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Proceedings of the American Mathematical Society

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

 

 

On the Arens product and commutative Banach algebras


Author: Pak Ken Wong
Journal: Proc. Amer. Math. Soc. 37 (1973), 111-113
MSC: Primary 46H99; Secondary 46J05
MathSciNet review: 0306912
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Abstract: The purpose of this note is to generalize two recent results by the author for commutative Banach algebras. Let A be a commutative Banach algebra with carrier space $ {X_A}$ and $ \pi $ the canonical embedding of A into its second conjugate space $ {A^{ \ast \ast }}$ (with the Arens product). We show that if A is a semisimple annihilator algebra, then $ \pi (A)$ is a two-sided ideal of $ {A^{ \ast \ast }}$. We also obtain that if A is a dense two-sided ideal of $ {C_0}({X_A})$, then $ \pi (A)$ is a two-sided ideal of $ {A^{ \ast \ast }}$ if and only if A is a modular annihilator algebra.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0306912-5
Keywords: Arens product, modular annihilator, annihilator algebra, carrier space
Article copyright: © Copyright 1973 American Mathematical Society