# What is a generalized limit?

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.

• Maybe it's the limit of a net. – Christian Remling Jun 15 '16 at 18:27
• Probably some context (like mentioning the paper where you saw this) could increase probability that this will be answered. – Martin Sleziak Jun 15 '16 at 18:28
• Link to article (no paywall, at time of writing) mathjournals.org/jot/1991-025-001/1991-025-001-001.html – Yemon Choi Jun 15 '16 at 18:51
• @PedroLauridsenRibeiro Oops. How about theta.ro/jot/archive/1991-025-001/1991-025-001-001.html ? (JOT's archives have a moving paywall, so anything older than 5 years should be free) – Yemon Choi Jun 15 '16 at 20:50
• @WłodzimierzHolsztyński How does that help? The set I in the question is precisely the indexing set for a convergent net, so we know full well that it makes sense to talk of convergence. The question, and the bit of Renault's paper that is referred to, is precisely talking about a functional which is supposed to extend "evaluation at the limit" when applied to the space of bounded nets – Yemon Choi Jun 15 '16 at 22:05

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.)

• Yes, I think your answer is probably what Renault meant. The phrase "generalized limit" is one I saw from authors/speakers who tended to also take Banach limits. Note however that the concept of Banach limit is problematic if $I$ does not come with a natural notion of a shift. – Yemon Choi Jun 15 '16 at 20:51
• Yes, I have seen the phrase generalized limit used for Banach limit several times. (And if there were no mention of nets in the question, that would be my first guess.) For example, Jerison, Meyer. The Set of all Generalized Limits of Bounded Sequences. Canadian Journal of Mathematics, 9: 79-89 (1956); DOI: 10.4153/CJM-1957-012-x and some related papers. – Martin Sleziak Jun 15 '16 at 20:55
• If I recall correctly, this is the notion of generalized limits used to define Dixmier traces, right? – Pedro Lauridsen Ribeiro Jun 15 '16 at 21:00
• @PedroLauridsenRibeiro It seems that a functional very similar to Banach limit is used in definition of Dixmier trace. But instead of shift-invariance this functional has property called scale invariance, i.e. $\omega(x_1,x_2,x_3,\dots)=\omega(x_1,x_1,x_2,x_2,x_3,x_3,\dots)$. – Martin Sleziak Jun 16 '16 at 4:02

My guess (based on terminology I've seen or heard from other analysts over the years) is that this just means taking a non-principal ultrafilter $U$ on $I$ and then defining $\mu(f)$ to be $\lim_{i\in U} f(x_i)$.

• But the ultrafilter should have something to do with the order structure on $I$? It should probably contain the filter given by all sets of the form $\{ i: i\ge i_0\}$ for $i_0\in I$. – user1688 Jun 15 '16 at 19:05
• @Anton good point, I am inclined to agree with you (but am rather brain-fried at the moment, so will have to think about it later) – Yemon Choi Jun 15 '16 at 19:10