In the proof of Lemma 1.3 in the paper "The ideal structure of a groupoid C\* algebra", Journal of Operator Theory 1991 by Jean Renault, I found the notion of a generalized limit of a net without any explanation or definition. A google search brought no results, so what is a generalized limit?


To be more precise, he takes a convergent net $x\_i$ with index set $I$ and then speaks of a generalized limit $\omega$ of $I$. He wants to show the continuity of a function $f$, and he applies $\omega$ to $f(x\_i)$ which then seems to be a number.



What is a generalized limit?
I tried to search for [renault ideal "generalized limit"](https://www.google.com/search?q=renault+ideal+%22generalized%20limit%22) 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](http://www.ams.org/books/surv/134/)*, AMS, 2007. Some parts of the book are freely available [here](http://www.ams.org/books/surv/134/surv134-endmatter.pdf). (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](https://books.google.com/books?id=ix4P1e6AkeIC&pg=PA82) 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](https://en.wikipedia.org/wiki/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.) 


