The (model-theoretic) ultrapowers had been used for studying elementary equivalnce of first-order structures. Then, they have been adapted to Banach spaces, which are, let me say, second-order creatures. Of course, one cannot expect that all the properties (propositions of the 'language of Banach spaces') can be persevered by ultrapowers. Indeed, an ultrapower of a reflexive space need not be reflexive.


But reflexivity is a somewhat complicated issue. What if we ask easier questions? What properties are preserved by ultrapowers? For instance, suppose that we are given a Banach space $X$ and each its subspace isomorphic to $X$ is complemented. Is such property preserved under ultrapowers?



Do (Banach) ultrapowers carry some sort of 'elementary equivalence'?
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.


