Dynamics: EQ-States vs. NESS-States

Question

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$.

Consider a state that relaxes towards equilibrium: $$\omega_T(A):=\omega\circ\tau^T(A)\stackrel{T\to\infty}{\to}\omega_\infty(A)$$

Then it relaxes in mean towards equilibrium, too: $$\langle\omega\rangle_T(A):=\frac{1}{T}\int_0^T\omega\circ\tau^s(A)\mathrm{d}s\stackrel{T\to\infty}{\to}\omega_\infty(A)$$ That is it has a unique steady-state: $\omega_\text{NESS}=\omega_\text{EQ}$

Intuitively, convergence implies convergence in mean but how to prove it here?

Answer 1

After a while it's not as hard...

In fact, local integrability is the key: $$\omega\circ\tau(A)\in\mathcal{C}(\mathbb{R}_+):\quad\omega_T(A)\stackrel{T\to\infty}{\to}\omega_\infty\implies\langle\omega\rangle_T\stackrel{T\to\infty}{\to}\omega_\infty$$ (See the thread: Asymptotic Convergence vs. Mean Convergence)