Bulletin of the
Korean Mathematical Society BKMS

It is shown that every almost positive linear mapping $h : \mathcal A \rightarrow \mathcal B$ of a Banach $*$-algebra $\mathcal A$ to a Banach $*$-algebra $\mathcal B$ is a positive linear operator when $h(rx) = r h(x) \, (r> 1)$ holds for all $x \in \mathcal A$, and that every almost linear mapping $h : \mathcal A \rightarrow \mathcal B$ of a unital $C^*$-algebra $\mathcal A$ to a unital $C^*$-algebra $\mathcal B$ is a positive linear operator when $h(2^n u^* y) = h(2^n u)^* h(y)$ holds for all unitaries $u \in \mathcal A$, all $y \in \mathcal A$, and all $n=0, 1, 2, \ldots$, 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 $h : \mathcal A \rightarrow \mathcal B$ of a unital $C^*$-algebra $\mathcal A$ to a unital $C^*$-algebra $\mathcal B$ is a positive linear operator. It is applied to investigate states, center states and center-valued traces.

Keywords: $C^*$-algebra, positive linear operator, state, Hyers-Ulam-Rassias stability, linear functional equation

MSC numbers: Primary 46L05, 47C15, 39B52