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Proceedings of the American Mathematical Society

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

 

 

On the ideal structure of the algebra of radial functions


Author: Alan Schwartz
Journal: Proc. Amer. Math. Soc. 26 (1970), 621-624
MSC: Primary 42.56
DOI: https://doi.org/10.1090/S0002-9939-1970-0265865-6
Correction: Proc. Amer. Math. Soc. 39 (1973), 288-294.
MathSciNet review: 0265865
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Abstract: Let $ L$ denote the convolution Banach algebra of integrable functions defined on $ {R^n}$ and let $ {L_r}$ consist of the subalgebra of radial functions. If $ I$ is a closed ideal of $ L$, the zero-set of $ I$ is defined by $ Z(I) = \{ y\vert\hat f(y) = 0{\text{ for all }}f \in I\} $ where $ \hat f$ is the Fourier transform of $ f$. The following theorem is proved. If $ {I_1}$ and $ {I_2}$ are closed ideals of $ {L_r}$ such that $ {I_1} \subset {I_2}$ ($ \subset $ denotes proper inclusion) then there is a closed ideal $ I$ such that $ {I_1} \subset I \subset {I_2}$.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0265865-6
Keywords: Convolution algebra, Fourier transform, ideal structure, radial functions, zero-sets
Article copyright: © Copyright 1970 American Mathematical Society