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Unital Banach algebras and their subalgebras

Published online by Cambridge University Press:  01 September 2007

TIANXUAN MIAO
TIANXUAN MIAO*
Affiliation:
Department of Mathematics, Lakehead University, Thunder Bay, ON P7B 5E1, Canada. email: [email protected]
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Abstract

Let A be a Banach algebra with a bounded approximate identity. We prove that if A is not unital, then there is a nonunital subalgebra B of A with a sequential bounded approximate identity. It follows that A must be unital if A is weakly sequentially complete and B** under the first Arens multiplication has a unique right identity for every subalgebra B of A with a sequential bounded approximate identity. As a consequence, we prove a result of Ülger that if A is both weakly sequentially complete and Arens regular, then A must be unital.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 2 , September 2007 , pp. 343 - 347
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

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