Lattice of self-adjoint bounded functionals on C*-algebra

Question

When it comes to real-valued signed measures on a measurable space $(X, \mathcal{B})$, there’s a lattice structure given by \begin{align*} (\mu \lor \nu) (A) & = \sup \{ \mu (A \cap B) + \nu (A \setminus B ) : B \in \mathcal{B} \} , \\ (\mu \land \nu) (A) & = \inf \{ \mu (A \cap B) + \nu (A \setminus B ) : B \in \mathcal{B} \} . \end{align*} Is there an analogous way to take lattice operations on the (real) space of self-adjoint bounded linear functionals on a C*-algebra?