(y)\xi_0, \pi_{\mathcal{H}}(y)\xi_0 \rangle \\
&= \sup_{\substack{y \in M \otimes_{\max} N^{\text{op}} \\ \|\pi_{\mathcal{H}}(y)\xi_0\| \le 1}} \langle \pi_{\mathcal{H}}(y^*|x|y)\xi_0, \xi_0 \rangle = \sup_{\substack{y \in M \otimes_{\max} N^{\text{op}} \\ \|\pi_{\mathcal{H}}(y)\xi_0\| \le 1}} \omega \circ \pi_{\mathcal{K}}(y^*|x|y) \\
&\le \|\pi_{\mathcal{K}}(|x|)\| \sup_{\substack{y \in M \otimes_{\max} N^{\text{op}} \\ \|\pi_{\mathcal{H}}(y)\xi_0\| \le 1}} \omega \circ \pi_{\mathcal{K}}(y^*y) = \|\pi_{\mathcal{K}}(|x|)\| \sup_{\substack{y \in M \otimes_{\max} N^{\text{op}} \\ \|\pi_{\mathcal{H}}(y)\xi_0\| \le 1}} \langle \pi_{\mathcal{H}}(y^*y)\xi_0, \xi_0 \rangle \\
&\le \|\pi_{\mathcal{K}}(|x|)\| = \|\pi_{\mathcal{K}}(x)\|
\end{align*}
$$
□

**Operations on bimodules.** Given $(\mathcal{H}, \pi_H)$ and $M-N$-bimodule and $(\mathcal{K}, \pi_\mathcal{K})$ an $N-P$-bimodule, we denote by $(\mathcal{H} \otimes_N \mathcal{K}, \pi_\mathcal{H} \otimes_N \pi_\mathcal{K})$ the *Connes tensor product* of these two bimodules, which is an $M-P$-bimodule [7, Appendix B.δ].

Given $(\mathcal{H}, \pi_\mathcal{H})$ an $M-N$-bimodule, we denote by $(\overline{\mathcal{H}}, \overline{\pi_\mathcal{H}})$ the *contragredient* bimodule associated to $(\mathcal{H}, \pi_\mathcal{H})$, which is an $N-M$-bimodule. It is defined in the following way: if $\overline{\mathcal{H}}$ is the conjugate Hilbert space of $\mathcal{H}$, we denote by $\xi \mapsto \bar{\xi}$ the canonical antilinear isometry from $\mathcal{H}$ to $\overline{\mathcal{H}}$. Then the $N-M$ bimodule structure on $\overline{\mathcal{H}}$ is given by $b \cdot \bar{\xi} \cdot a \doteq \overline{a^* \cdot \xi \cdot b^*}$ [7, Appendix B.δ].

These two operations behaves well with respect to weak containment:

**Proposition 2.2** ([13, Proposition 2.2.1]). Let $\mathcal{H}$ and $\mathcal{K}$ be two $M-N$-bimodules such that $\mathcal{H} \prec \mathcal{K}$.
Then:

1. $\overline{\mathcal{H}} \prec \overline{\mathcal{K}}$

2. For $\mathcal{L}$ any $P-M$-bimodule, $\mathcal{L} \otimes_M \mathcal{H} \prec \mathcal{L} \otimes_M \mathcal{K}$

3. For $\mathcal{L}$ any $N-P$-bimodule, $\mathcal{H} \otimes_N \mathcal{L} \prec \mathcal{K} \otimes_N \mathcal{L}$

**Classical bimodules.** In this paragraph we refer to [9] and [7, Chapter 5 Appendix B] for details. Let $M$ be a von Neumann algebra. A fundamental example of $M-M$-bimodule is given by the standard form $(M, L^2M, J, L^2M^+)$ of $M$, where $L^2M$ is a Hilbert space, $J$ the canonical antilinear involution, and $L^2M^+$ the positive cone of $L^2M$. The Hilbert space $L^2M$ is canonically endowed with a bimodule structure: for $\xi \in L^2M$ and $a \in M$, we denote by $a\xi$ (resp. $\xi a$) the corresponding left (resp. right) multiplication. This defines the *standard bimodule*. The standard $M-M$-bimodule is a neutral element for the Connes tensor product: $L^2(M) \otimes_M \mathcal{H}$, $\mathcal{H} \otimes_M L^2(M)$ and $\mathcal{H}$ are unitarily equivalent.

Each automorphism $\theta \in \mathrm{Aut}(M)$ defines an $M-M$-bimodule $L^2(\theta) \doteq (L^2M, \pi_\theta)$ through $\pi_\theta(a \otimes b^{\mathrm{op}})\xi \doteq \theta^{-1}(a)\xi b$ for all $a, b \in M, \xi \in L^2M$. In particular, $L^2(\mathrm{id})$ is the standard bimodule.

Given $(X, \mu)$ a measure space, a Borel map $\chi : (X, \mu) \to \mathrm{Aut}(M)$ defines an $M-M$-bimodule $L^2(\chi) \doteq (L^2(X, \mu) \otimes L^2M, \pi_\chi)$ through $\pi_\chi(a \otimes b^{\mathrm{op}})\xi(x) \doteq \chi(x)^{-1}(a)\xi(x)b$ for all $a, b \in M, x \in X$, $\xi \in L^2(X, \mu) \otimes L^2M$. In particular, if $\alpha : G \curvearrowleft M$ is a continuous action of a locally compact group $G$ on $M$, each Borel subset $A \subset G$ defines a bimodule $L^2(\alpha|_A)$.

Here are some standard properties of these bimodules:

**Proposition 2.3** ([7, Appendix B.δ]).

1. Let $\theta \in \mathrm{Aut}(M)$. Then $\overline{L^2(\theta)} \simeq L^2(\theta^{-1})$.

2. Let $\theta_1, \theta_2 \in \mathrm{Aut}(M)$. Then $L^2(\theta_1) \otimes_M L^2(\theta_2) \simeq L^2(\theta_1 \circ \theta_2)$.

3. Let $\alpha : G \curvearrowleft M$ be a continuous action of $G$ on $M$, $A$ be a Borel subset of $G$ and $h \in G$. Then $L^2(\alpha_h) \otimes_M L^2(\alpha|_A) \simeq L^2(\alpha|_{h \cdot A})$

**Proposition 2.4.** Let $(X, \nu)$ be a $\sigma$-finite measure space and $\chi : (X, \nu) \to \mathrm{Aut}(M)$ a Borel map.
Then

$$L^2(\chi) \sim \bigoplus_{\theta \in \mathrm{essim} \, \chi} L^2(\theta),$$

where $\mathrm{essim}(\chi)$ is the essential range of $\chi$ (i.e. the support of the pushforward measure $\chi_*\nu$).

*Proof.* Let $a, b \in M$. Then

$$\|\pi_\chi(a \otimes b^{\mathrm{op}})\| = \underset{x \in (X, \nu)}{\mathrm{ess \, sup}} \|\pi_{\chi(x)}(a \otimes b^{\mathrm{op}})\| = \sup_{\theta \in \mathrm{essim} \, \chi} \|\pi_\theta(a \otimes b^{\mathrm{op}})\|.$$

The first equality is a general result about decomposable operators (see [3, Section 1.H]). As for the second, the inequality $\mathrm{ess \, sup}_{x \in (X, \nu)} \|\pi_{\chi(x)}(a \otimes b^{\mathrm{op}})\| \le \sup_{\theta \in \mathrm{essim} \, \chi} \|\pi_\theta(a \otimes b^{\mathrm{op}})\|$ is a simple
consequence of the definitions. For the other direction, let $\theta \in \text{essim}(\chi)$. Let $(\mathcal{U}_i)_{i \in I}$ be a basis of neighborhoods of $\theta$ in Aut($M$). For all $i \in I$, $\nu(\chi^{-1}(\mathcal{U}_i)) > 0$, and by $\sigma$-finiteness of $\nu$ we can take Borel sets $\mathcal{V}_i \subset \chi^{-1}(\mathcal{U}_i)$ with $0 < \nu(\mathcal{V}_i) < \infty$. Define $\eta_i = \frac{1}{\nu(\mathcal{V}_i)^{1/2}} \nu_i \in L^2(X, \nu)$. For any $\xi \in L^2 M$ and $a, b \in M$,

$$ \langle \pi_{\chi}(a \otimes b^{\text{op}}) \xi \otimes \eta_i, \xi \otimes \eta_i \rangle \xrightarrow{i \to \infty} \langle \pi_{\theta}(a \otimes b^{\text{op}}) \xi, \xi \rangle. $$

But for all $i \in I$, $\|\pi_{\chi}(a \otimes b^{\text{op}})\| \|\xi\|^2 = \|\pi_{\chi}(a \otimes b^{\text{op}})\| \|\xi \otimes \eta_i\|^2 \ge \langle \pi_{\chi}(a \otimes b^{\text{op}}) \xi \otimes \eta_i, \xi \otimes \eta_i \rangle$. So $\|\pi_{\theta}(a \otimes b^{\text{op}})\| \le \|\pi_{\chi}(a \otimes b^{\text{op}})\|$, and more generally $\sup_{\theta \in \text{essim}(\chi)} \|\pi_{\theta}(a \otimes b^{\text{op}})\| \le \|\pi_{\chi}(a \otimes b^{\text{op}})\|$. $\square$

As a consequence, if $\alpha : G \curvearrowleft M$ is a continuous action, $A \subset G$ is a Borel subset and $g$ is in the support of $\mu_{|A}$, then $L^2(\alpha_g) \prec L^2(\alpha_{|A})$. Note that if $\theta \in \text{Aut}(M)$ is such that $L^2(\text{id}) \prec L^2(\theta)$, then by propositions 2.2 and 2.3, $L^2(\text{id}) \simeq \overline{L^2(\text{id})} \prec \overline{L^2(\theta)} \simeq L^2(\theta^{-1})$, and therefore $L^2(\text{id}) \sim L^2(\theta)$. In particular, if $\alpha : G \curvearrowleft M$ is a continuous action, the set $H$ of elements $h \in G$ such that $L^2(\text{id}) \prec L^2(\alpha_h)$ is a subgroup of $G$.

**Lemma 2.5.** Let $\alpha : G \curvearrowleft M$ be a continuous action of a locally compact group on a von Neumann algebra. Let $H = \{g \in G \mid L^2(\text{id}) \prec L^2(\alpha_g)\} = \{g \in G \mid L^2(\text{id}) \sim L^2(\alpha_g)\}$ and $U$ be an open subset of $G$. Then $L^2(\alpha_{|U})$ and $L^2(\alpha_{|U})$ are weakly equivalent.

*Proof.* Using proposition 2.4 and the fact that $\text{essim}(\alpha_{|UH}) = \alpha(UH)$ because $U$ is open, we obtain:

$$ \begin{aligned} L^2(\alpha_{|UH}) &\sim \bigoplus_{\theta \in UH} L^2(\theta) \sim \bigoplus_{g \in U, h \in H} L^2(\alpha_{gh}) \sim \bigoplus_{g \in U, h \in H} L^2(\alpha_g) \otimes_M L^2(\alpha_h) \\ &\sim \bigoplus_{g \in U, h \in H} L^2(\alpha_g) \otimes_M L^2(\text{id}) \sim \bigoplus_{g \in U} L^2(\alpha_g) \\ &\sim L^2(\alpha_{|U}). \end{aligned} $$

$\square$

**Unitary implementation of automorphisms.** Given $\varphi \in M_+^*$, we denote by $\varphi^{1/2}$ the unique vector in $L^2 M^+$ such that for all $x \in M$, $\varphi(x) = \langle x\varphi^{1/2}, \varphi^{1/2} \rangle$. When $\theta$ is an automorphism of $M$, there exists a unique unitary $U_\theta \in B(L^2 M)$ such that for all $x \in M$, $\theta(x) = U_\theta x U_\theta^*$: we denote by $\kappa : \text{Aut}(M) \to \mathcal{U}(B(L^2 M))$, $\theta \mapsto U_\theta$ this unitary implementation. When $\alpha : G \curvearrowleft M$ is a continuous action on $M$, the unitary implementation extends $\alpha$ in a continuous action $\tilde{\alpha} : G \curvearrowleft B(L^2 M)$ through $\tilde{\alpha}_g(x) := \kappa(\alpha_g)x\kappa(\alpha_g)^*$. For $a, b \in M$, $X \subset G$, $\tilde{\alpha}_X(\pi_{\text{id}}(a \otimes b^{\text{op}})) = \frac{1}{\mu(X)} \int_X \pi_{\text{id}}(\alpha_h(a) \otimes \alpha_h(b)^{\text{op}}) d\mu(h)$ (cf [9, Thm 3.2]). In particular, $\tilde{\alpha}_X(\pi_{\text{id}}(a \otimes 1)) = \pi_{\text{id}}(\alpha_X(a) \otimes 1)$ and $\tilde{\alpha}_X(\pi_{\text{id}}(1 \otimes b^{\text{op}})) = \pi_{\text{id}}(1 \otimes \alpha_X(b)^{\text{op}})$. Note also that for all $a, b \in M_\alpha$, $\pi_{\text{id}}(a \otimes b^{\text{op}}) \in B(L^2 M)_{\tilde{\alpha}}$.

**Spectral gaps.** Let $M$ be a $\Pi_1$ factor. Given an $M$-$M$-bimodule $\mathcal{K}$, it is particularly interesting to check if $\mathcal{K}$ weakly contains $L^2(\text{id})$. For any $\varepsilon > 0$ and any finite subset $F$ of $M$, consider the subset of Bimod($M, M$) defined by

$$ W(\varepsilon, F) \doteq \left\{ M-M\text{-bimodules } (\mathcal{H}, \pi_{\mathcal{H}}) \mid \exists \xi \in \mathcal{H}, \sum_{x \in F} \|x \cdot \xi - \xi \cdot x\|_2^2 < \varepsilon \|\xi\|_2^2 \right\}. $$

Then (see [2, Proposition 13.3.7]) $L^2(\text{id}) \not\prec (\mathcal{K}, \pi_\mathcal{K})$ if and only if there exists $\varepsilon$ and $F$ such that $(\mathcal{K}, \pi_\mathcal{K}) \notin W(\varepsilon, F)$. This proposition allows us to interpret $L^2(\text{id}) \not\prec (\mathcal{K}, \pi_\mathcal{K})$ as a spectral gap property. Applied to bimodules associated to automorphisms, it gives:

**Proposition 2.6.** Let $M$ be a $\Pi_1$ factor, and $A$ be a subset of Aut($M$). Then the following statements are equivalent:
1. There exists $\varepsilon > 0$ and $a_1, ..., a_n$ in $M$ such that for all $\theta$ in $A$ and for all $\xi$ in $L^2 M$,

$$ \sum_i \|\theta(a_i)\xi - \xi a_i\|_2^2 \geq \varepsilon \|\xi\|_2^2. $$

2. $L^2(\text{id}) \not\subset \bigoplus_{\theta \in A} L^2(\theta)$.

In particular:

* Applied to $A = \{\theta_0\}$ a single automorphism, item 1 is a spectral gap property because it says that 0 is not in the closure of the spectrum of the operator $\sum_{i=1}^n |\lambda(\theta_0(a_i)) - \rho(a_i)|^2$. Then proposition 2.6 is the analog of the classical fact that a group representation has spectral gap if and only if it does not weakly contain the trivial representation.

* Using Proposition 2.4, if $(X, \nu)$ is a $\sigma$-finite measure space and $\chi : (X, \nu) \to \text{Aut}(M)$ is a Borel map, then $L^2(\text{id}) \not\subset L^2(\chi)$ if and only if there exists $\varepsilon > 0$ and $a_1, ..., a_n$ in $M$ such that for all $\theta$ in $\text{essim}(\chi)$ and for all $\xi$ in $L^2 M$,

$$ \sum_i \|\theta(a_i)\xi - \xi a_i\|_2^2 \geq \varepsilon \|\xi\|_2^2. $$

Recall that, when $M$ is a $\Pi_1$ factor, $\theta \notin \overline{\text{Inn}}(M)$ (i.e. $\theta$ is topologically outer) if and only if $\theta$ exhibits the spectral gap properties of Proposition 2.6:

**Theorem 2.7** ([6, Theorem 3.1]). Let $M$ be a factor, and let $\theta \in \text{Aut}(M)$. Consider the two following statements:

1. $L^2(\text{id}) \not\subset L^2(\theta)$ ($\theta$ is spectrally outer).

2. $\theta \notin \overline{\text{Inn}}(M)$ ($\theta$ is topologically outer).

Then 1 $\Rightarrow$ 2, and if $\theta$ is a trace-preserving automorphism of a type $\Pi$ factor, then 2 $\Rightarrow$ 1.

However there exist topologically outer automorphisms that are not spectrally outer, for instance trace scaling automorphisms of the injective $\Pi_\infty$ factor (see [11, Introduction] for examples of such automorphisms on type III factors).

*Proof.* 1 $\Rightarrow$ 2. Let $\theta \in \overline{\text{Inn}}(M)$ and choose a net $(u_i)_{i \in I}$ of unitaries such that $\text{Ad}(u_i) \to \theta$. Denote by $\hat{1}$ the standard cyclic unit vector in $L^2(M)$. Consider the net of states $(\omega_i)_{i \in I}$ defined by $\omega_i(x) = \langle xu_i^* \hat{1}, u_i^* \hat{1} \rangle$ for $x \in \mathcal{B}(L^2(M))$. Then for $a, b \in M$,

$$ \omega_i(\pi_\theta(a \otimes b^{\text{op}))) = \langle \theta^{-1}(a)u_i^* \hat{1}b, u_i^* \hat{1} \rangle = \langle u_i\theta^{-1}(a)u_i^* \hat{1}b, \hat{1} \rangle \xrightarrow{i \to \infty} \langle a\hat{1}b, \hat{1} \rangle. $$

Therefore, any weak-* accumulation point $\omega$ of $(\omega_i)_{i \in I}$ will verify $\omega(\pi_\theta(a \otimes b^{\text{op}))) = \langle a\hat{1}b, \hat{1} \rangle$ for all $a, b \in M$ and therefore, by Proposition 2.1, $L^2(\text{id}) \not\subset L^2(\theta)$.

2 $\Rightarrow$ 1(for $\Pi_1$ factors). This is [6, Theorem 3.1].

2 $\Rightarrow$ 1(for trace preserving actions on semifinite factors). Let $\theta \in \text{Aut}(M)$ a trace-preserving automorphism of the $\Pi_\infty$ factor $(M, \tau)$ such that $L^2(\text{id}) \not\subset L^2(\theta)$. Take $p_1 \in M$ a projection with $\tau(p_1) = 1$: $\tau(\theta(p_1)) = \tau(p_1)$ so there exists a unitary $u_1 \in M$ such that $\text{Ad}(u_1) \circ \theta(p_1) = p_1$. Then $\text{Ad}(u_1) \circ \theta$ stabilizes $(1-p_1)M(1-p_1)$, and therefore we can find $p_2 < 1-p_1$ with $\tau(p_2) = 1$ and $u_2$ a unitary in $M$ such that $\text{Ad}(u_2u_1) \circ \theta(p_1) = p_1$ and $\text{Ad}(u_2u_1) \circ \theta(p_2) = p_2$. Inductively, we may consider a partition of the unit $(p_n)_{n \in \mathbb{N}}$ and a unitary $u \in \mathcal{U}(M)$ such that $\text{Ad}(u) \circ \theta(p_i) = p_i$ for all $i \in \mathbb{N}$. Put $\beta = \text{Ad}(u) \circ \theta \in \text{Aut}(M)$. Then $L^2(\text{id}) \not\subset \mathcal{H}(\beta_{p_i M p_i})$ for $i \in \mathbb{N}$, and thus, applying 2$\Rightarrow$1 in the $\Pi_1$ setting, $\beta_{p_i M p_i} \in \overline{\text{Inn}}(p_i M p_i)$. Therefore, there exists a sequence of unitaries $(v_n^i)_{n \in \mathbb{N}}$ in $p_i M p_i$ such that $v_n^i \xrightarrow{n \to \infty} \beta_{p_i M p_i}$ for all $i \in \mathbb{N}$. Taking $v_n = \sum_{i \in \mathbb{N}} v_n^i$, we get $\beta \in \overline{\text{Inn}}(M)$. But then $\theta = \text{Ad}(u^*) \circ \beta$ is in $\overline{\text{Inn}}(M)$ too. $\square$
Take $M$ a factor: given an element $\theta \in \mathrm{Aut}(M) \setminus \overline{\mathrm{Inn}}(M)$, $\theta$ admits a neighborhood $\mathcal{O}$ consisting of topologically outer automorphisms. If $M$ is a $\Pi_1$ factor, we can apply Theorem 2.7 and Proposition 2.6 to each $\beta \in \mathcal{O}$ independently, which gives us $\varepsilon_\beta > 0$ and $\{a_1^\beta, \dots, a_{n_\beta}^\beta\}$ characterizing the spectral gap of $\beta$. It is a natural question to ask if we can obtain a uniform spectral gap, i.e. obtaining spectral gap with the same $\varepsilon > 0$ and the same finite set $\{a_1, \dots, a_n\}$ for all automorphisms in a neighborhood of $\theta$. More specifically, is it possible to establish, for some families of factors (not necessarily of type $\Pi_1$), whether one of the following uniform spectral gap properties is verified?

**P1** For all $\theta \in \mathrm{Aut}(M) \setminus \overline{\mathrm{Inn}}(M)$, there is a neighborhood $\mathcal{V}$ of $\underline{p}(\theta)$ in $\underline{\mathrm{Out}}(M)$ such that

$$L^2(\mathrm{id}) \not\subset \bigoplus_{\beta \in \underline{p}^{-1}\mathcal{V}} L^2(\beta).$$

Equivalently in the $\Pi_1$ factor case, there exist $a_1, \dots, a_n \in M$, $\varepsilon > 0$ such that for all $\beta \in \underline{p}^{-1}(\mathcal{V})$ and $\xi \in L^2 M$,

$$\sum_{i=1}^{n} \|\beta(a_i)\xi - \xi a_i\|_2^2 \geq \varepsilon \|\xi\|_2^2.$$

**P2** For all neighborhoods $\mathcal{V}$ of 1 in $\underline{\mathrm{Out}}(M)$,

$$L^2(\mathrm{id}) \not\subset \bigoplus_{\beta \in \mathrm{Aut}(M) \setminus \underline{p}^{-1}(\mathcal{V})} L^2(\beta).$$

Equivalently in the $\Pi_1$ factor case, there exist $a_1, \dots, a_n \in M$, $\varepsilon > 0$ such that for all $\beta \notin \underline{p}^{-1}(\mathcal{V})$ and $\xi \in L^2 M$,

$$\sum_{i=1}^{n} \|\beta(a_i)\xi - \xi a_i\|_2^2 \geq \varepsilon \|\xi\|_2^2.$$

Note that P2 implies P1.

The relevance of these properties is supported by the following results:

**Theorem 2.8 ([10, Lemma 4]).** Let $M$ be a full $\Pi_1$ factor. Then $M$ has P2.

Using Proposition 2.4 and noticing that separability assumptions can be avoided in the proof, we can rephrase [12, Lemma 1] as the following generalization of Jones’ result:

**Theorem 2.9 ([12, Lemma 1]).** Let $M$ be a full $\sigma$-finite factor. Then $M$ has P2.

The proposition below can be easily adapted from [12, Proof of Theorem A]:

**Proposition 2.10.** Let $\alpha: G \curvearrowleft M$ be a continuous topologically outer (i.e. $\alpha^{-1}(\overline{\mathrm{Inn}}(M)) = 1$) action of a locally compact group on a factor $M$ that has P1. Then $\alpha$ is strictly outer.

It is the combination of Theorem 2.9 and Proposition 2.10 that allows Marrakchi and Vaes to show that outer actions on full factors are strictly outer. Jones’s result 2.8 strongly relies on the theorem by Connes that gives a spectral gap characterization of fullness [6, Theorem 3.1], and thus its proof can’t be generalized to arbitrary factors. We don’t even know if the weaker property P1 holds for non-full factors. However, Theorem C asserts that such a uniform spectral gap can be obtained at least for locally compact group actions.

# 3 Proof of the theorems

First we establish a prolongation result for states on bimodules.
**Lemma 3.1.** Let $M$ and $N$ be two von Neumann algebras. Let $(\mathcal{H}, \pi_{\mathcal{H}})$ be an $M$-$N$-bimodule endowed with a binormal state $\varphi \in B(\mathcal{H})^*$. Let $(\mathcal{K}, \pi_{\mathcal{K}})$ be another $M$-$N$-bimodule endowed with a state $\omega \in B(\mathcal{K})^*$ such that:

* $\omega \circ \lambda_{\mathcal{K}} = \varphi \circ \lambda_{\mathcal{H}}$.

* $\omega \circ \rho_{\mathcal{K}} = \varphi \circ \rho_{\mathcal{H}}$.

* there exist $*$-strongly dense subsets $M_0 \subset M$ and $N_0 \subset N$ such that $\omega \circ \pi_{\mathcal{K}} = \varphi \circ \pi_{\mathcal{H}}$ on $M_0 \odot N_0^{\text{op}}$.

Then $\omega \circ \pi_{\mathcal{K}} = \varphi \circ \pi_{\mathcal{H}}$ on $M \odot N^{\text{op}}$.

*Proof.* Let $a \in M$, $b \in N$. Consider two nets $(a_i)_{i \in I} \in (M_0)^I$ and $(b_i)_{i \in I} \in (N_0)^I$ such that $a_i \xrightarrow{i \to \infty} a$ and $b_i \xrightarrow{i \to \infty} b$. Then:

$$ \begin{aligned} & |\omega(\pi_{\mathcal{K}}(a_i \otimes b_i^{\text{op}})) - \omega(\pi_{\mathcal{K}}(a \otimes b^{\text{op}}))| \\ & = |\omega(\pi_{\mathcal{K}}((a_i - a) \otimes b_i^{\text{op}})) + \omega(\pi_{\mathcal{K}}(a \otimes (b_i - b)^{\text{op}}))| \\ & \le \|a_i - a\|_{\omega \circ \lambda_{\mathcal{K}}} \|(b_i^{\text{op}})^*\|_{\omega \circ \rho_{\mathcal{K}}} + \|a\|_{\omega \circ \lambda_{\mathcal{K}}} \|(b_i^{\text{op}} - b^{\text{op}})^*\|_{\omega \circ \rho_{\mathcal{K}}} \\ & \le \|a_i - a\|_{\varphi \circ \lambda_{\mathcal{H}}}^{\sharp} \|b_i^{\text{op}}\|_{\varphi \circ \rho_{\mathcal{H}}}^{\sharp} + \|a\|_{\varphi \circ \lambda_{\mathcal{H}}}^{\sharp} \|(b_i - b)^{\text{op}}\|_{\varphi \circ \rho_{\mathcal{H}}}^{\sharp} \xrightarrow{i \to \infty} 0 \end{aligned} $$

$$ \begin{aligned} & |\omega(\pi_{\mathcal{K}}(a_i \otimes b_i^{\text{op}})) - \varphi(\pi_{\mathcal{H}}(a \otimes b^{\text{op}}))| \\ & = |\varphi(\pi_{\mathcal{H}}(a_i \otimes b_i^{\text{op}})) - \varphi(\pi_{\mathcal{H}}(a \otimes b^{\text{op}}))| \\ & \le |\varphi(\pi_{\mathcal{H}}((a_i - a) \otimes b_i^{\text{op}})) + \varphi(\pi_{\mathcal{H}}(a \otimes (b_i - b^{\text{op}})))| \\ & \le \|a_i - a\|_{\varphi \circ \lambda_{\mathcal{H}}}^{\sharp} \|b_i^{\text{op}}\|_{\varphi \circ \rho_{\mathcal{H}}}^{\sharp} + \|a\|_{\varphi \circ \lambda_{\mathcal{H}}}^{\sharp} \|(b_i - b)^{\text{op}}\|_{\varphi \circ \rho_{\mathcal{H}}}^{\sharp} \xrightarrow{i \to \infty} 0. \end{aligned} $$

□

*Proof of Theorem C.* Let $g \in G$, and denote by $\mathcal{B}$ a basis of open neighborhoods of $g$ in $G$. Assume that for all $\mathcal{O} \in \mathcal{B}$, $L^2(\text{id}) \prec \bigoplus_{\theta \in \mathcal{O}} L^2(\theta)$ or equivalently that $L^2(\text{id}) \prec L^2(\alpha_{|\mathcal{O}})$. Because $M$ is $\sigma$-finite, we can choose $\varphi \in M_*^+$ a faithful normal state on $M$. Using Proposition 2.1 we choose for each $\mathcal{O} \in \mathcal{B}$ a state $\omega_{\mathcal{O}} \in B(L^2(\alpha_{|\mathcal{O}}))^*$ such that $\omega_{\mathcal{O}}(\pi_{\alpha_{|\mathcal{O}}}(a \otimes b^{\text{op}})) = \langle a\varphi^{1/2}b, \varphi^{1/2} \rangle$ for all $a, b \in M$. It induces a net $(\tilde{\omega}_{\mathcal{O}})_{\mathcal{O} \in \mathcal{B}}$ of states in $B(L^2 M)^*$ defined by $\tilde{\omega}_{\mathcal{O}}(x) \doteq \omega_{\mathcal{O}}(1_{\mathcal{O}} \otimes x)$ for $x \in B(L^2 M)$. Consider $\tilde{\omega}$ a weak-$*$ accumulation point of $(\tilde{\omega}_{\mathcal{O}})_{\mathcal{O} \in \mathcal{B}}$. Let $a \in M_{\alpha}$, $b \in M$: for any $\varepsilon > 0$, we can find $\Omega \in \mathcal{B}$ small enough such that for all $h \in \Omega$, $\|\alpha_g^{-1}(a) - \alpha_h^{-1}(a)\| \le \varepsilon$. Then:

$$ \begin{aligned} & |\tilde{\omega}(\pi_{\alpha_g}(a \otimes b^{\text{op}})) - \langle a\varphi^{1/2}b, \varphi^{1/2} \rangle| \\ & \le \limsup_{\mathcal{O} \to \infty} |\omega_{\mathcal{O}}(1_{\mathcal{O}} \otimes \pi_{\alpha_g}(a \otimes b^{\text{op}})) - \omega_{\mathcal{O}}(\pi_{\alpha_{|\mathcal{O}}}(a \otimes b^{\text{op}}))| \\ & \le \limsup_{\mathcal{O} \to \infty} |\omega_{\mathcal{O}}([1_{\mathcal{O}} \otimes \pi_{\alpha_g}(a \otimes 1) - \pi_{\alpha_{|\mathcal{O}}}(a \otimes 1)](\pi_{\alpha_{|\mathcal{O}}}(1 \otimes b^{\text{op}})))| \\ & \le \limsup_{\mathcal{O} \to \infty} \|(1_{\mathcal{O}} \otimes \pi_{\alpha_g}(a \otimes 1) - \pi_{\alpha_{|\mathcal{O}}}(a \otimes 1))\| \|\pi_{\alpha_{|\mathcal{O}}}(1 \otimes b^{\text{op}})\| \\ & \le \|1_{\mathcal{O}} \otimes \pi_{\alpha_g}(a \otimes 1) - \pi_{\alpha_{|\mathcal{O}}}(a \otimes 1)\| \|\pi_{\alpha_{|\mathcal{O}}}(1 \otimes b^{\text{op}})\| \\ & \le \varepsilon \|1 \otimes b^{\text{op}}\|. \end{aligned} $$

This being true for all $\varepsilon > 0$, we get that for all $a \in M_{\alpha}$ and $b \in M$,

$$ \tilde{\omega}(\pi_{\alpha_g}(a \otimes b^{\text{op}})) = \langle a\varphi^{1/2}b, \varphi^{1/2} \rangle. \quad (3.1) $$

Now we smooth the state $\tilde{\omega}$. Let $\mathcal{C}$ be a base of open neighborhoods of 1 in $G$, and consider $\omega$ an accumulation point of $(\tilde{\omega} \circ \tilde{\alpha}_{\nu})_{\nu \in \mathcal{C}}$.

$$ \text{For } b \in M: \quad |\omega(\pi_{\alpha_g}(1 \otimes b^{\text{op}})) - \langle \varphi^{1/2}b, \varphi^{1/2} \rangle| $$
$$
\begin{align*}
&\leq \limsup_{\mathcal{V} \to \infty} \left| \tilde{\omega} \circ \tilde{\alpha}_{\mathcal{V}}(\pi_{\alpha_g}(1 \otimes b^{\text{op}))) - \langle \varphi^{1/2}b, \varphi^{1/2} \rangle \right| \\
&\leq \limsup_{\mathcal{V} \to \infty} \left| \langle \varphi^{1/2}(\alpha_{\mathcal{V}}(b) - b), \varphi^{1/2} \rangle \right| \\
&= 0.
\end{align*}
$$

$$
\begin{align*}
\text{For } a \in M: \quad & \left| \omega(\pi_{\alpha_g}(a \otimes 1)) - \langle a\varphi^{1/2}, \varphi^{1/2} \rangle \right| \\
& \le \limsup_{\mathcal{V} \to \infty} \left| \tilde{\omega} \circ \tilde{\alpha}_{\mathcal{V}}(\pi_{\alpha_g}(a \otimes 1)) - \langle a\varphi^{1/2}, \varphi^{1/2} \rangle \right| \\
& \le \limsup_{\mathcal{V} \to \infty} \left| \langle (\alpha_g \nu_{g^{-1}}(a) - a)\varphi^{1/2}, \varphi^{1/2} \rangle \right| \\
& = 0.
\end{align*}
$$

$$
\begin{align*}
\text{For } a, b \in M_{\alpha}: \quad & \left| \omega(\pi_{\alpha_g}(a \otimes b^{\text{op}))) - \langle a\varphi^{1/2}b, \varphi^{1/2} \rangle \right| = \left| \omega(\pi_{\text{id}}(\alpha_{g^{-1}}(a) \otimes b^{\text{op}))) - \langle a\varphi^{1/2}b, \varphi^{1/2} \rangle \right| \\
& \le \limsup_{\mathcal{V} \to \infty} \left| \tilde{\omega} \left( \frac{1}{\mu(\mathcal{V})} \int_{\mathcal{V}} \tilde{\alpha}_h[\pi_{\text{id}}(\alpha_{g^{-1}}(a) \otimes b^{\text{op}})] d\mu(h) \right) - \langle a\varphi^{1/2}b, \varphi^{1/2} \rangle \right| \\
& \le \limsup_{\mathcal{V} \to \infty} \left| \frac{1}{\mu(\mathcal{V})} \int_{\mathcal{V}} \tilde{\omega} \left( \tilde{\alpha}_h[\pi_{\text{id}}(\alpha_{g^{-1}}(a) \otimes b^{\text{op}})] \right) d\mu(h) - \langle a\varphi^{1/2}b, \varphi^{1/2} \rangle \right| \\
& \qquad \qquad \qquad \quad \text{(since } \pi_{\text{id}}(\alpha_{g^{-1}}(a) \otimes b^{\text{op}}) \in B(L^2 M)_{\tilde{\alpha}} \text{)} \\
& \le \limsup_{\mathcal{V} \to \infty} \left| \frac{1}{\mu(\mathcal{V})} \int_{\mathcal{V}} \tilde{\omega} \left( \pi_{\alpha_g} \left( \alpha_{ghg^{-1}}(a) \otimes \alpha_h(b)^{\text{op}} \right) \right) d\mu(h) - \langle a\varphi^{1/2}b, \varphi^{1/2} \rangle \right| \\
& \le \limsup_{\mathcal{V} \to \infty} \frac{1}{\mu(\mathcal{V})} \int_{\mathcal{V}} \left| \langle \alpha_{ghg^{-1}}(a)\varphi^{1/2}\alpha_h(b) - a\varphi^{1/2}b, \varphi^{1/2} \rangle \right| d\mu(h) \\
& \qquad \qquad \qquad \quad \text{(by (3.1), since } \alpha_{ghg^{-1}}(a) \in M_{\alpha} \text{)} \\
& \le \limsup_{\mathcal{V} \to \infty} \sup_{h \in \mathcal{V}} \left( \|\alpha_{ghg^{-1}}(a) - a\| \|\alpha_h(b)\| + \|a\| \|\alpha_h(b) - b\| \right) \\
& = 0.
\end{align*}
$$

Now we can apply lemma 3.1: $\omega \circ \pi_{\alpha_g} = \varphi \circ \pi_{\text{id}}$ on $M \odot M^{\text{op}}$ and thus $L^2(\alpha_g)$ weakly contains $L^2(\text{id})$ by proposition 2.1. $\square$

The following result explains how structure on the algebra of intertwiners of the action bi-module can be used to locate the relative commutant $M' \cap (M \rtimes_{\alpha} G)$ in the crossed product by a subgroup $M \rtimes_{\alpha} H$.

**Lemma 3.2.** Let $\alpha: G \curvearrowleft M$ be a continuous action of a locally compact group on a factor. Let $H < G$ be a closed subgroup such that $1 \overline{\otimes} L^\infty(H \setminus G) \subset \mathcal{Z}(\text{End}(L^2(\alpha)))$. Then

$M' \cap (M \rtimes_{\alpha} G) \subset M \rtimes_{\alpha} H.$

*Proof.* Recall that

$$
\mathrm{End}(L^2(\alpha)) = \alpha(M)' \cap (M^{\mathrm{op}} \otimes 1)' \cap (B(L^2 M) \bar{\otimes} B(L^2 G)) = \alpha(M)' \cap (M \bar{\otimes} B(L^2 G))
$$

and that the crossed product $M \rtimes_{\alpha} G$ can be realized as the algebra of fixed points in $M \bar{\otimes} B(L^2 G)$ under the action $\alpha \otimes \text{Ad}(\rho)$ (cf [14, p. X.1.22]). Using the $\alpha \otimes \text{Ad}(\rho)$ invariance of $\alpha(M)$, we obtain that $M' \cap (M \rtimes_{\alpha} G) = \text{End}(L^2(\alpha))^{\alpha \otimes \text{Ad}(\rho)}$. Then:

$$
\begin{align*}
M' \cap (M \rtimes_{\alpha} G) &= (\text{End}(L^2(\alpha)) \cap \mathcal{Z}(\text{End}(L^2(\alpha))))'^{\alpha \otimes \text{Ad}(\rho)} \\
&\subset \alpha(M)' \cap (1 \overline{\otimes} L^{\infty}(H \setminus G))' \cap (M \rtimes_{\alpha} G)
\end{align*}
$$
$$ \begin{aligned} & \subset (1 \otimes L^\infty(H \setminus G))' \cap M \rtimes_\alpha G \\ & = M \rtimes_\alpha H \end{aligned} \qquad (\text{by [4, Prop. 2.2].}) $$

<figure><img src="image_5.png" /></figure>

**Proof of Theorem A.** Using Theorem C we show that if $H = \{g \in G \mid L^2(\mathrm{id}) \prec L^2(\alpha_g)\}$, then

$$ \mathbb{C}\bar{\otimes}L^\infty(G/H) \subset \mathcal{Z}(\mathrm{End}(L^2(\alpha))). $$

Noticing that $H$ is a normal subgroup of $G$, lemma 3.2 concludes the proof of Theorem A.

Set $\mathcal{N} = \mathrm{End}(L^2(\alpha))$. Take $K$ a compact subset of $G$. We want to establish that $1 \otimes 1_{KH} \in \mathcal{Z}(\mathcal{N})$. Because $M$ is a factor, $\mathcal{N} \cap (1 \bar{\otimes} L^\infty G)' = (1 \bar{\otimes} L^\infty G)$ and consequently $\mathcal{Z}(\mathcal{N}) \subset 1 \otimes L^\infty(G)$. Consider $A \subset G$ such that $z(1 \otimes 1_{KH}) = 1 \otimes 1_A$, where $z(\cdot)$ is the central support in $\mathcal{N}$. Let $L$ be the support of $\mu_{|A}$, and take $g_0 \in L$: then $L^2(\alpha_{g_0}) \prec L^2(\alpha_L) \sim L^2(\alpha_{|A}) \sim L^2(\alpha_{|KH})$ (because $L^2(\alpha_{|A})$ and $L^2(\alpha_{|KH})$ are quasi-equivalent by construction).

Now assume that $g_0 \notin KH$. Take any $g \in KH$: $g_0^{-1}g \notin H$. By Theorem C there exists an open neighborhood $U_g$ of $g$ such that $L^2(\mathrm{id}) \prec L^2(\alpha_{|g_0^{-1}U_g})$. Taking the tensor product by $L^2(\alpha_{g_0})$ we get an open covering $(U_g)_{g \in G}$ of $KH$ such that for all $g \in G$, $L^2(\alpha_{g_0}) \prec L^2(\alpha_{|U_g})$. Denote by $q: G \to G/H$ the quotient map, and recall that it is continuous and open. Hence, our open covering induces an open covering $q(K) \subset \bigcup_{g \in G} q(U_g)$. We extract a finite covering $q(K) \subset \bigcup_{i=1}^n U_{g_i}$, and pull it back: $KH \subset \bigcup_{i=1}^n U_{g_i}H$. For $1 \le i \le n$, $L^2(\alpha_{g_0}) \prec L^2(\alpha_{|U_{g_i}}) \sim L^2(\alpha_{|U_{g_i}H})$ (lemma 2.5), hence $L^2(\alpha_{g_0}) \prec \bigoplus_{i=1}^n L^2(\alpha_{|U_{g_i}H}) \sim L^2(\alpha_{\bigcup_{i=1}^n U_{g_i}H})$, and therefore $L^2(\alpha_{g_0}) \prec L^2(KH)$, which is a contradiction.

$$ \text{Thus, } g_0 \in KH, \text{ so } L \subset KH, \mu(A \setminus KH) = 0 \text{ and finally } 1 \otimes 1_{KH} \in \mathcal{Z}(\mathcal{N}). \quad \square $$

**Proof of Corollary B.** Here $\alpha: G \rightsquigarrow M$ is a topologically outer action of a locally compact group on a type II factor $M$. Denote by $H$ the subgroup of spectrally inner automorphisms in $G$. By Theorem 2.7, $H$ only contains trace-scaling automorphisms. By Theorem A, $M' \cap (M \rtimes_\alpha G) \subset M' \cap (M \rtimes_\alpha H)$. By Connes-Takesaki's relative commutant theorem [14, XII.1.7 in combination with XII.1.1], $M' \cap (M \rtimes_\alpha H) = \mathbb{C}$, which completes the proof. $\square$

**Acknowledgments.** We are very grateful to our advisor Amine Marrakchi for suggesting this question and for his help during this work. We thank Cyril Houdayer, Sven Raum and Stefaan Vaes for their comments on a first draft of this article.

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