Jordan decomposition functional $C^*$-algebra [closed]

The elements of $S$ are either states, or negatives of states. So a convex combination $\psi$ of elements in $S$ looks like $$ \psi=\sum_{j=1}^nt_j\phi_j+\sum_{j=n+1}^m t_j (-\phi_j), $$ where $\phi_j$ is a state and $t_j\geq0$ for all $j$, and $\sum_jt_j=1$. Recall that a convex combination of states is a state. Let $$ \lambda=\sum_{j=1}^nt_j,\quad \mu=\sum_{j=n+1}^m t_j, $$ and $$ \omega_1=\tfrac1\lambda\,\sum_{j=1}^nt_j\phi_j,\qquad \omega_2=\tfrac1\mu\,\sum_{j=n+1}^mt_j\phi_j. $$ Then $\omega_1$ and $\omega_2$ are states, $\lambda+\mu=1$, and $$ \psi=\lambda\omega_1-\mu\omega_2, $$