Normed algebras of differentiable functions on compact plane sets: completeness and semisimple completions

Heiko Hoffmann. "Normed algebras of differentiable functions on compact plane sets: completeness and semisimple completions." Studia Mathematica 207.1 (2011): 19-45. <http://eudml.org/doc/285683>.

@article{HeikoHoffmann2011,
abstract = {We continue the study of the completeness and completions of normed algebras of differentiable functions Dⁿ(K) (where K is a perfect, compact plane set), initiated by Bland, Dales and Feinstein [Studia Math. 170 (2005) and Indian J. Pure Appl. Math. 41 (2010)]. We prove new characterizations of the completeness of D¹(K) and results concerning the semisimplicity of the completion of D¹(K). In particular, we prove that semi-rectifiability is necessary for the completion of D¹(K) to be semisimple in the case where K lies on a rectifiable, injective curve. Furthermore, we answer a question posed by Dales and Feinstein and show that another question posed by them has an affirmative answer in some special cases. As compared with the approach taken by Bland, Dales and Feinstein, which comes from the theory of function algebras, we move within an operator-theoretic framework by investigating the mapping properties of certain derivation operators.},
author = {Heiko Hoffmann},
journal = {Studia Mathematica},
keywords = {differentiable functions on planar compacta; completeness of algebras; semi-simplicity; geodesical boundedness},
language = {eng},
number = {1},
pages = {19-45},
title = {Normed algebras of differentiable functions on compact plane sets: completeness and semisimple completions},
url = {http://eudml.org/doc/285683},
volume = {207},
year = {2011},
}

TY - JOUR
AU - Heiko Hoffmann
TI - Normed algebras of differentiable functions on compact plane sets: completeness and semisimple completions
JO - Studia Mathematica
PY - 2011
VL - 207
IS - 1
SP - 19
EP - 45
AB - We continue the study of the completeness and completions of normed algebras of differentiable functions Dⁿ(K) (where K is a perfect, compact plane set), initiated by Bland, Dales and Feinstein [Studia Math. 170 (2005) and Indian J. Pure Appl. Math. 41 (2010)]. We prove new characterizations of the completeness of D¹(K) and results concerning the semisimplicity of the completion of D¹(K). In particular, we prove that semi-rectifiability is necessary for the completion of D¹(K) to be semisimple in the case where K lies on a rectifiable, injective curve. Furthermore, we answer a question posed by Dales and Feinstein and show that another question posed by them has an affirmative answer in some special cases. As compared with the approach taken by Bland, Dales and Feinstein, which comes from the theory of function algebras, we move within an operator-theoretic framework by investigating the mapping properties of certain derivation operators.
LA - eng
KW - differentiable functions on planar compacta; completeness of algebras; semi-simplicity; geodesical boundedness
UR - http://eudml.org/doc/285683
ER -