Extensions of Banach spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:53:30Z http://mathoverflow.net/feeds/question/69317 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69317/extensions-of-banach-spaces Extensions of Banach spaces jap 2011-07-02T08:40:31Z 2011-07-02T20:48:29Z <p>I am looking for an answer to the following questions:</p> <p>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 $E_2$?</p> <p>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?</p> <p>Thanks in advance.</p> http://mathoverflow.net/questions/69317/extensions-of-banach-spaces/69321#69321 Answer by Theo Buehler for Extensions of Banach spaces Theo Buehler 2011-07-02T09:53:38Z 2011-07-02T20:48:29Z <p>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. <a href="http://dx.doi.org/10.1007/b80626" rel="nofollow">Nicolas Monod's thesis</a> Corollary&nbsp;4.2.4 for a detailed proof).</p> <p>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 <a href="http://www.jstor.org/stable/2315346" rel="nofollow">Whitley's note</a> 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.</p> <p>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).</p> <p>Two final remarks:</p> <ul> <li><p>A very interesting procedure for producing non-split extensions of Banach spaces is the <em>twisted sum construction</em> due to <a href="http://www.ams.org/mathscinet-getitem?mr=542869" rel="nofollow">Kalton-Peck</a> (I recently learned about this from Bill Johnson in <a href="http://mathoverflow.net/questions/66345/example-of-a-compact-set-that-isnt-the-spectrum-of-an-operator" rel="nofollow">this thread</a>).</p></li> <li><p>Basically, you're asking about the <a href="http://en.wikipedia.org/wiki/Ext_functor#Ext_and_extensions" rel="nofollow">Yoneda Exts</a> in the <a href="http://en.wikipedia.org/wiki/Exact_category" rel="nofollow">exact category</a> 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 <a href="http://dx.doi.org/10.1016/j.exmath.2009.04.004" rel="nofollow">self-advertisement</a>.</p></li> </ul>