Projective Banach spaces - MathOverflow

Projective Banach spaces

2011-03-30T10:04:25Z

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?

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!)

Claim: The scalar field (say $\mathbb C$, but also works for $\mathbb R$) is not projective.

Proof: 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 contractive $g:\mathbb C\rightarrow c_0$ 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.

Question: Are there any projective Banach spaces?

It seems to me that the problem is insisting upon contractive morphisms. In the proof above, if we just need, for each $\epsilon>0$, to find $g$ with $\|g\|<1+\epsilon$, 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).

Answer by Bill Johnson for Projective Banach spaces

2011-03-30T10:22:31Z

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).

Answer by Theo Buehler for Projective Banach spaces

2011-03-30T10:28:34Z

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.

On the other hand: It is a theorem due to Köthe and Pełczyński 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.

The references for the second paragraph are:

A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. MR0126145.
G. Köthe, Hebbare lokalkonvexe Räume, Math. Ann. 165 (1966), 181–195. MR0196464.