Complemented subspaces of Banach spaces $C(K\times L)$

We prove that, for every compact spaces $K_1,K_2$ and compact group $G$, if both $K_1$ and $K_2$ map continuously onto $G$, then the Banach space $C(K_1 \times K_2)$ contains a complemented subspace isometric to the Banach space $C(G)$. Consequently, $C(K_1\times K_2)$ contains a complemented copy of $C([0,1])$ for every non-scattered $K_1,K_2$. Also, answering a question of Alspach and Galego, we get that $C(\beta\omega\times\beta\omega)$ contains a complemented copy of $C([0,1]^\kappa)$ for every cardinal number $1\le\kappa\le{\mathfrak c}$ and hence a complemented copy of $C(K)$ for every metric compact space $K$. On the other hand, for the pointwise topology, we show that $C_p(\beta\omega\times\beta\omega)$ contains no complemented copy of $C_p(2^\omega)$.