Metrically and Topologically Projective Ideals of Banach Algebras

In the present paper, necessary conditions for the metric and topological projectivity of closed ideals of Banach algebras are given. In the case of commutative Banach algebras, a criterion for the metric and topological projectivity of ideals admitting a bounded approximate identity is obtained. The main result of the paper is as follows: a closed ideal of an arbitrary $C^*$-algebra is metrically or topologically projective if and only if it admits a self-adjoint right identity.