Let $X\in E$, where $E$ is a Banach function space on $(0,1)$. Consider the interval $[-X,X]$ and generate it to a closed lattice ideal $I$ of $E$. We may renorm this ideal $I$ such that $[-X,X]$ is the unit ball. Why is $I$ lattice-isomorphic to $C(K)$? Here, $K$ is a compact space.
2026-02-23 13:42:58.1771854178
closed lattice ideal is isomorphic to $C(K)$
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There is something wrong here. Take $E=L_1(0,1)$. Then your construction yields a Banach-space embedding of an infinite-dimensional $C(K)$-space into $L_1$, which is impossible as the latter space is weakly sequentially complete, whilst the former one is not.
Probably, under some extra condition the problem aims at applying Kakutani's theorem on representing archimedean lattices as $C(K)$-spaces. (See Theorem 1.11 here.)