Uniform approximation on manifolds

Abstract

It is shown that if A is a uniform algebra generated by a family Φ of complex-valued C 1 functions on a compact C 1 manifold-with-boundary M , the maximal ideal space of A is M , and E is the set of points where the differentials of the functions in Φ fail to span the complexified cotangent space to M , then A contains every continuous function on M that vanishes on E. This answers a 45-year-old question of Michael Freeman who proved the special case in which the manifold M is two-dimensional. More general forms of the theorem are also established. The results presented strengthen results due to several mathematicians.

A peak point theorem for uniform algebras on real-analytic varieties

“…We will make use of a recent result of Izzo that will enable us to reduce approximation on a variety to approximation on the union of the exceptional set and the singular set of the variety. This type of theorem has a long history, going back to work of John Wermer [17] and Michael Freeman [9] in the 1960's -for a detailed account, see [12]. Theorem 2.4 (Izzo, [12]).…”

“…This type of theorem has a long history, going back to work of John Wermer [17] and Michael Freeman [9] in the 1960's -for a detailed account, see [12]. Theorem 2.4 (Izzo, [12]). Let A be a uniform algebra on a compact Hausdorff space X, and suppose that the maximal ideal space of A is X.…”

Uniform algebras invariant under every homeomorphism

“…The basic idea of the proofs of Theorems 1.5, 1.7, and 1.8 is to determine the maximal ideal space M A of A, and then to invoke an abstract uniform algebra generalization given in [11] of a well-known approximation result of Hörmander and Wermer [8]. For the reader's convenience, we quote here the result we will use.…”

“…Step 3 goes through as before except that we must allow for the possibility that M γ is a zero-dimensional manifold. To handle that case we need a zero-dimensional form of [11,Theorem 1.3]. Fortunately the needed result is an immediate consequence of theŠilov idempotent theorem as we next observe.…”

Uniform algebras on the sphere invariant under group actions

“…Anderson and Izzo [1,2] Anderson et al [3][4][5] have proved theorems showing that under a variety of additional hypotheses the peak point conjecture does hold. The proofs of the positive results of the present paper will be based on the following peak point theorem of Anderson and Izzo [2] concerning the essential set of a counterexample to the peak point conjecture (whose proof in turn relies on [11]). (The essential set E for a uniform algebra A on a space X is the smallest closed subset of X such that A contains every continuous function on X that vanishes on E. Note that A contains every continuous function whose restriction to E lies in the restriction of A to E. The uniform algebra A is said to be essential if E = X ).…”