Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-15T01:18:33.659Z Has data issue: false hasContentIssue false
  • English
  • Français

On James' Quasi-Reflexive Banach Space as a Banach Algebra

Published online by Cambridge University Press:  20 November 2018

Alfred D. Andrew  and
William L. Green
Alfred D. Andrew
Affiliation:
The Georgia Institute of Technology, Atlanta, Georgia
William L. Green
Affiliation:
The Georgia Institute of Technology, Atlanta, Georgia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [4] and [5], R. C. James introduced a non-reflexive Banach space J which is isometric to its second dual. Developing new techniques in the theory of Schauder bases, James identified J**, showed that the canonical image of J in J** is of codimension one, and proved that J** is isometric to J.

In Section 2 of this paper we show that J, equipped with an equivalent norm, is a semi-simple (commutative) Banach algebra under point wise multiplication, and we determine its closed ideals. We use the Arens multiplication and the Gelfand transform to identify J**, which is in fact just the algebra obtained from J by adjoining an identity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1080 - 1101
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Bonsall, F. F. and Duncan, J., Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete 80 (Springer-Verlag, New York-Heidelberg-Berlin, 1973).CrossRefGoogle Scholar
2. Casazza, P. G., Lin, Bor-Luh, and Lohman, R. H., On James' quasi-reflexive Banach space, Proc. Amer. Math. Soc. 67 (1977), 265–271.Google Scholar
3. 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
4. James, R. C., Bases and reflexivity of Banach spaces, Ann. Math. 52 (1951), 518–527.Google Scholar
5. James, R. C., A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 174–177.Google Scholar
6. Larsen, R., An introduction to the theory of multipliers, Die Grundlehren der mathematischen Wissenschaften 175 (Springer-Verlag, New York-Heidelberg-Berlin, 1971).CrossRefGoogle Scholar
7. Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I, sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebeite 92 (Springer-Verlag, Berlin-Heidelberg-New York, 1977).Google Scholar
8. Namioka, J., Partially ordered linear topological spaces, Mem. Amer. Math. Soc. 24 (1957).Google Scholar