Nonlinear functionals on C([0, 1] × [0, 1])

Let M be a compact Hausdorff space. Let L(M) denote the Banach space of continuous functions f on M. We are interested in functionals Φ on L(M) with the following properties: (i) |Φ(f)| ≦ ‖f‖ for every f ∈ L(M), (ii)Φ(f + g) = Φ(f) + Φ(g) whenever fg = 0, (iii)Φ(f + α) = Φ(f) +α for every f ∈ L(M) and every real number α.