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

Abstract

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.