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The Second Conjugates of Certain Banach Algebras

Published online by Cambridge University Press:  20 November 2018

Pak-Ken Wong
Pak-Ken Wong*
Affiliation:
Seton Hall University, South Orange, New Jersey
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Let A be a Banach algebra and A** its second conjugate space. Arens has denned two natural extensions of the product on A to A**. Under either Arens product, A** becomes a Banach algebra. Let A be a semisimple Banach algebra which is a dense two-sided ideal of a B*-algebra B and R** the radical of (A**, o). We show that A** = Q ⊕ R**, where Q is a closed two-sided ideal of A**, o). This was inspired by Alexander's recent result for simple dual A*-algebras (see [1, p. 573, Theorem 5]). We also obtain that if A is commutative, then A is Arens regular.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 5 , 01 October 1975 , pp. 1029 - 1035
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Alexander, F. E., The dual and bidual of certain A*-algebras, Proc. Amer. Math. Soc. 38 (1973), 571–576.Google Scholar
2. Arens, R., The adjoint of a linear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848.Google Scholar
3. Barnes, B. A., Banach algebras which are ideals in a Banach algebra, Pacific J. Math. 38 (1971), 1–7.Google Scholar
4. Barnes, B. A., Modular annihilator algebras, Can. J. Math. 18 (1966), 566–578.Google Scholar
5. Civin, P. and Yood, B., The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847–870.Google Scholar
6. Gohberg, I. C. and Krein, M. G., Introduction to the theory of linear nonself-adjoint operators, Transi. Math. Monographs, vol. 18 (Amer. Math. Soc, Providence, R. I., 1969).Google Scholar
7. Rickart, C. E., General theory of Banach algebras (University Series in Higher Math., Princeton, N.J. 1960).Google Scholar
8. Tomiuk, B. J. and Wong, P. K., The Arens product and duality in B*-algebras, Proc. Amer. Math. Soc. 25 (1970), 529–535.Google Scholar
9. Wong, P. K., Modular annihilator A*-algebras, Pacific J. Math. 37 (1971), 825–834.Google Scholar
10. Wong, P. K., On the Arens products and certain Banach algebras, Trans. Amer. Math. Soc. 180 (1973), 837–848. \\m Anoie on annihilator and complemented Banach algebras, J. Austral. Math. Soc. (to appear).Google Scholar
12. Yood, B., Ideals in topological rings, Can. J. Math. 16 (1964), 28–45.Google Scholar