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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
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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

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