Do (Banach) ultrapowers carry some sort of 'elementary equivalence'? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T17:08:08Z http://mathoverflow.net/feeds/question/94254 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94254/do-banach-ultrapowers-carry-some-sort-of-elementary-equivalence Do (Banach) ultrapowers carry some sort of 'elementary equivalence'? Olaf Kummers 2012-04-16T22:31:03Z 2012-04-18T18:24:25Z <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/94254/do-banach-ultrapowers-carry-some-sort-of-elementary-equivalence/94256#94256 Answer by Juris Steprans for Do (Banach) ultrapowers carry some sort of 'elementary equivalence'? Juris Steprans 2012-04-16T23:05:54Z 2012-04-16T23:05:54Z <p>There is currently a great deal of activity in answering your question. The subject is called "continuous logic" and you can learn quite a bit more about it by searching through the work of Ward Henson or Itai Ben Yaacov. The article [Model theory for metric structures, based on continuous logic][1]http://www.math.uiuc.edu/~henson/cfo/UCLA09text.pdf might be good place to start. But if you are specifically interested in Banach spaces, perhaps the work of David Sherman is a better starting point.</p> http://mathoverflow.net/questions/94254/do-banach-ultrapowers-carry-some-sort-of-elementary-equivalence/94417#94417 Answer by Bill Johnson for Do (Banach) ultrapowers carry some sort of 'elementary equivalence'? Bill Johnson 2012-04-18T15:10:53Z 2012-04-18T18:24:25Z <p>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 $1\le p \not= 2 &lt; \infty$ 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.</p> <p>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.</p> <p>Therefore property $K$ is not preserved under ultrapowers.</p>