Reference: B.A. Barnes. Proc. Amer. Math. Soc. 126 (1998), 1055-1061.

Note that $B$ is injective in the kernel of $I-AB$.

I have found that Theorem 4.2 in [S. Janson. Interpolation of subcouples and quotient couples. Ark. Mat. 31 (1993), 307-338] looks like a result more general than my conjecture, but for the real interpolation method. I need to understand this result.

I cannot see why Pisier's example included in "Tauberian operators" is a counterexample to the previous question.

I suppose that, in the concrete version $E_n\subset L_2\{-1,1\}^n$, the proportionality of dimension of $E_n$ implies that the $E_n$'s are not uniformly complemented in $L_p$ for $p<2$, and that the independence of the sum $E=\oplus E_n$ implies that $E$ is isomorphic to $\ell_2$ in $L_p$ for $p<2$. Is $E$ also a subspace of $L_q$ isomorphic to $\ell_2$ for $q>2$?

@Yemon Choi: Thank you. I will look at the paper you mention.

The case in which I am interested is $A=L_1(G)^{**}$ with one of the Arens products induced by the convolution algebra $L_1(G)$, where $G$ is a LCA group.