Ergodic actions and deviation from invariance

Question

Let $M$ be a von Neumann algebra and let $(\phi_t)$ be an ergodic point-$\sigma$-weakly continuous one-parameter group of automorphisms $\phi_t\in \mathrm{Aut}(M)$, i.e., $\Vert\omega-\omega\circ\phi_t\Vert \to 0$ as $t\to0$ for all $\phi\in M_*$. Given an arbitrary normal state $\omega$ on $M$, is it true that $$ \sup_{t\ge0} \Vert\omega-\omega\circ\phi_t\Vert = \limsup_{t\to\infty} \Vert\omega-\omega\circ\phi_t\Vert\ ? $$ The left-hand side quantifies how much $\omega$ deviates from being an invariant state, and it is clear that the right-hand side is dominated by the left-hand side. As a side question, does it matter if $M$ is abelian or a factor?

Answer 1

No, this equality does not hold in general. I first show the positive result that the equality holds up to a factor $2$ and then give two examples (an abelian and a factorial one) to show that the constant $2$ is optimal.

We first prove that

$$\sup_{s \in \mathbb{R}} \|\omega - \omega \circ \phi_s\| \leq 2 \, \limsup_{t \to +\infty} \| \omega - \omega \circ \phi_t\| \; .$$

To prove this, write $\kappa = \limsup_{t \to +\infty} \| \omega - \omega \circ \phi_t\|$ and let $s \in \mathbb{R}$ and $\delta > 0$ be arbitrary. Take $t_0 > 0$ such that $\| \omega - \omega \circ \phi_t\| \leq \kappa + \delta$ for all $t \geq t_0$. Take $N > 0$ arbitrary and define the state $\eta$ by

$$\eta(a) = \frac{1}{N} \int_{t_0}^{t_0+N} \omega(\phi_t(a)) \, dt \; .$$

Then $\|\omega - \eta\| \leq \kappa + \delta$. We also get that $\|\eta - \eta \circ \phi_s\| \leq 2 |s| / N$. Then

$$\|\omega - \omega \circ \phi_s\| \leq 2 (\kappa + \delta) + \frac{2 |s|}{N} \; .$$

Since $\delta > 0$ and $N > 0$ are arbitrary, it follows that $\|\omega - \omega \circ \phi_s\| \leq 2 \kappa$.

As for the counterexamples, first fix any mixing probability measure preserving action $\mathbb{R} \curvearrowright^\phi (X,\mu)$. Denote $M = L^\infty(X,\mu)$.

Take $\varepsilon > 0$. We'll construct a faithful normal state $\omega$ on $M$ such that $\|\omega - \omega \circ \phi_1\| > 2 - \varepsilon$ and

$$\limsup_{t \to +\infty} \|\omega - \omega \circ \phi_t\| \leq 1 + 2 \varepsilon \; .$$

Applying the Rokhlin lemma to the ergodic transformation $\phi_1$, we can choose a Borel set $Y \subset X$ such that $Y$ and $\phi_1(Y)$ are disjoint and

$$\mu(X \setminus (Y \cup \phi_1(Y))) < \varepsilon/2 \; .$$

Define the normal state $\omega'$ on $M$ by

$$\omega'(F) = \mu(Y)^{-1} \int_Y F(y) \, d\mu(y) \; .$$

Choose a faithful state $\omega$ on $M$ such that $\|\omega - \omega'\| < \varepsilon / 2$.

Denote by $\eta \in L^1(X,\mu)$ the function $\eta = \mu(Y)^{-1} 1_Y$. Then,

$$\|\omega' - \omega' \circ \phi_1\| = \|\eta - \phi_1(\eta)\|_1 = 2 \; .$$

Therefore, $\|\omega - \omega \circ \phi_1\| > 2 - \varepsilon$. On the other hand by mixing,

$$\limsup_{t \to +\infty} \|\omega' - \omega' \circ \phi_t\| = \limsup_{t \to +\infty} \|\eta - \phi_t(\eta)\|_1 = \mu(Y)^{-1} \limsup_{t \to +\infty} (\mu(Y \cap \phi_t(X\setminus Y)) + \mu((X \setminus Y) \cap \phi_t(Y))) = 2 \mu(X \setminus Y) \leq 1 + \varepsilon \; .$$

Then also $\limsup_{t \to +\infty} \|\omega - \omega \circ \phi_t\| \leq 1 + 2 \varepsilon$.

To give a factorial counterexample, realize the hyperfinite II$_1$ factor $R$ with its tracial state $\tau$ as the completion of the CAR algebra associated with the Hilbert space $L^2(\mathbb{R})$, with generators $a(\xi)$, $\xi \in L^2(\mathbb{R})$, such that $\xi \mapsto a(\xi)$ is linear and

$$a(\xi) a(\eta) + a(\eta) a(\xi) = 0 \quad\text{and}\quad a(\xi)^* a(\eta) + a(\eta) a(\xi)^* = \langle \eta, \xi \rangle \, 1 \; .$$

Define $\mathbb{R} \curvearrowright^\phi R$ by $\phi_t(a(\xi)) = a(\lambda_t \xi)$, where $\lambda_t$ is the unitary of left translation by $t$. Again, $\mathbb{R} \curvearrowright^\phi R$ is a mixing action.

Define the orthogonal unit vectors $\xi_n \in L^2(\mathbb{R})$ by $\xi_n = 1_{[n,n+1)}$. Then $p_n = a(\xi_n)^* a(\xi_n)$ is a commuting family of orthogonal projections in $R$ with $\tau(p_n) = 1/2$ for all $n \in \mathbb{Z}$. Denote by $A$ the von Neumann algebra generated by these projections $p_n$.

Fix $\varepsilon > 0$. We again construct a faithful normal state $\omega$ on $R$ such that $\|\omega - \omega \circ \phi_1\| > 2 - \varepsilon$ and

$$\limsup_{t \to +\infty} \|\omega - \omega \circ \phi_t\| \leq 1 + 2 \varepsilon \; .$$

Note that $\phi_1(A) = A$. By the Rokhlin lemma, we can choose a projection $p \in A$ such that $p \perp \phi_1(p)$ and $\tau(p + \phi_1(p)) > 1-\varepsilon /2$. We define the state $\omega'$ on $R$ by $\omega'(a) = \tau(p)^{-1} \tau(pap)$ and choose a faithful normal state $\omega$ on $R$ such that $\|\omega - \omega'\| < \varepsilon/2$.

We can now basically repeat the computation above. We only have to be careful with the computation of $\|p - \phi_t(p)\|_1$ for $t$ large. But by construction, $p$ will almost commute with $\phi_t(p)$ and the projections $p$ and $\phi_t(p)$ will be almost independent for large $t$. So, we still have that

$$\lim_{t \to +\infty} \|p - \phi_t(p)\|_1 = 2 \tau(p) \tau(1-p) \; .$$