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Bounded approximate identities and tensor products

Published online by Cambridge University Press:  17 April 2009

J.R. Holub
J.R. Holub
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA.
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Abstract

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The question has been raised [R.J. Loy, Bull. Austral. Math. Soc. 5 (1970), 253–260] as to whether the existence of a bounded (left) approximate identity in the tensor product A ⊗αB of Banach algebras A and B (for a a crossnorm on A ⊗ B ) implies the existence of a bounded (left) approximate identity in A and B. This is known [David A. Robbins, Bull. Austral. Math. Soc. 6 (1972), 443–445] to be the case for α equal to the greatest crossnorm. This paper answers the general question affirmatively.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 7 , Issue 3 , December 1972 , pp. 443 - 445
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Lamadrid, Jesús Gil de, “Measures and tensors”, Trans. Amer. Math. Soc. 114 (1965), 98–121.CrossRefGoogle Scholar
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[3]Robbins, David A., “Existence of a bounded approximate identity in a tensor product”, Bull. Austral. Math. Soc. 6 (1972), 443–445.CrossRefGoogle Scholar
[4]Schatten, Robert, A theory of cross-spaces (Annals of Mathematics Studies, 26. Princeton University Press, Princeton, Nev Jersey; 1950).Google Scholar