Let $\Omega$ be a compact metric space and $C(\Omega,\mathbb{R})$ the space of borel continuous real valued functions. I would like to know if there is any real Banach space $V$ such that its dual space (topological)is exactly $C(\Omega,\mathbb{R})$.

My main interest is when $\Omega$ is an infinite cartesian product as for example, $\Omega=E^{\mathbb{Z}^d}$, where $E$ is a compact metric space. If the answer for this case is also negative are we in a better situation if $E$ is a finite set ?

Thanks for any comment or reference.

Edition. Remove the superfluous hypothesis pointed out by Philip.

  • $\begingroup$ In the infinite Cartesian product example you mention, I don't think you have a separable metric space. Or are you taking some kind of restricted direct product? $\endgroup$ – user16299 Feb 10 '12 at 3:00
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    $\begingroup$ What do you mean by "Borel continuous"? Are they continuous, or only Borel? If you mean continuous, then e.g. if $\Omega=[0,1]$ or another connected space, then the only extreme points of the unit ball of $C(\Omega,\mathbb R)$ will be the constant functions $f=1$ and $f=-1$, and by the Krein-Milman theorem this implies that $C(\Omega,\mathbb R)$ is not a dual space. If $\Omega$ is the one point compactification of the integers then you get a space isomorphic to the space of convergent sequences, which I believe is not a dual space but don't know how to show it. $\endgroup$ – Jonas Meyer Feb 10 '12 at 3:02
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    $\begingroup$ In fact: in the complex-valued case, C(X) for X compact Hausdorff is a dual Banach space precisely when it is an abelian von Neumann algebra, which happens if and only if X is extremely disconnected. I have a feeling that the only Von Neumann algebras that are separable in norm topology are the finite dimensional ones. Now all this should carry over to the real-valued case, implying that $\Omega$ is finite... $\endgroup$ – user16299 Feb 10 '12 at 3:03
  • $\begingroup$ @Yemon: Your feeling about separability of von Neumann algebras is correct. There's maybe an easier way to show it than this, but every infinite dimensional C*-algebra contains an element with infinite spectrum, and with Borel functional calculus of a self-adjoint element with infinite spectrum it isn't hard to show that the von Neumann algebra generated by such an element is nonseparable. $\endgroup$ – Jonas Meyer Feb 10 '12 at 3:09
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    $\begingroup$ @Yemon: your "if and only if" is not exactly right. There are extremely disconnected compact spaces $X$ such that $C(X)$ is not a von Neumann algebra. The right iff is given by hyperstonean spaces; these are extremely disconnected but they have the additional property that $C(X)$ has normal states. $\endgroup$ – Martin Argerami Feb 22 '12 at 7:41

Firstly, let me just point out that every compact metric space is automatically separable.

Secondly, note that $c_0$ does not embed into any separable dual space, hence neither does any Banach space containing a subspace isomorphic to $c_0$. Since every $C(K)$ space ($K$ compact Hausdorff) contains a subspace isomorphic to $c_0$, no $C(K)$ space embeds isomorphically into a separable dual space. In particular, since metrizability of $K$ is equivalent to $C(K)$ being norm separable, the answer to your question is always no.

For a reference for all of the above claims, look up $C(K)$ and $c_0$ in the index of Albiac and Kalton's book Topics in Banach Space Theory.

  • $\begingroup$ Hi Philip thanks a lot for your help. I will have a look at the reference you provided. $\endgroup$ – Leandro Feb 10 '12 at 3:20

The result is false in general. For example, consider $\Omega=\{\frac 1 n: n\in \mathbb N\}\cup \{0\}$, then $C(\Omega,\mathbb R)=c_0$ which is not a dual space. Edit: I'm adding some references: My claim follows from Proposition 4.4.1 of Albion and Kalton's book.

  • $\begingroup$ hi azarel, you mean that $c_0$ has no a predual space ? could you elaborate more or give me a reference ? $\endgroup$ – Leandro Feb 10 '12 at 3:00
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    $\begingroup$ $c_0$ is a pre-dual space (it is the predual of its dual), however it is not a dual space (which is what you mean to say). $\endgroup$ – Martin Wanvik Feb 10 '12 at 3:01
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    $\begingroup$ In your example, $C(\Omega,\mathbb R)\cong c$, the space of real convergent sequences, not $c_0$. It is easier to see that $c_0$ is not a dual space, because its unit ball has no extreme points. I don't know how to show that $c$ is not a dual space. $\endgroup$ – Jonas Meyer Feb 10 '12 at 3:04
  • $\begingroup$ Note that $c_0=c_0\oplus \mathbb R=c$ $\endgroup$ – azarel Feb 10 '12 at 3:45
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    $\begingroup$ @azarel: No, $c_0\neq c$. $\endgroup$ – Jonas Meyer Feb 10 '12 at 4:16

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