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On Maximal Regular Ideals and Identities in the Tensor Product of Commutative Banach Algebras

Published online by Cambridge University Press:  20 November 2018

L. J. Lardy  and
J. A. Lindberg Jr.
L. J. Lardy
Affiliation:
Syracuse University, Syracuse, New York
J. A. Lindberg Jr.
Affiliation:
Syracuse University, Syracuse, New York
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Let A1and A2be commutative Banach algebras and A1 ⊙ A2 their algebraic tensor product over the complex numbers C.There is always a t least one norm, namely the greatest cross-norm γ (2), on A1 ⊙ A2 that renders it a normed algebra. We shall write A1 ⊗αA2 for the α-completion of A1⊙ A2when αis an algebra norm on A1⊙ A2.Gelbaum (2; 3), Tomiyama (9), and Gil de Lamadrid (4) have shown that for certain algebra norms α on A1⊙ A2 every complex homomorphism on A1 ⊙ A2 is α-continuous. In § 3 of this paper, we present a condition on an algebra norm α which is equivalent to the α-continuity of every complex homomorphism on A1⊙ A2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 639 - 647
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Butzer, P. L., On two-dimensional Bernstein polynomials, Can. J. Math. 5 (1953), 107–113.Google Scholar
2. Gelbaum, B. R., Tensor products of Banach algebras, Can. J. Math. 11 (1959), 297–310.Google Scholar
3. Gelbaum, B. R., Tensor products and related questions, Trans. Amer. Math. Soc. 103 (1962), 525–548.Google Scholar
4. Gil de Lamadrid, J., Topological modules: Banach algebras, tensor products, algebras of kernels, Trans. Amer. Math. Soc. 126 (1967), 361–419.Google Scholar
5. Jacobson, N., Structure of rings, Amer. Math. Soc. Colloq. Publ., Vol. 37 (Amer. Math. Soc, Providence, R.I., 1956).Google Scholar
6. Kaplansky, I., Ring isomorphisms of Banach algebras, Can. J. Math. 6 (1954), 374–381.Google Scholar
7. Rickart, C. E., General theory of Banach algebras (Van Nostrand, Princeton, N.J., 1960).Google Scholar
8. Smith, H. A., Proper tensor products of commutative Banach algebras, Notices Amer. Math. Soc. 11 (1964), 668; see also, Tensor products of topological algebras, Dissertation, University of Pennsylvania, 1964.Google Scholar
9. Tomiyama, J., Tensor products of commutative Banach algebras, Tôhoku Math. J. 12 (1960), 147–154.Google Scholar