On uncomplemented isometric copies of c0c0 in spaces of continuous functions on products of the two-arrows space II

Let LL be the two-arrows space. It is a separable, Hausdorff, compact space. Let (⨁n=2∞C(Ln))c0 be the c0c0-sum of the sequence of Banach spaces of all continuous scalar (real or complex) functions on nn-fold products of LL. We show that for every subspace XX of the Banach space (⨁n=2∞C(Ln))c0 isomorphic to c0c0 the quotient space (⨁n=2∞C(Ln))c0/X does not contain any subspace isomorphic to c0(Γ)c0(Γ) for any uncountable set ΓΓ. The space (⨁n=2∞C(Ln))c0 contains an uncomplemented subspace isometric to c0c0.