I've read a thesis where there is an example on ergodic mean, where however there is one point that is not fully clear to me, so hopefully someone can help me solve it.
Let's consider a quasi-local algebra $\mathcal{B}$ defined as norm closure of $\otimes_i \mathcal{A}_i$, where each $\mathcal{A}_i = \mathcal{A}$ is the algebra of bounded operators on a Hilbert space $\mathcal{H}$ (the index $i$ runs over the set of integer numbers). The element $x_i \in \mathcal{A}_i$ can be written as follows: $x_i = \otimes_{j<i}\mathcal{I}_j \otimes x \, \otimes_{k>i}\mathcal{I}_k$ where $\mathcal{I}$ is the identity. Let's consider the automorphism $\tau: \tau(x_i) = x_{i+1} $, and let's consider the tau-invariant state $\omega$. Then, in the thesis it is written that $ \frac{1}{2N + 1} \sum_{i=-N}^{N} x_i$ converges weakly to $\omega(x)\mathcal{I}$ as $N \to +\infty$.
My doubt is the following. Let's suppose that there are $y, z \in \mathcal{A}$ such that $[x, y] = z$. Then we can consider the operator $\sum_{i=-N}^{N} y_i$ such that $[\frac{1}{2N + 1} \sum_{i=-N}^{N} x_i, \sum_{j=-N}^{N} y_j] = \frac{1}{2N + 1} \sum_{i=-N}^{N} z_i$. If we take the limit $N \to \infty$ this commutator does not go to zero despite the fact that before we said that $ \frac{1}{2N + 1} \sum_{i=-N}^{N} x_i$ converges weakly to the identity operator. How is that possible? Maybe the operator $\sum_{j=-N}^{N} y_j$ does not converges weakly?