I am looking for an answer to the following questions:
Are there infinite-dimensional Banach spaces $X$ and $Y$ for which there are non-split extensions $0 \to X \to E_1 \to Y \to 0$ and $0 \to X \to E_2 \to Y \to 0$ such that $X$ is complemented in $E_1$ but non-complemented in $E_2$?
Also, let $\Delta \subset X \times X$ be the diagonal. Is $\Delta$ complemented in $X \times X$? Is the extension $ 0 \to \Delta \to X \times X \to X \times X/{\Delta} \to 0 $ split-exact?
Thanks in advance.
I don't understand the questions for the following reason: If the image of $X$ is complemented in $E_1$ then the extension is split. Indeed, if $P$ is a projection of $E_1$ onto the image of $X$ then $1-P$ is a projection onto an isomorph of $Y$ by the open mapping theorem (see e.g. Nicolas Monod's thesis Corollary 4.2.4 for a detailed proof).
First question: If you're asking about a pair of extensions of $Y$ by $X$ with $E_1$ split and $E_2$ non-split, take $X = c_0$ and $Y = \ell^{\infty}/c_0$. Then $E_2: 0 \to X \to \ell^{\infty} \to Y \to 0$ is not split by Phillips' lemma (see Whitley's note in the Monthly for a simple proof), and $E_1: 0 \to X \to X \oplus Y \to Y \to 0$ is split by definition.
Second question: Yes, $(x, y) \mapsto \left(\frac{1}{2}(x+y), \frac{1}{2}(x+y)\right)$ is a projection of $X \oplus X$ onto $\Delta$. I recommend you to prove that this sequence is isomorphic to the obvious extension $0 \to X \to X \oplus X \to X \to 0$ (inclusion into the first summand, projection onto the second).
Two final remarks:
A very interesting procedure for producing non-split extensions of Banach spaces is the twisted sum construction due to Kalton-Peck (I recently learned about this from Bill Johnson in this thread).
Basically, you're asking about the Yoneda Exts in the exact category of Banach spaces with the exact structure consisting of all kernel-cokernel pairs. If you're interested in such abstract nonsense, please allow me a bit of self-advertisement.
