Quasianalyticity in certain Banach function algebras

Abstract

Let X be a perfect, compact subset of the complex plane. We consider algebras of those functions on X which satisfy a generalised notion of differentiability, which we call F-differentiability. In particular, we investigate a notion of quasianalyticity under this new notion of differentiability and provide some sufficient conditions for certain algebras to be quasianalytic. We give an application of our results in which we construct an essential, natural uniform algebra A on a locally connected, compact Hausdorff space X such that A admits no non-trivial Jensen measures yet is not regular. This construction improves an example of the first author (2001).Let X be a perfect, compact subset of the complex plane C. We consider those normed algebras consisting of complex-valued, continuously complexdifferentiable functions on X, denoted D(1) (X). These algebras were introduced by Dales and Davie in [9] and further investigated, for example, in [2] and [10]. The algebra D(1) (X) need not be complete, and the completion of D(1) (X) need not be a Banach function algebra in general.Bland and the first author [2] introduced F -differentiation, which generalises the usual complex-differentiation, and considered normed algebras ofF (X). * This paper contains work from the second author's PhD thesis. † The second author was supported by EPSRC grant number EP/L50502X/1

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

“…We are almost ready to construct the examples mentioned in the introduction. First we need the following result, which is taken from [3] (see also [4,8]). Proposition 4.…”