Localization for uniform algebras generated by smooth functions on two-manifolds

Abstract

Conditions are given under which a uniform algebra on a two‐manifold must contain all continuous functions. In particular, it is shown that, if A is a uniform algebra generated by smooth functions on a compact smooth two‐manifold M such that the maximal ideal space of A is M, and every continuous function on M is locally a uniform limit of functions in A, then A = C(M). This gives an affirmative answer to a special case of a question from the Proceedings of the Symposium on Function Algebras held at Tulane University in 1965. It is also shown that, on every smooth manifold of dimension at least four, there exists a uniform algebra that is nonlocal. Additional related results are also established.

Localization for uniform algebras generated by real-analytic functions

“…It is an open question (posed in [7], p. 348) whether or not this result continues to hold if we replace the condition that each function f in C(X) is locally in A by the condition that each function f in C(M A ) is locally uniformly approximable by functions in A, i.e., that for each f ∈ C(M A ) and each x ∈ M A there exists a neighborhood U of x with f U ∈ A| U . The second author [11] established this for algebras on two-manifolds generated by continuously differentiable functions. For two-manifolds, via the device of pushing forward measures by functions in the algebra to the complex plane, one may reduce approximation problems to questions about the algebra R(K) (see [9]) consisting of uniform limits on a compact set K in the plane of rational functions with poles off K. The algebra R(K) is known to be local -a function locally in A belongs to A (see [10, II.10]) -and a theorem of Alexander [1] provides a generalization of this fact: if K is the union of a countable collection {K n } of compact sets with R(K n ) = C(K n ) for each n, then R(K) = C(K).…”

“…For two-manifolds, via the device of pushing forward measures by functions in the algebra to the complex plane, one may reduce approximation problems to questions about the algebra R(K) (see [9]) consisting of uniform limits on a compact set K in the plane of rational functions with poles off K. The algebra R(K) is known to be local -a function locally in A belongs to A (see [10, II.10]) -and a theorem of Alexander [1] provides a generalization of this fact: if K is the union of a countable collection {K n } of compact sets with R(K n ) = C(K n ) for each n, then R(K) = C(K). In [11] a more general result on R(K) is established, prompting the following definition: Definition 1.1. If A is a uniform algebra on a compact space X, we say that A has the countable approximation property if for each f ∈ C(X) there exists a countable collection {M n } of compact subsets of X with ∪ ∞ n=1 M n = X and f Mn ∈ A Mn for each n. Note that the sets M n in the preceding definition are allowed to depend on f .…”

“…Also, if A has the property that each f ∈ C(X) is locally uniformly approximable by functions in A, then A clearly has the countable approximation property. We can now state the main result of [11]:…”

Nonlocal uniform algebras on three-manifolds

“…The space on which Kallin's nonlocal uniform algebra is defined cannot be embedded in a three-manifold, so the approach used to construct nonlocal uniform algebras on four-manifolds in [Izzo 2010] does not work for three-manifolds. Instead we use an approach to nonlocal uniform algebras due to Sidney [1968].…”

“…In [Izzo 2010], we studied localization for uniform algebras generated by smooth functions on two-manifolds. The results there suggest that perhaps these uniform algebras are always local.…”