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By introducing a new notion of approximate biprojectivity we show that nilpotent ideals in approximately amenable or pseudo-amenable Banach algebras, and nilpotent ideals with the nilpotency degree larger than two in biflat Banach algebras cannot have the special property which we call 'property (B)' (Definition 5.2 below) and hence, as a consequence, they cannot be boundedly approximately complemented in those Banach algebras.2010 Mathematics subject classification: primary 46H20; secondary 46B28.

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