The subject of the paper is the nonlinearity of the quasi-states on commutative \(C^*\)-algebras \(C(X)\). A complex-valued function \(\rho\) on a unital \(C^*\)-algebra \(A\) is called a quasi-state if it is a state on each unital \(C^*\)-subalgebra generated by a selfadjoint element and if \(\rho(a+ib)=\rho(a)+i\rho(b)\) for every self-adjoint elements \(a\) and \(b\).
The author first establishes the correspondence between the quasi-states on \(C(X)\) and the so-called quasi-measures on \(X\) with total mass 1, where a quasi-measure is defined on the class of compact sets and that of open sets in \(X\). For linear states, this correspondence reduces to the familiar Riesz representation theorem. Therefore, to construct a nonlinear quasi-state on \(C(X)\) amounts to constructing a quasi-measure which is not (the restriction of) a regular Borel measur on \(X\). Indeed, such a quasi-measure is constructed given that \(X\) is the unit square in the plane.