On $C(K)$ spaces embeddable into the Banach space $c_0$

Problem 1. Characterize compact Hausdorff spaces $$K$$ for which the Banach space $$C(K)$$ of continuous real-valued functions embeds into the Banach space $$c_0$$.

Since $$c_0$$ has separable dual, such $$K$$ must me countable. So, we can make Problem 1 more precise:

Problem 2. Is it true that for every compact countable space $$K$$ the Banach space $$C(K)$$ is isomorphic to a subspace of $$c_0$$?

Another possible option:

Problem 3. Let $$K$$ be a compact Hausdorff space. Is it true that the Banach space $$C(K)$$ is isomorphic to $$c_0$$ if $$C(K)$$ is isomorphic to a subspace of $$c_0$$?

• Problem 3: $K$ finite makes trivial counterexamples. Surprisingly (?) the answer is yes for $K$ infinite according to Tomek Kania's answer. – YCor Jun 3 '19 at 6:32

1 Answer

The Szlenk index is the answer.

A space $$C(K)$$, where $$K$$ is infinite compact Hausdorff space, is embeddable into $$c_0$$ if and only if $$K$$ is homeomorphic to an ordinal below $$\omega^\omega$$ and if this is the case (and $$K$$ is infinite) the space itself is isomorphic to $$c_0$$.

So the answer to problem 2 is no however the answer to problem 3 is yes.

For details see Rosenthal's chapter in the Handbook of Banach spaces.

• Thank you very much for the answer. This is exactly what I need! – Taras Banakh Jun 3 '19 at 5:27
• Rosenthal, Haskell P. The Banach spaces C(K). Handbook of the geometry of Banach spaces, Vol. 2, 1547-1602, North-Holland, Amsterdam, 2003. – YCor Jun 3 '19 at 6:23