# Function Algebras, and a resounding NO! to the Peak Point Conjecture

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 $\Lambda$ of continuous functions defined on a compact set $X$ which (i) is closed with respect to pointwise multiplication and addition, (ii) contains the constant functions and separates points of $X$, and (iii) is closed as a subspace of $C(X)$ where, here, $C(X)$ denotes the space of continuous functions defined on $X$ equipped with the sup norm: $\|f\|=\sup_{x\in X}|f(x)|$. Associated to such an $A$ is the collection $M=\mathcal{M}_A$ of all nonzero homomorphisms $\varphi:A\to\mathbb{C}$; one easily verifies that every maximal ideal of $A$ is the kernel of some element of $M$ and vice versa, whereby the space $\mathcal{M}_A$ is called the maximal ideal space associated to $A$. Also:

Definition: A point $p$ in $X$ is said to be a peak point of $A$ provided there exists a function $f\in A$ so that $f(p)=1$ and $|f|<1$ on $X\setminus\{p\}$.

One problem of importance in the realm of function algebras is to characterize $C(X)$ with respect to such algebras $A$ of $X$. To quote Anderson and Izzo:

A central problem in the subject of uniform algebras is to characterize $C(X)$ among the uniform algebras on $X$.

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).