Extensions of Banach spaces 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.
 A: 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:


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*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.
