Partial regularity and t-analytic sets for Banach function algebras

Abstract

In this note we introduce the notion of $t$-analytic sets. Using this concept, we construct a class of closed prime ideals in Banach function algebras and discuss some problems related to Alling's conjecture in $H^\infty$. A description of all closed $t$-analytic sets for the disk-algebra is given. Moreover, we show that some of the assertions in Daoui et al. (Proc. Am. Math. Soc. 131:3211-3220, 2003) concerning the $O$-analyticity and $S$-regularity of certain Banach function algebras are not correct. We also determine the largest set on which a Douglas algebra is pointwise regular.Comment: 18 pages, 1 figur

Regularity of 𝑅(𝑋) does not pass to finite unions

“…The natural uniform algebra A is regular if and only if every point of X is a point of continuity, and this is also equivalent to the condition that every point of X is an R-point ( [6,7]).…”

Regularity points and Jensen measures for $R(X)$

“…It is standard that the natural Banach function algebra A is regular if and only if every point of X is a point of continuity, and this is also equivalent to the condition that every point of X is an R-point ( [8,9] Let A be a uniform algebra on X, and let ϕ be a character on A. A representing measure µ for ϕ is a regular, Borel probability measure supported on X such that…”

“…It is standard that the natural Banach function algebra A is regular if and only if every point of X is a point of continuity, and this is also equivalent to the condition that every point of X is an R-point ( [8,9]).…”

“…We remark that if f ∈ A(K) and f is constant on the line segment forming the left edge of K, then an easy application of the reflection principle (see, for example, [13, Chapter IX, Theorem 2.1]) shows that f is constant on K. A similar compact set K was used in [8,Example 4.2].…”

“…In Section 3 we give examples to show that they can differ even for natural uniform algebras such as R(X), for suitable compact plane sets X. These examples are inspired by an example in [8], where the first author and Mortini also studied these types of regularity points. (In that paper, these points were also described as regularity points of types I and II).…”