Why care about ultrafilters (in operator algebras)?

Question

I recently learned about filters and ultrafilters and the notion of convergence with respect to an ultrafilter in a topological space. I have to say, at first I could not appreciate their potential. I think the two central theorems are the following:

Proposition: Let $X$ be a set. Then any $S\subset\mathcal{P}(X)$ having the finite intersection property is included in some ultrafilter.

Proposition: Let $X$ be a topological space. Then $X$ is compact if-f every ultrafilter converges to at least one point in $X$. Also, $X$ is Hausdorff if-f every ultrafilter converges at most to one point in $X$.

I realized that ultrafilters are a powerful tool when I saw a very simple proof of Tychonoff's theorem using ultrafilters. I always thought that Tychonoff's theorem had a difficult proof and I was amazed to see such a simple proof using ultrafilters.

I would like to ask the following question: how are ultrafilters used in operator algebras? What are some interesting/ standard applications they have in this field? What books or papers would you recommend to someone that wants to understand the use of ultrafilters in operator algebras?

Bonus question: What about topology, in general? To what cause would someone use ultrafilters besides proving that a space is compact or Hausdorff? Are there any interesting constructions based on ultrafilters?

Answer 1

The use of ultrafilters in topology is extremely well established. Bourbaki would be a standard reference. Filters in general are fundamental tools in set theory and analysis, albeit not so common in the standard pedagogical route. One can even construct the real numbers in terms of filters of rational numbers.

As you point out, ultrafilters are a convenient tool when proving various results. It is a way avoid the cumbersome details of sequences and nets (when sequences are not enough). The set-theoretic formalism of filters encapsulates things quite nicely. However, of course, ultrafilters are highly non-constructive. You will never see one that is not principal. So, their use is only theoretical.

Ultrafilters are very cool, no doubt. However, one should be aware of other cool options to prove results, e.g., Tychonoff's theorem, without ultrafilters. For instance, the characterisation of compactness in terms of the closedness of the projection mapping with respect to all spaces leads to a beautiful proof of Tychonoff's theorem [Tychonoff’s theorem in a category, Proc. Amer. Math. Soc.]. That proof is applicable in a much wider context than the ultrafilter proof since it is not set-theoretic based.

I hope this somewhat answers your question.

Answer 2

Here is an application of ultra-filters that is very much in the spirit of operator algebras. The ultra-filters in question are not quite on topological spaces, but instead, in the context of meet semi-lattices.

For example, if $X$ is a topological space, then the set of all subsets of $X$, denoted $\mathscr P(X)$, is a meet semi-lattice relative to the order of inclusion and the usual ultra-filters in $X$ correspond precisely to ultra-filters in the meet semi-lattice $\mathscr P(X)$.

With these disclaimers, here we go: many fundamental examples of C$^*$-algebras possess a set $S$ of generators consisting of partial isometries. Moreover the set $S$ is often closed under multiplication and adjoint, in which case it forms an inverse semigroup.

Since the algebraic structure of inverse semigroups is way easier to understand as compared to C$^*$-algebras, the question arises as to what extent the given C$^*$-algebra may be reconstructed entirely in terms of the generating inverse semigroup $S$.

As it turns out the answer is generally negative because one can easily obtain examples of non-isomorphic C$^*$-algebras generated by isomorphic inverse semigroups of partial isometries.

Nevertheless, there is a construction of a specific C$^*$-algebra from any given inverse semigroup $S$, called the tight C$^*$-algebra, which is able to recover some of the most important C$^*$-algebra in applications, such as Cuntz-algebras and a host of other examples.

The contruction of the tight C$^*$-algebra can be made from a certain dynamical system on the closure of the space of all ultra-filters of the idempotent semi-lattice of the given inverse semigroup.