A former professor of mine (who I’ve discussed here previously) still to this day does some of the best work of anyone I’ve ever known. When he and I were *colleagues* (we existed at very-near geographical locations), I didn’t know the capacity of his amazing work, rather I just knew his work was amazing and that he for all intents and purposes should’ve been at a bigger, more prominent school.

Now, I know a bit more.

One of the things for which he was pretty *famous* (subjectively, *natürlich*) was proving a 50-ish year old unsolved problem in manifold theory; another of his fortes, though, is in the study of function algebras. That’s where this little journey takes us.

A **function algebra** is a family of continuous functions defined on a compact set which (i) is closed with respect to pointwise multiplication and addition, (ii) contains the constant functions *and* separates points of , and (iii) is closed as a subspace of where, here, denotes the space of continuous functions defined on equipped with *the sup norm*: . Associated to such an is the collection of all nonzero homomorphisms ; one easily verifies that every maximal ideal of is the kernel of some element of and vice versa, whereby the space is called the **maximal ideal space** associated to . Also:

*Definition*: A point in is said to be a *peak point* of provided there exists a function so that and on .

One problem of importance in the realm of function algebras is to characterize with respect to such algebras of . To quote Anderson and Izzo:

A central problem in the subject of uniform algebras is to characterize among the uniform algebras on .

One attempt at satisfying this necessity was the so-called *peak point conjecture*, which was strongly believed to be true until it was shown to be definitively untrue. The purpose of this entry is to focus a little on topics related thereto including the conjecture itself, the counterexample and its construction, and the related results (including work done by Izzo and various collaborators).

Continue reading →

### Like this:

Like Loading...