On uncomplemented isometric copies of c0 in spaces of continuous functions on products of the two-arrows space

Let L be the two-arrows space. It is a Hausdorff, compact and separable space. First we construct for every n⩾2 an isometric to c0 subspace Xn of C(Ln), the Banach space of all real or complex continuous functions on n-fold product of L, such that inf{‖P‖:P:C(Ln)→Xn  is a projection}⩾n+2. Next we find an uncomplemented subspace Y of C(LN) isometric to c0 such that the quotient space C(LN)/Y is isomorphic to a subspace of l∞. The space C(LN) itself is isometric to a subspace of l∞.