I tried to search for renault ideal "generalized limit" to see whether I will find some related works where the definition of this notion is included.
I found this thesis: Groupoid Crossed Products by Geoff Goehle, https://arxiv.org/abs/0905.4681
It also uses this notion and includes the definition. (Since it is from a closely related area, it is a reasonable guess that this notion is used in the same way.)
If $I$ is a directed set then a generalized limit $\omega$ on $\ell^\infty(I)$ is defined here as:
A generalized limit is a norm one extension of the ordinary limit functional on the subspace $c_0$ of $\ell^\infty(I)$ consisting of those nets $\{a_i\}$ such that $\lim_I a_i$ exists.
Another source where this notion is defined is the book Dana P. Williams: Crossed Products of $C^*$-Algebras, AMS, 2007. Some parts of the book are freely available here. (This includes the part I quoted below.)
In order to give his proof, we need to recall the notion of a generalized limit.
Let $D$ be a directed set and let $c_D$ be the set of bounded $D$-convergent nets in $\mathbb C$.
That is, $c_D$ is the set of bounded nets $x=\{x_d\}_{d\in D}$ indexed by $D$ such that $\lim x_d$ exists.
... Note that $c_D$ is a subspace of the Banach space $\ell^\infty(D)$ of bounded functions on $D$ with the sup norm: $\|x\| := \sup_{d\in D} |x_d|$
The linear functional $\gamma$ sending $x\in c_D$ to $\lim x_d$ is of norm $1$.
Any norm $1$ extension $\Gamma$ of $\gamma$ to such that $\Gamma (x)>0$ if $x_d > 0$ for all $d$ is called a generalized limit over $D$.
The book then gives a detailed proof that such functional exists. A reference given there is Theorem III.7.1 from
John B. Conway, A course in functional analysis, Graduate texts in mathematics, vol. 96, Springer-Verlag, New York, 1985. However, if you check this theorem in Conway's book, it is analogous result with $I=\mathbb N$ and it additionally requires the functional to be shift-invariant; i.e., it is a proof that Banach limit exists. (Admittedly, the proofs are rather similar.)
In the other words, a generalized limit $\omega \colon \ell^\infty(I) \to \mathbb K$ is:
- linear;
- $|\omega (x)| \le \sup_{i\in I} |x_i|$;
- If $\lim_i x_i=l$, then $\omega(x)=l$, i.e., it extends the usual limit of nets on the directed set $I$.
Notice that for real $x$ we have $\inf_{i\in I} x_i \le \omega(x) \le \sup_{i\in I} x_i$, and thus $x\ge0 \Rightarrow \omega(x)\ge0$, i.e., such functional will necessary be positive.
It is also clear that any convex combination of generalized limits is again a generalized limit.
One natural way to obtain such generalized limit is to use Hahn-Banach theorem and extend the limit from the space of convergent nets on $I$ to $\ell_\infty(I)$. This would be rather straightforward in the case $\mathbb K=\mathbb R$. If $\mathbb K=\mathbb C$ then we take a real generalized limit $\omega_1$ and we define $\omega=\omega_1+i\omega_1$. Some additional work is needed to show that $\|L\|\le 1$. Details can be found in the references above.
As mentioned in Yemon Choi's answer, another possible approach is take any ultrafilter $\mathcal U$ containing the tail filter of the directed set $I$, i.e., the filter consisting of all sets of the form $[i_0,\infty)=\{i\in I; i\ge i_0\}$ for $i_0\in I$. If we have such ultrafilter, then the $\mathcal U$-limit extends the usual limit of the net. (As far as I can say, this seems again rather straightforward in the real case. In the complex case, we can use the same approach as before to show that the norm of $\mathcal U$-limit is equal to $1$.)
However, not every generalized limit can be realized in this way. (In the case $I=\mathbb N$ example of a generalized limit which is not an ultralimit is a Banach limit, which was already mentioned above.)
