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. 2010 Mathematics Subject Classification: Primary 46J20;

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... ry 46B45, 46J35. [35] c Instytut Matematyczny PAN, 2010 40 A. BIRD By Lemma 2.3, and by the definition of V p as the closure of c 00 , we have: By Proposition 2.4, we have: By [2, Remark 4.16] the map λ : X p → C : (α n ) n∈N → lim n→∞ α n , is a linear functional; λ is also multiplicative and hence is a character of X p , and so λ is continuous and λ = 1. This observation gives an alternative explanation for the boundedness of λ. Let S B denote the state space of a unital Banach * -algebra B. Corollary 4.6. The Banach * -algebras V p , W p , and X p are * -semisimple. Proposition 4.7. The sequence (χ n ) is a bounded approximate identity of norm-1 projections in the Banach * -algebra V p , and is contained in c 00 . Proof. We may calculate from the definition that χ n Wp = 1 for all n ∈ N. We note that by a remark following [5, 4.1.34], Proposition 4.7 implies that the Banach sequence algebra V p is a strong Ditkin algebra, and spectral synthesis holds. By an application of Proposition 2.5(i), we have: Corollary 4.8. The closed ideals of V p are all of the form I Vp (ζ), for some ζ ⊆ N and each has an approximate identity.