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Completions of normed algebras of differentiable functions

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Bland W., Feinstein J.
Studia Mathematica
|
2005
|
tom 170
|
nr 1
89-111
EN
We look at normed spaces of differentiable functions on compact plane sets, including the spaces of infinitely differentiable functions considered by Dales and Davie in [7]. For many compact plane sets the classical definitions give rise to incomplete spaces. We introduce an alternative definition of differentiability which allows us to describe the completions of these spaces. We also consider some associated problems of polynomial and rational approximation.
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Quasicompact endomorphisms of commutative semiprime Banach algebras

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Feinstein J., Kamowitz H.
Banach Center Publications
|
2010
|
tom 91
|
nr 1
159-167
EN
This paper is a continuation of our study of compact, power compact, Riesz, and quasicompact endomorphisms of commutative Banach algebras. Previously it has been shown that if B is a unital commutative semisimple Banach algebra with connected character space, and T is a unital endomorphism of B, then T is quasicompact if and only if the operators Tⁿ converge in operator norm to a rank-one unital endomorphism of B. In this note the discussion is extended in two ways: we discuss endomorphisms of commutative Banach algebras which are semiprime and not necessarily semisimple; we also discuss commutative Banach algebras with character spaces which are not necessarily connected. In previous papers we have given examples of commutative semisimple Banach algebras B and endomorphisms T of B showing that T may be quasicompact but not Riesz, T may be Riesz but not power compact, and T may be power compact but not compact. In this note we give examples of commutative, semiprime Banach algebras, some radical and some semisimple, for which every quasicompact endomorphism is actually compact.
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