If Banach space $C(K)$ is decomposable (non trivial) then $C(K)$ is isomorphic to some of the sumands subspaces 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.
 A: 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.)
