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Existence of a bounded approximate identity in a tensor product

Published online by Cambridge University Press:  17 April 2009

David A. Robbins
David A. Robbins
Affiliation:
Duke University, Durham, North Carolina, USA.
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Abstract

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It has been shown that the existence of a (left) approximate identity in the tensor product A ⊗ B of Banach algebras A and B, where α is an admissible algebra norm on A ⊗ B, implies the existence of approximate identities in A and B. The question has been raised as to whether the boundedness of the approximate identity in A ⊗αB implies the boundedness of the approximate identities in A and B. This paper answers the question affirmatively with a being the greatest cross-norm.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 6 , Issue 3 , June 1972 , pp. 443 - 445
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Loy, R.J., “Identities in tensor products of Banach algebras”, Bull. Austral. Math. Soc. 2 (1970), 253–260.CrossRefGoogle Scholar
[2]Schatten, Robert, A theory of cross-spaces (Annals of Mathematics Studies, 26. Princeton University Press, Princeton, New Jersey, 1950).Google Scholar