Hyperfinite-dimensional subspaces of the nonstandard hull of $c_{0}$

Abstract:

Let ${\hat c_0}$ be the nonstandard hull of the Banach space ${c_0}$ formed with respect to an ${\aleph _1}$-saturated extension. Then ${\hat c_0}$ is not isometrically isomorphic to any hyperfinite-dimensional subspace of ${\hat c_0}$ and hence not to any hyperfinite-dimensional Banach space. This gives a negative answer to the question posed by Ward Henson: “Does every Banach space have a nonstandard hull which is isometrically isomorphic to a hyperfinite-dimensional Banach space?” As a consequence of the result, no ultrapower of ${c_0}$ is isometrically isomorphic to an ultraproduct of finite-dimensional Banach spaces.