It is well-known that if a real Banach space $E$ is "almost metrically projective" then $E$ is isometrically isomorphic to some $\ell^1(\Gamma)$. We say $E$ is "almost metrically projective" if whenever $T$ is a bounded linear map from $E$ into a quotient Banach space $X/Y$, then for every $\epsilon > 0$ there is an almost norm preserving lifting of $T$, i.e., an operator $\tilde{T}:E\to X$ such that $q\tilde{T}=T$ and $\| \tilde{T}\|<=(1+\epsilon)\| T \|$, where $q:X \to X/Y$ is the canonical quotient map.

This result is due to Grothendieck (Canadian J. M., 1955), and proofs can be found in the books of Semadeni (Banach Spaces of Continuous Functions, 1971) and Lacey (Isometric Theory of Classical Banach Spaces, 1974). However, these proofs deal only with real scalars. Grothendieck states (Remark 6, p. 559) "There is little doubt ... that the theorems in this article are valid for complex scalars as well as real scalars. It is not clear, however, that this extension is immediate."

Has a complete proof for complex scalars ever appeared in the literature?

`$P_{1+\epsilon}$`

space for all ϵ>0. This is OK for complex scalars. $\endgroup$ – Fred Dashiell Jun 4 '13 at 18:42`$E^* \in P_{1+\epsilon}$`

for all $\epsilon > 0$ implies $E^* \in P_1$ ok for complex scalars OK in complex case for dual spaces. So I see what you were saying. $\endgroup$ – Fred Dashiell Jun 4 '13 at 18:491more comment