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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A structure theorem for discontinuous derivations of Banach algebras of differential functions


Author: Viet Ngo
Journal: Proc. Amer. Math. Soc. 102 (1988), 507-513
MSC: Primary 46J15
DOI: https://doi.org/10.1090/S0002-9939-1988-0928969-1
MathSciNet review: 928969
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Abstract: Let $ D:{C^n}\left[ {0,1} \right] \to \mathcal{M}$ be a derivation from the Banach algebra of $ n$ times continuously differentiable functions on $ \left[ {0,1} \right]$ into a Banach $ {C^n}\left[ {0,1} \right]$-module $ \mathcal{M}$. If $ D$ is continuous then it is completely determined by $ D\left( z \right)$ where $ z\left( t \right) = t,0 \leq t \leq 1$. For the case when $ D$ is discontinuous we show that $ D\left( f \right)$ is determined by $ D\left( z \right)$ for all $ f$ in an ideal $ \mathcal{T}{\left( D \right)^2}$ of $ {C^n}\left[ {0,1} \right]$ where its closure $ \overline {\mathcal{T}{{\left( D \right)}^2}} $ is of finite codimension.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0928969-1
Article copyright: © Copyright 1988 American Mathematical Society

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