Localization for uniform algebras generated by real-analytic functions

Abstract

It is shown that if A is a uniform algebra generated by real-analytic functions on a suitable compact subset K of a real-analytic variety such that the maximal ideal space of A is K, and every continuous function on K is locally a uniform limit of functions in A, then A = C(K). This gives an affirmative answer to a special case of a question from the

Analytic discs and uniform algebras generated by real-analytic functions

“…The general idea of the proof of Theorem 1.5 is to use Theorem 2.6 to reduce approximation on a variety V to approximation on successively smaller and smaller sets. In [4] and [5] this was done by considering the support of an arbitrary annihilating measure. As noted in the introduction, it seems that in the present setting dealing with an individual annihilating measure does not work and that we must, in effect, consider all the annihilating measures at once.…”

“…However, it is not obvious that the union of the exceptional set and the singular set, even locally, must itself be contained in a subvariety of V of dimension less than that of V . To get around this difficulty, we proceed as in [4] and [5] treating the exceptional set and singular set separately, introducing a filtration of V into exceptional sets and singular sets of decreasing dimensions. We show by induction on decreasing dimension of the exceptional sets that E A must lie in the singular set.…”

“…In particular different lemmas must be used. Moreover, while the proofs in [4] and [5] were carried out via duality working with an individual annihilating measure for the algebra A, it seems that that approach does not work here and that we must instead consider all annihilating measures simultaneously. That can be done conveniently by using the notion of essential set which was introduced by Herbert Bear [9] and whose definite is recalled below in the next section.…”

“…In [5] Anderson and the present author used an argument similar to the one used to prove the peak point theorem in [4] to prove a localization result for uniform algebras generated by real-analytic functions. That result also follows from Theorem 1.1.…”

“…Much of the proof of Theorem 1.1 is essentially a repetition of the proofs given in [4] and [5]. We will, nevertheless, present the proof in full detail as we believe that the proof should be given explicitly in the literature, and changes from the earlier proofs are needed.…”