Hyperfinite-dimensional subspaces of the nonstandard hull of 𝑐₀

Abstract

Let c ^ 0 {\hat c_0} be the nonstandard hull of the Banach space c 0 {c_0} formed with respect to an ℵ 1 {\aleph _1} -saturated extension. Then c ^ 0 {\hat c_0} is not isometrically isomorphic to any hyperfinite-dimensional subspace of c ^ 0 {\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 {c_0} is isometrically isomorphic to an ultraproduct of finite-dimensional Banach spaces.