Jordan decomposition for linear functionals on $C^*$-algebras

Question

When $X$ is a locally compact Hausdorff space, there is a Jordan decomposition theorem for the dual of $C_0(X,R)$. If we consider a unital algebra $A$, $\tau$ is a linear functional on $A$, does there exist a Jordan decomposition for $\tau$?

Answer 1

I guess you're looking for the lemma in this paper, "every bounded linear functional can be expressed by a finite linear combination of states". I believe Takeda and Grothendieck obtained this result independently.