The essential set of function algebras

On the local behavior of function algebras

“…L In this paper we shall give complete proofs of theorems announced recently in [8], These results are concerned with properties of interpolation sets and essential set with respect to a function algebra, and some of them are regarded as generalization of those results established in several literatures [4], [7] and [9]. Our main results are Theorem 1 and Theorem 3, which state that a ^-interpolation set is not compatible with the essential set of the function algebra.…”

“…By using these results, in some cases we can determine the essential set of the function algebra from those essential sets of the restriction algebras of countable closed partitions (Theorem 4). Theorem 3, together with Theorem 4, has been previously treated by Mullins [9] under the assumptions that the representing space X of the function algebra coincides with M A and M A is metrizable.…”

“…For a subset F in X, we shall denote by A\F the restricted algebra of A to F. Let F be a closed subset of X. If A\F is closed in C(F) 9 A\F is regarded as a function algebra on F. Then F is called an interpolation set for A when A\F=C(F).…”

Characterizations for approximately normal algebras

“…For A having metrizable ideal space M=M(A), Mullins [4] has characterized the essential set on M(A) by Em = M~P where P=\xEM: 3 a closed nbhd Vx of x with A\ Vx = C(Vx)}. Such a characterization fails in general for XÇZ M (A), as he points out.…”

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

“…The following theorem, which is related to the theorem of Rainwater mentioned above, seems to be due independently to Chalice [6], Gamelin and Wilken [9] and Mullins [12]. Theorem 1.2 [6,9,12]. Let A be a uniform algebra on a compact space X.…”