A reflexive Banach space $E$ has an unconditional finite dimensional expansion of the identity iff $E$ has the approximation property and $E$ is a subspace of a space with an unconditional basis. More results are given in the non-reflexive case. The results are applied to show that the non-complementation of $C(E,F)$ in $L(E,F)$ is equivalent to $C(E,F)\neq L(E,F)$ in certain cases such as: $E$ is reflexive, $E$ or $F$ has the b.a.p, and $F$ is a subspace of a space with an unconditional basis.

This paper provides a unified approach to some perturbation problems recently considered by the author in joint work with S. Kantorovitz. The key result is that the semigroup of unbounded operators $\{J^{\alpha}\}$ formed by the fractional powers of a closed operator $J$ can be approximated in a canonical way by a certain family of bounded semigroups.