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).

There are two well-known results serve as necessary conditions for a function algebra A over a compact set X to be equal to C(X), namely:

  1. A = C(X) if \mathcal{M}(A)=X. In particular, this means that the only nonzero multiplicative linear functionals on A are the point evaluations.
  2. A = C(X) if every point of X is a peak point for A.

Having stated these two conditions, one can easily describe the peak point conjecture:

Peak Point Conjecture. For a compact set X, a function algebra A is identically equal to C(X) 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 A over X, the algebra A^r denotes the algebra in which every function f of A has a square root. Of course, every function f^r of A^r need not have a square root in A^r, so the process of attaching squasre roots for functions in A^r to A^r 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 A_\gamma of X_\gamma as \gamma runs through ordinals less than some arbitrary \gamma_0. After much verifying, one can allow \gamma=\Omega to be the first uncountable ordinal, whereby the collection A_\Omega is reached. A_\Omega is called the universal root algebra of A.

Theorem 2.4. Let A be a function algebra on X. The universal root algebra of A is a function algebra \widetilde{A} on \widetilde{X} with an associated continuous map \widetilde{\pi}:\widetilde{X}\to X so that:

  1. For h\in A, h\circ\widetilde{\pi} defines an embedding of A in \widetilde{A}
  2. Every function in \widetilde{A} has a square root in \widetilde{A}
  3. \widetilde{A}\cap\widetilde{\pi}*(C(X))=\widetilde{\pi}*(A)
  4. \widetilde{\pi}^{-1}(\partial_A)=\partial_{\widetilde{A}} where, here, \partial_B denotes the Silov boundary of B.
  5. \widetilde{A}|\widetilde{\pi}^{-1}(x) is dense in C\left(\widetilde{\pi}^{-1}(x)\right) for each x\in X.
  6. If X=\mathcal{M}_A, then X=\mathcal{M}_{\widetilde{A}}. \square

The main results of this theorem that are of fundamental importance moving forward are: (i) \widetilde{A} is non-trivial whenever A is non-trivial, which follows directly from 1 above; (ii) \partial_{\widetilde{A}}\neq\widetilde{X} whenever \partial_A\neq X by way of 4 above; (iii) by 2 above, \mathcal{M}_{\widetilde{A}} 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 A_\omega (here, \omega is the first infinite ordinal) can be formed inductively given a countable, dense subset F_1 of A_1 and defining F_{k+1} in A_{k+1} to be the collection of all polynomials in \mathbb{Z}[x]\cap F_k along with the collection of all functions of the form p_f, f\in F_k, where here, p_f:X\to\mathbb{C}^{F_k} is the so-called f coordinate function. Along with such F_{k+1} being countable and dense in A_{k+1}, F_{k+1} is also closed under the taking of square roots; moreover, by defining the embedding of F_k into A_\omega by F_k^*, we have that every function in F_n^* has a square root in F_{n+1}^* and hence that \cup_n F_n^* is countable, closed under taking of square roots, and dense in A_\omega. By way of this construction, the following result holds:

Theorem 2.5. For a function algebra A on X with countable dense subset F, then there exists a function algebra \widetilde{A} on \widetilde{X} with an associated projection \widetilde{\pi}:\widetilde{X}\to X so that items 1., 3., 4., 5., and 6. of 2.4 hold along with (2′) there exists a countable dense subset \widetilde{F} of \widetilde{A} which is closed under taking of square roots and contains F\circ\widetilde{\pi}. \square

As a result of Theorem 2.5, \widetilde{A} is a separable function algebra, \widetilde{A}\neq C(X) if A\neq C(X) by way of Theorem 2.4, and \mathcal{M}_A 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:

Lemma.

  1. Let A be a function algebra on X and define E=\left\{f^2\,:\,f\in A\right\}. If E is dense in A, then every probability measure \mu which represents a point x\in X with respect to A is a so-called Jensen measure, i.e. a measure for which

    \displaystyle \int\log|f|\,d\mu \geq \log|f(x)| for all f\in A.

  2. Each of the function algebras \widetilde{A} 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 f and a sequence \{f_k\} in A so that (f_k)^{2^n}\to f in A as k\to\infty. Because \mu represents a point x\in X, f_k(x)=\int f_k\,d\mu, whereby

\displaystyle |f_k(x)|\leq\int|f_k|\,d\mu \implies |f(x)|^{2^{-n}}\leq\int|f|^{2^{-n}}\,d\mu as k\to\infty.

Defining the sequence h_n=2^n\left(|f|^{2^{-n}}-1\right), one obtains a decreasing sequence of functions which converges pointwise to \log|f|, whereby taking limits yields the result. \square

And finally, as a result:

Theorem 2.8. Let A be a function algebra with X=\mathcal{M}_A such that every point has a unique Jensen measure on X. Then for \widetilde{A} the universal root algebra in theorems 2.4 and 2.5,

  1. every representing measure for \widetilde{A} on X is a point mass.
  2. \widetilde{X}=\mathcal{M}_{\widetilde{A}}.
  3. \widetilde{A}\neq C(\widetilde{X}) if A\neq C(X).

Moreover, if A is separable and if \widetilde{A} is as in 2.5, then 4. \widetilde{A} is separable, and 5. every point of \widetilde{X} is a peak point. \square

Note, then, that by considering for A the example here (that paper uses the notation R(X) instead of A), 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 \widetilde{A}\neq C(\widetilde{X}) for which \widetilde{X}=\mathcal{M}_{\widetilde{A}} and for which every point in \widetilde{X} is a peak point for \widetilde{A}.

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.

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