A closed ideal A closed ideal I of a commutative C*-algebra A and a closed ideal J of I such that J isn't an ideal of A.

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Let A = C([0,1]), I = Af and J = Cf+Af2,where f(t) = t;0 £ t £ 1. Then J is an ideal of I and I is an ideal of A; but f Î J and f.f[1/2] Ï J (otherwise, there exist l Î C and g Î A such that f.f[1/2] = lf+gf2. So limt® 0 t[1/2] = l+limt® 0tg(t). Therefore l = 0 and t[1/2] = tg contradicting the continuity of g. Thus J isn't an ideal of A.

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On 22 Feb 2001, 00:19.