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In Beauzamy (Banach spaces) book appears this statement without proof: "if $X\oplus Y$ is isomorphic to Banach space $C(K)$ then either $X$ or $Y$ is isomorphic to $C(K)$'' where $X$, $Y$ are Banach spaces and $C(K)$ is the Banach space of continuous real-valued functions defined on the compact Hausdorff topological space $K$, endowed with sup norm.

I would appreciate some help about how to prove it.

Aligomez
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It seems to be a non-trivial result.

A Banach space $X$ is primary if whenever $X$ is isomorphic to $Y \oplus Z$ (for some Banach spaces $Y$ and $Z$), then $X$ is isomorphic to $Y$ or to $Z$.

In the paper: J. Lindenstrauss and A. Pełczynski, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249, it is shown that $C[0,1]$ is primary. Any $C(K)$ space with $K$ uncountable metric and compact is isomorphic to $C[0,1]$ (this is known as Miljutin's Theorem; a proof can be found in Albiac and Kalton's Topics in Banach Space theory). So, $C(K)$ is primary whenever $K$ is uncountable metric and compact.

In the papers: D.E. Alspach and Y. Benyamini, Primariness of spaces of continuous functions on ordinals, Israel J. Math. 27 (1977), 64–92 and P. Billard, Sur la primarité des espaces C(α), Studia Math. 62 (2) (1978), 143–162 (French), it is shown that $C(K)$ is primary whenever $K$ is countable and compact.

The above (with a different proof that $C[0,1]$ is primary) is contained in Rosenthal's article in The Handbook of the Geometry of Banach Spaces, Vol II.

I don't know of any references for the case where $K$ is uncountable, compact, Hausdorff, but not metrizable. (In this case, $C(K)$ is not separable.)

David Mitra
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  • Thank you, David Mitra, this results and references are very important for me. I was thinking in the fact that Banach spaces isomorphic to complemented subspaces of $C(K)$ contains a subspace isomorphic to $c_0$. But It appears impossible to go further. – Aligomez May 03 '19 at 16:21
  • In case $K$ not metrizable uncountable is false: I. Schlackow in "centripetal operators and Koszmider Spaces", 2008; proved that for $K$ weakly Koszmider space, the space $C(K)$ is not isomorphic to any of its proper subspaces. Infinite compact Hausdorff spaces weakly Koszmider were constructed first by Koszmider in 2004. Schlakow and Fajardo constructions of an example of Koszmider not weakly Koszmider spaces can be adapted in order to get a weakly Koszmider space $K$ such that $C(K)$ is decomposable. – Aligomez Jun 08 '19 at 18:44