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It is known that there exist some isomorphism between $L_\infty$ and $\ell_\infty$, which is not explicit at all.

Could someone tell me whether there exist an isometric isomorphism between $L_\infty$ and $\ell_\infty$?

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Side note (sorry): existence of this isomorphism depends essentially on the Axiom of Choice. Which is why you found it to be "not explicit at all". –  GEdgar Oct 5 '12 at 14:11

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up vote 7 down vote accepted

They aren't isometric.

We have $\ell_\infty = C(K)$ and $L_\infty = C(L)$ for $K = \beta \mathbb N$ and $L$ the Stone space of the Lebesgue algebra on $[0,1]$. By the Banach-Stone theorem an isometry would involve a homeomorphism between $K$ and $L$. Such a homeomorphism can't exist: For example, $K = \beta\mathbb{N}$ is separable while $L$ isn't separable, being the Stone space of an atomless probability algebra. In fact, every countable subset of $L$ is nowhere dense.


A more explicit and probably more elementary approach is outlined in these two exercises from page 268 in M. Fabian, P. Habala et al. Functional Analysis and Infinite-Dimensional Geometry, Springer 2001, depending on recognizing points of differentiability of the norms:

Exercises 8.30 and 8.31

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Where can one find a proof that such $L$ is not separable? –  Nate Eldredge Oct 5 '12 at 21:24
    
Could you give a reference for separability and non-separability of $K$ and $L$ compactifications. I'm not familiar with Stone-Cech compactification. –  no identity Oct 5 '12 at 22:54
    
@Nate: See e.g. Fremlin, measure theory, volume 3. Combine 322F with 316I to get that every meager set in the Stone space of a probability algebra is nowhere dense. 316L says that atoms correspond to isolated points in the Stone space so that countable unions of points are meager and hence nowhere dense. See also the last paragraph on page 68 of loc. cit. –  commenter Oct 6 '12 at 2:08
    
@Norbert: For non-separability of $L$ see my comment to Nate. The separability of the Stone-Cech compactification follows from the definition of a compactification: $\mathbb{N}$ is dense in $\beta\mathbb{N}$. –  commenter Oct 6 '12 at 2:10
    
Sorry, I meant to say: "countable unions of points in the Stone space of an atomless probability algebra are meager". –  commenter Oct 6 '12 at 2:36

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