Approximate complementation and its applications in studying ideals of Banach algebras - Zhang, Yong (Mathematica Scandinavica, 2003)
We show that a subspace of a Banach space having the approximation property inherits this property if and only if it is approximately complemented in the space. For an amenable Banach algebra a closed left, right or two-sided ideal admits a bounded right, left or two-sided approximate identity if and only if it is bounded approximately complemented in the algebra. If an amenable Banach algebra has a symmetric diagonal, then a closed left (right) ideal J has a right (resp. left) approximate identity (p_α) such that, for every compact subset K of J, the net (a· p_α) (resp. (p_α· a)) converges to a uniformly for a K if and only if J is approximately complemented in the algebra.