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Let me first recall the definition of an $\mathcal{L}^{\infty}$-space:

A Banach space $X$ is called an $\mathcal{L}^{\infty}$-space if there is a net $(X_{\lambda})_{\lambda \in \Lambda}$ (directed by inclusion) of finite-dimensional subspaces such that each one is $(1+\varepsilon)$-isomorphic to some $\ell_{\infty}^{n}$ and $X = \overline{\bigcup X_{\lambda}}$.

I am asking for a proof or reference to a proof that $C(K)$-spaces enjoy this property. A natural idea for a proof would be to consider a net of finite open covers of $K$, directed by refinement, and for every such cover take a subspace spanned by subordinate partition of unity. However, I don't have a clue how to choose "compatible" partitions of unity, i.e. that the subspace spanned by the one coming from finer cover actually contains the other one.

I would be glad if someone just opened my eyes to see some obvious argument.

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    $\begingroup$ There is a detailed argument in Defant-Floret, Tensor Norms and Operator Ideals, see the Lemma on page 51, and the paragraph following the proof. $\endgroup$
    – Martin
    Jun 2, 2013 at 10:27
  • $\begingroup$ Thank you very much. At first I was quite confused by the fact that if we take for every finite-dimensional subspace some appropriate superspace then they will not necessarily be directed by inclusion, but then I realized that they, in fact, will. $\endgroup$ Jun 2, 2013 at 11:42

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There is a detailed argument in Defant-Floret, Tensor Norms and Operator Ideals, see the Lemma on page 51, and the paragraph following the proof.

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