It is well known that on the Banach space $l^{\infty}\left(\mathbb{N}\right)$ of bounded functions $f:\mathbb{N}\to\mathbb{C}$ with the sup-norm, there exists a (non-unique) Banach limit. This is a linear functional $L$ that is positive, bounded and shift-invariant. It can be seen that such a functional satisfies, for every real-valued $f\in l^{\infty}\left(\mathbb{N}\right)$, $$\liminf_{n\to\infty} f\left(n\right) \leq L\left(f\right) \leq \limsup_{n\to\infty}f\left(n\right).$$
I was wondering what if we replace $\mathbb{N}$ by a directed set. More specifically, let $\mathbf{N}$ be a directed set and consider the Banach space $l^{\infty}\left(\mathbf{N}\right)$ of bounded functions (nets) $f:\mathbf{N}\to\mathbb{C}$ with the sup-norm. While it seems that there is no a natural analog to the shift-invariance property, there are notions of liminf and limsup for nets (see for instance here). Then my question is as follows.
Let $\mathbf{N}$ be a directed set. Does there exist a bounded linear functional $L$ on $l^{\infty}\left(\mathbf{N}\right)$ such that $$\liminf_{n\in\mathbf{N}} f\left(n\right) \leq L\left(f\right) \leq \limsup_{n\in\mathbf{N}}f\left(n\right)$$ for every real-valued net $f\in l^{\infty}\left(\mathbf{N}\right)$?
When reading the standard proofs that a Banach limit exists for $l^{\infty}\left(\mathbb{N}\right)$, there is an essential use of partial sums. Namely, we are looking at the subspace of those elements in $l^{\infty}\left(\mathbb{N}\right)$ whose finite partial sums are uniformly bounded, and using Hahn-Banach theorem to extend a seminorm from it to the whole space. For general nets (say uncountable) it seems that this should not work anymore.
