A unital commutative Banach algebra with A unital commutative Banach algebra with a maximal ideal M of codimension 1 and a Banach A-module X such that H2(A,X) = 0 but H2(M,X) ¹ 0.

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Let A = C2 with the product (a,b)(a¢,b¢) = (aa¢,ab¢+a¢b). M = {0} ÅC, being the kernel of the character f: A ® C defined by f(z,w) = z, is a maximal ideal of codimension 1. Regard X = C as an annihilator A-module. By [B&D&L, Proposition 2.2], H2(A,X) = {0}. If m((0,w1),(0,w2)) = w1w2, then m Î Z2(M,X), but m Ï N2(M,X) ( otherwise w1w2 = m((0,w1),(0,w2)) = (d1 l)((0,w1),(0,w2)) = (0,w1).l((0,w2)) - l((0,w1).(0,w2)) + l(0,w1).(0,w2) = 0 - l(0) + 0 = 0, for all w1, w2 Î C, a contradiction).
Thus H2(M,X) ¹ 0.

Ref.
[B&D&L] W.G. Bode, H.G. Dales and Z. Lykova, Algebraic and strany splittings of extensions of Banach algebras, mem. Amer. Math. Soc. 137 (1999).

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On 22 Feb 2001, 00:20.