Can one view $\ell_{1}^{4}$, $\mathbb{R}^{4}$ equipped with the $\ell_{1}$-norm, as a space of continuous functions on any extremally disconnected space?
Existence of extremally disconnected space $X$ for which $C(X)$ is $\ell_1^4$
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general-topology
functional-analysis
banach-spaces
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0No. The only possibility is a four point space. However, by looking at the combinatorial structure of the unit ball you'll see that $\ell^{1}(4)$ is not isometrically isomorphic to $\ell^{\infty}(4)$. Only $\ell^{1}(1)$ and $\ell^{1}(2)$ are isometrically isomorphic to the corresponding $\ell^{\infty}$-spaces, so only in dimensions $1$ and $2$ such a thing is possible. – 2011-10-01
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1Are this and [your other question](http://math.stackexchange.com/q/68983/) by any chance motivated by [this question](http://mathoverflow.net/questions/76179/) on MO? Then Bill Johnson already answered it in full... – 2011-10-01