Normed algebras of differentiable functions on compact plane sets |
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Authors: | H G Dales J F Feinstein |
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Institution: | 1.Department of Pure Mathematics,University of Leeds,Leeds,UK;2.School of Mathematical Sciences,University of Nottingham,University Park, Nottingham,UK |
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Abstract: | We investigate the completeness and completions of the normed algebras (D
(1)(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D
(1)(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of
connected, compact plane sets X for which the completeness of (D
(1)(X), ‖ · ‖) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for
all star-shaped, compact plane sets, and for all Jordan arcs in ℂ. |
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