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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Locally $ B\sp{\ast} $-equivalent algebras. II


Author: Bruce A. Barnes
Journal: Trans. Amer. Math. Soc. 176 (1973), 297-303
MSC: Primary 46K05
DOI: https://doi.org/10.1090/S0002-9947-1973-0320762-X
MathSciNet review: 0320762
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Abstract: Let A be a locally $ {B^\ast}$-equivalent Banach $ ^\ast$-algebra. Then A possesses a unique norm $ \vert \cdot \vert$ with the property that $ \vert{a^\ast}a\vert = \vert a{\vert^2}$ for all $ a \in A$. Let B be the $ {B^\ast}$-algebra which is the completion of A in the norm $ \vert \cdot \vert$. In this paper it is shown that there exists a closed $ {B^\ast}$-equivalent $ ^\ast$-ideal of A which contains the maximal GCR ideal of B. In particular, when B is a GCR algebra, then $ A = B$.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0320762-X
Article copyright: © Copyright 1973 American Mathematical Society