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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Norming $ C(U)$ and related algebras


Author: B. E. Johnson
Journal: Trans. Amer. Math. Soc. 220 (1976), 37-58
MSC: Primary 46J10; Secondary 02H25
MathSciNet review: 0415326
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Abstract: The first result of the paper is that the question of defining a submultiplicative seminorm on the commutative unital $ {C^\ast}$ algebra $ C(\Omega )$ is equivalent to that of putting a nontrivial submultiplicative seminorm on the algebra of infinitesimals in some nonstandard model of C. The extent to which the existence of such a norm on one $ C(\Omega )$ implies the existence for others is investigated. Using the continuum hypothesis it is shown that the algebras of infinitesimals are isomorphic and that if such an algebra has a submultiplicative norm (or, equivalently, seminorm) then, for any totally ordered field $ \mathfrak{k}$ containing R, the R-algebra of infinitesimals in $ \mathfrak{k}$ has a norm. A result of Allan is extended to show that in the particular case when $ \mathfrak{k}$ is a certain field of Laurent series in several (possibly infinitely many) unknowns then the infinitesimals have a submultiplicative seminorm.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0415326-6
PII: S 0002-9947(1976)0415326-6
Article copyright: © Copyright 1976 American Mathematical Society