Monotonicity of comonotonically maximative functionals on compacta

Determine whether every comonotonically maximative functional I: C(X) -> R on the Banach lattice of continuous real-valued functions over an arbitrary compact Hausdorff space X is monotone; specifically, show or refute that I(f) ≤ I(g) whenever f ≤ g in C(X) under the sole assumption that I(f ∨ g) = I(f) ∨ I(g) for all comonotonic functions f, g ∈ C(X).

Background

The paper studies functionals on C(X) that preserve maxima of comonotonic functions and addition of constants, introducing the functor S of normalized, monotone, comonotonically maximative, plus-homogeneous functionals and showing it is homeomorphic to the capacity functor M via the max-plus integral. Within this context, the authors recall that while maximative functionals are evidently monotone, the analogous implication for comonotonically maximative functionals is nontrivial.

It is known that the monotonicity of comonotonically maximative functionals holds for finite compacta, and similar results have been established under additional conditions. However, for general compact Hausdorff spaces, whether comonotonically maximative implies monotone remains unresolved, and this uncertainty impacts the possibility of simplifying characterizations and categorical structures discussed in the paper.

References

Evidently, each maximative functional is monotone. The problem is not so plain for comonotonically maximative functionals. It is known that each comonotonically maximative functional is monotone for finite compacta. The implication was proved for any compactum with some additional conditions on functional. But generally the problem is still open (see [20] for more details).

— On the functor of comonotonically maxitive functionals (2407.18345 - Radul, 25 Jul 2024) in Section 2 (Preliminaries and Definitions)