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Completion of norms for $ C(X,\,Q)$


Author: Edith H. Luchins
Journal: Proc. Amer. Math. Soc. 28 (1971), 478-480
MSC: Primary 46.55
DOI: https://doi.org/10.1090/S0002-9939-1971-0273412-9
MathSciNet review: 0273412
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Abstract: Let $ C(X,Q)$ denote the algebra of all continuous quaternion-valued functions vanishing at infinity on a locally compact Hausdorff space $ X$. Under the natural norm (the sup norm) and under the spectral radius norm, $ r(f)$, which is equivalent to the sup norm, $ C(X,Q)$ is a Banach algebra. Let $ \delta (f)$ be any multiplicative norm for $ C(X,Q)$; i.e., one under which it is a normed algebra. It is shown that $ \delta (f)$, whether or not it is complete, majorizes the natural norm and $ r(f)$. Under certain conditions on the radical of the completion of $ \delta (f),\delta (f)$ is equivalent to the natural norm and $ r(f)$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0273412-9
Keywords: Real Banach algebra, quaternion-valued continuous functions, sup, minimal, incomplete and equivalent norms, radical, semisimple, strictly semisimple
Article copyright: © Copyright 1971 American Mathematical Society

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