It is shown that every almost positive linear mapping h : $mathcal{A} ightarrowmathcal{B}$ of a Banach *-algebra $mathcal{A}$ to a Banach *-algebra $mathcal{B}$ is a positive linear operator when h(rx) = rh(x) (r > 1) holds for all $xinmathcal{A}$, and that every almost linear mapping h : $mathcal{A} ightarrowmathcal{B}$ of a unital C*-algebra $mathcal{A}$ to a unital C*-algebra $mathcal{B}$ is a positive linear operator when h($2^nu*y$) = h($2^nu$)*h(y) holds for all unitaries $uin mathcal{A}$, all $y in mathcal{A}$, and all n = 0, 1, 2, ..., by using the Hyers-Ulam-Rassias stability of functional equations. Under a more weak condition than the condition as given above, we prove that every almost linear mapping h : $mathcal{A} ightarrowmathcal{B}$ of a unital C*-algebra $mathcal{A}$ A to a unital C*-algebra $mathcal{B}$ is a positive linear operator. It is applied to investigate states, center states and center-valued traces.