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

J. F. Feinstein¹ ; H. Yang¹

¹ School of Mathematical Sciences The University of Nottingham University Park, Nottingham, NG7 2RD, UK

Studia Mathematica, Tome 235 (2016) no. 1, pp. 1-15

Résumé

We discuss two types of “regularity point”, points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset (2000). 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 has no non-trivial 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.

Détail

DOI : 10.4064/sm8351-7-2016

Keywords: discuss types regularity point points continuity r points banach function algebras which introduced first author somerset even natural uniform algebras compact plane sets nbsp these types regularity point different method constructing swiss cheese sets regular has non trivial jensen measures original construction appears first author previous work approach constructing sets general allows obtain additional properties particular construction example swiss cheese set property set points discontinuity has positive area