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).
There are two well-known results serve as necessary conditions for a function algebra over a compact set to be equal to , namely:
- if . In particular, this means that the only nonzero multiplicative linear functionals on are the point evaluations.
- if every point of is a peak point for .
Having stated these two conditions, one can easily describe the peak point conjecture:
Peak Point Conjecture. For a compact set , a function algebra is identically equal to if and only if both the two conditions above are satisfied.
As the title of this entry suggests, the peak point conjecture was eventually disproven (in Brian James Cole’s 1968 doctoral thesis), and rather than doing so with soft-analysis arguments, he disproved the conjecture directly by constructing a function algebra over a space which satisfies the two hypotheses but fails to satisfy the conclusion. Some of the details are given below using the same notation/numbering from Cole, beginning with a slew of facts needed before proceeding.
First, given a function algebra over , the algebra denotes the algebra in which every function of has a square root. Of course, every function of need not have a square root in , so the process of attaching squasre roots for functions in to is repeated, and re-repeated, and re-re-repeated ad infinitum using transfinite induction. This process constructs a so-called system of root extensions over an algebra of as runs through ordinals less than some arbitrary . After much verifying, one can allow to be the first uncountable ordinal, whereby the collection is reached. is called the universal root algebra of .
Theorem 2.4. Let be a function algebra on . The universal root algebra of is a function algebra on with an associated continuous map so that:
- For , defines an embedding of in
- Every function in has a square root in
- where, here, denotes the Silov boundary of .
- is dense in for each .
- If , then .
The main results of this theorem that are of fundamental importance moving forward are: (i) is non-trivial whenever is non-trivial, which follows directly from 1 above; (ii) whenever by way of 4 above; (iii) by 2 above, decomposes into one-point parts and supports no non-trivial point derivation.
Next, note that moving forward, it would behoove us to have algebras whose underlying spaces are metrizable (i.e., separable algebras). To this end, note that a countable, dense subset of (here, is the first infinite ordinal) can be formed inductively given a countable, dense subset of and defining in to be the collection of all polynomials in along with the collection of all functions of the form , , where here, is the so-called coordinate function. Along with such being countable and dense in , is also closed under the taking of square roots; moreover, by defining the embedding of into by , we have that every function in has a square root in and hence that is countable, closed under taking of square roots, and dense in . By way of this construction, the following result holds:
Theorem 2.5. For a function algebra on with countable dense subset , then there exists a function algebra on with an associated projection so that items 1., 3., 4., 5., and 6. of 2.4 hold along with (2′) there exists a countable dense subset of which is closed under taking of square roots and contains .
As a result of Theorem 2.5, is a separable function algebra, if by way of Theorem 2.4, and consists of one-point parts and supports no non-trivial bounded point derivations by way of 2′. Moreover, after stating two lemmas, the peak point conjecture is all but history:
- Let be a function algebra on and define . If is dense in , then every probability measure which represents a point with respect to is a so-called Jensen measure, i.e. a measure for which
for all .
- Each of the function algebras in theorems 2.4 and 2.5 above has the property that every representing measure is a Jensen measure.
Proof (Sketch). The second part of the lemma is immediate from the first. For the first part, choose an arbitrary function and a sequence in so that in as . Because represents a point , , whereby
Defining the sequence , one obtains a decreasing sequence of functions which converges pointwise to , whereby taking limits yields the result.
And finally, as a result:
Theorem 2.8. Let be a function algebra with such that every point has a unique Jensen measure on . Then for the universal root algebra in theorems 2.4 and 2.5,
- every representing measure for on is a point mass.
- if .
Moreover, if is separable and if is as in 2.5, then 4. is separable, and 5. every point of is a peak point.
Note, then, that by considering for the example here (that paper uses the notation instead of ), Cole has constructed what is precisely a disproof of the peak point conjecture: In particular, Cole’s Theorem 2.5 applied to the example in the linked article yields a function algebra for which and for which every point in is a peak point for .
As mentioned before, I’d like to extend this exposition by one or two more to maybe touch on some of the advancements and generalizations (by Izzo and others) of the peak point conjecture. Of course, if you’d like to read more from Cole’s actual point of view, see his thesis here.