Regularity points and Jensen measures for R(X)Tools Feinstein, Joel and Yang, H. (2016) Regularity points and Jensen measures for R(X). Studia Mathematica, 235 (1). pp. 115. ISSN 17306337
AbstractWe discuss two types of `regularity point', points of continuity and Rpoints for Banach function algebras, which were introduced by the first author and Somerset in [16]. We show that, even for the natural uniform algebras R(X) (for compact plane sets X), these two types of regularity point can be different. We then give a new method for constructing Swiss cheese sets X such that R(X) is not regular, but such that R(X) has no nontrivial Jensen measures. The original construction appears in the first author's previous work. Our new approach to constructing such sets is more general, and allows us to obtain additional properties. In particular, we use our construction to give an example of such a Swiss cheese set X with the property that the set of points of discontinuity for R(X) has positive area.
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