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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.

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I don't understand the questions for the following reason: if $X$ is complemented in $E_1$ then the extension is necessarily split by the open mapping theorem. 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 $0 \to X \to \ell^{\infty} \to Y \to 0$ is not split by Phillips' lemma, and $0 \to X \to X \times Y \to Y \to 0$ is split by definition. Second question: yes, $(x, y) \to \left(\frac{1}{2}(x+y), \frac{1}{2}(x+y)\right)$ is a projection of $X \times X$ onto $\Delta$. –  Theo Buehler Jul 2 '11 at 10:08
    
Theo, you may as well post this as an answer so that the OP can accept it –  Yemon Choi Jul 2 '11 at 10:43
    
Thank you Theo for the clarification and for the answer. I accept your answer as final. –  jap Jul 2 '11 at 11:10
    
Yemon: Okay, done. It initially was an answer, but I then decided to move it to the comments because I was unsure how the community would react, as this question is barely "research level". I tried to supplement the comment with a bit of information. –  Theo Buehler Jul 2 '11 at 11:35

1 Answer 1

up vote 3 down vote accepted

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.

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Thank you Theo for the detailed explanation. –  jap Jul 2 '11 at 15:20

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