An amalgamation of the Banach spaces associated with James and Schreier, Part II: Banach-algebra structure

Abstract

The James-Schreier spaces, defined by amalgamating James' quasi-reflexive Banach spaces and Schreier space, can be equipped with a Banach-algebra structure. We answer some questions relating to their cohomology and ideal structure, and investigate the relations between them. In particular we show that the James-Schreier algebras are weakly amenable but not amenable, and relate these algebras to their multiplier algebras and biduals.

An amalgamation of the Banach spaces associated with James and Schreier, Part I: Banach-space structure

“…This block right shift is bounded on c 00 , · Vp under exactly the same condition that the usual right shift is bounded on c 00 , · Sp (see Corollary 3.17(ii)), as the following proposition shows; this result will be important in the study [9] of V p as a Banach algebra.…”

“…The original motivation behind these spaces was to produce a new example of a Banach sequence algebra with a bounded approximate identity, in analogy with Andrew and Green's study of the James space as a Banach algebra [3]. This idea turned out to be successful, as essentially all results about the James space as a Banach algebra carry over to our new spaces; see [9] for details.Having thus reached our initial goal, we soon realized that a serious problem was lurking in the background, namely: how can we distinguish the James-Schreier spaces from the James spaces? Obviously, if they were isomorphic, our findings would be of no interest.…”

“…The original motivation behind these spaces was to produce a new example of a Banach sequence algebra with a bounded approximate identity, in analogy with Andrew and Green's study of the James space as a Banach algebra [3]. This idea turned out to be successful, as essentially all results about the James space as a Banach algebra carry over to our new spaces; see [9] for details.…”