Say that a Banach space $X$ has property $K$ (for Kummers) provided every subspace of $X$ that is isomorphic to $X$ is complemented. The classical separable, infinite dimensional spaces that have property $K$ include $\ell_2$, $c_0$, and $\ell_2 \oplus c_0$; the first obviously, the second because $c_0$ is separably injective, and the last is not hard to show. It is known that $\ell_p$ for $1\le p \not= 2 < \infty$ does not have property $K$, and it is not hard to show that other possible examples, such as $\ell_2(c_0)$ and $c_0(\ell_2)$, fail property $K$. Ultrapowers of $\ell_2$ of course have property $K$. I don't know about ultrapowers of $c_0$. Are they all injective (which clearly implies property $K$)? Probably not, but I did not try to check the literature.

However, every (complex) HI space [Gowers-Maurey] is not isomorphic to any proper subspace and hence has property $K$. Some real HI spaces have the same property. Now there are HI spaces that contain uniformly complemented copies of $\ell_1^n$ for all $n$. Probably the original Gowers-Maurey space has this property, but, as Thomas Schlumprecht pointed out to me, it is clear that the HI space $AD$ of Argyros-Delyiani does, and $AD$ has property $K$. Since $\ell_1$ is a dual space, there is an ultrapower $Y$ of $AD$ that contains complemented copies of $\ell_1$ and hence $Y$ is isomorphic to $Y \oplus \ell_1$. But $\ell_1$ contains an uncomplemented copy of $\ell_1$ [Bourgain] and hence $Y$ contains an uncomplemented copy of itself.

Therefore property $K$ is not preserved under ultrapowers.

Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72--104? $\endgroup$ – Yemon Choi Apr 16 '12 at 22:40thistrue?" is an easier question to answer well than "What kinds of thing might be true?" $\endgroup$ – Yemon Choi Apr 16 '12 at 23:512more comments