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.



Extensions of Banach spaces
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](http://dx.doi.org/10.1007/b80626) 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](http://www.jstor.org/stable/2315346) 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](http://www.ams.org/mathscinet-getitem?mr=542869) (I recently learned about this from Bill Johnson in [this thread](https://mathoverflow.net/questions/66345/example-of-a-compact-set-that-isnt-the-spectrum-of-an-operator)).
* Basically, you're asking about the [Yoneda Exts](http://en.wikipedia.org/wiki/Ext_functor#Ext_and_extensions) in the [exact category](http://en.wikipedia.org/wiki/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](http://dx.doi.org/10.1016/j.exmath.2009.04.004).


