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Such an ideal does not exist.

Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is almost convergent, and therefore $I$-convergent, to $1/2$. So, $$\mathbb N=\{n\in\mathbb N\colon|x_n-1/2|\ge1/2\}\in I, $$ and hence $I$ is the powerset of $\mathbb N$. So, every sequence is $I$-convergent, and therefore almost convergent, to every real limit, which is of course absurd.

Such an ideal does not exist.

Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is almost convergent, and therefore $I$-convergent, to $1/2$. So, $$\mathbb N=\{n\in\mathbb N\colon|x_n-1/2|\ge1/2\}\in I, $$ and hence $I$ is the powerset of $\mathbb N$. So, every sequence is $I$-convergent, and therefore almost convergent, to every real limit, which is of course absurd.

This consideration also shows that the Cesàro convergence -- which is implied by the almost-convergence -- is also not the $I$-convergence, for any ideal $I$.

Such an ideal does not exist.

Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is almost convergent to $1/2$. So, $$\mathbb N=\{n\in\mathbb N\colon|x_n-1/2|\ge1/2\}\in I, $$ and hence $I$ is the powerset of $\mathbb N$. So, every sequence is almost convergent, to every real limit, which is of course absurd.

Such an ideal does not exist.

Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is almost convergent, and therefore $I$-convergent, to $1/2$. So, $$\mathbb N=\{n\in\mathbb N\colon|x_n-1/2|\ge1/2\}\in I, $$ and hence $I$ is the powerset of $\mathbb N$. So, every sequence is $I$-convergent, and therefore almost convergent, to every real limit, which is of course absurd.

Such an ideal does not exist.

Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is almost convergent to $1/2$. So, $$\mathbb N=\{n\in\mathbb N\colon|x_n-1/2|\ge1/2\}\in I, $$ and hence $I$ is the powerset of $\mathbb N$. So, every sequence is almost convergent, to every real limit, which is of course absurd.

Such an ideal does not exist.

Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is almost convergent to $1/2$. So, $$\mathbb N=\{n\in\mathbb N\colon|x_n-1/2|\ge1/2\}\in I, $$ and hence $I$ is the powerset of $\mathbb N$. So, every sequence is almost convergent, to every real limit, which is of course absurd.