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Proceedings of the American Mathematical Society

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A new topology on $ B^{\ast} $-algebras arising from the Arens products


Author: Edith A. McCharen
Journal: Proc. Amer. Math. Soc. 37 (1973), 77-83
MSC: Primary 46H05; Secondary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1973-0310638-1
MathSciNet review: 0310638
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Abstract: A locally convex topology $ \mu $ is defined on a Banach algebra A. This topology arises naturally from considerations of the Arens products on the second conjugate space $ {A^{ \ast \ast }}$ of A. The main result states that if A is a $ {B^\ast}$-algebra on which the mapping $ (a,b) \to ab$ is $ \mu $-continuous for $ \left\Vert a \right\Vert \leqq 1$, then the completion of A with respect to the uniformity generated by $ \mu $ is linearly isomorphic to $ {A^{ \ast \ast }}$. An example is included which shows that this continuity condition does not hold in general as announced by P. C. Shields.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0310638-1
Keywords: Arens products, locally convex completion of a $ {B^\ast}$-algebra
Article copyright: © Copyright 1973 American Mathematical Society

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