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 $a \in K$ if and only if $J$ is approximately complemented in the algebra.