Projective Banach spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T22:31:20Z http://mathoverflow.net/feeds/question/60050 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60050/projective-banach-spaces Projective Banach spaces Matthew Daws 2011-03-30T10:04:25Z 2011-03-30T10:28:34Z <p>Injective Banach spaces, with morphisms as contractive linear maps, have been classically studied (and are $C(K)$ spaces with $K$ Stonian). But what about projectives?</p> <p>So $P$ will be projective if given Banach spaces $E$ and $F$ and a quotient map (aka metric surjection) $\psi:E\rightarrow F$, given any contractive $\phi:P\rightarrow F$, we can lift this to a contractive $\varphi:P\rightarrow E$ with $\psi\varphi = \phi$. (If someone can make a nice commutative diagram, go ahead and edit this!)</p> <p><strong>Claim:</strong> The scalar field (say $\mathbb C$, but also works for $\mathbb R$) is not projective.</p> <p><strong>Proof:</strong> Let $f:c_0\rightarrow\mathbb C$ be the contractive functional $f(x) = \sum_{n=1}^\infty 2^{-n} x_n$ for $x=(x_n)\in c_0$. This induces an isometric isomorphism $c_0 / \ker(f) \cong \mathbb C$. So if $\mathbb C$ is projective, we can find a <em>contractive</em> $g:\mathbb C\rightarrow c_0$ with $fg=1$. That is, $g=(g_n)\in c_0$ is a norm-one vector with $f(g) = \sum 2^{-n} g_n =1$, which is impossible.</p> <blockquote> <p><strong>Question:</strong> Are there any projective Banach spaces?</p> </blockquote> <p>It seems to me that the problem is insisting upon <em>contractive</em> morphisms. In the proof above, if we just need, for each $\epsilon>0$, to find $g$ with $\|g\|&lt;1+\epsilon$, then this is no problem. Is anything known in this generalised setting? (It's easy to see that then $\ell_1(\Gamma)$ is always projective, if I allow myself this wiggle-room).</p> http://mathoverflow.net/questions/60050/projective-banach-spaces/60051#60051 Answer by Bill Johnson for Projective Banach spaces Bill Johnson 2011-03-30T10:22:31Z 2011-03-30T10:22:31Z <p>Matthew, you answered your own question with your claim--there are no isometrically projective Banach space (since, for example, a contractively complemented subspace of a projective space is again projective).</p> http://mathoverflow.net/questions/60050/projective-banach-spaces/60052#60052 Answer by Theo Buehler for Projective Banach spaces Theo Buehler 2011-03-30T10:28:34Z 2011-03-30T10:28:34Z <p>You essentially answered your first question yourself: the ground field is a (contractive) retract of any nonzero Banach space by Hahn-Banach. If there were a non-zero projective Banach space in your sense then the ground field would be projective as well.</p> <p>On the other hand: It is a theorem due to K&ouml;the and Pe&#322;czy&#324;ski that every projective Banach space (lifting over surjective maps in the additive category of Banach spaces) is isomorphic to $\ell^1{(S)}$ for some $S$. I don't know how well the norm of the isomorphism is controlled, but as all these spaces already satisfy your relaxed condition, you won't find any others.</p> <p>The references for the second paragraph are:</p> <ul> <li>A. Pe&#322;czy&#324;ski, <em>Projections in certain Banach spaces</em>, Studia Math. <strong>19</strong> (1960), 209–228. MR0126145.</li> <li>G. K&ouml;the, <em>Hebbare lokalkonvexe R&auml;ume</em>, Math. Ann. <strong>165</strong> (1966), 181–195. MR0196464.</li> </ul>