  In this paper, on the basis of a (Fenchel) duality theory on the continuous
level, we derive an $\textit{a posteriori}$ error identity for arbitrary
conforming approximations of the primal formulation and the dual formulation of
the scalar Signorini problem. In addition, on the basis of a (Fenchel) duality
theory on the discrete level, we derive an $\textit{a priori}$ error identity
that applies to the approximation of the primal formulation using the
Crouzeix-Raviart element and to the approximation of the dual formulation using
the Raviart-Thomas element, and leads to quasi-optimal error decay rates
without imposing additional assumptions on the contact set and in arbitrary
space dimensions.
