Definition 2

Let A be an involutive algebra. A linear form $\lambda$ on A is said to be positive if $\lambda (a^*a)\in \mathbf{R}_+$ for every $a\in A$.

If A is an involutive Banach algebra, one denotes by $A'_+$ the set of continuous positive linear forms on A.

Let A be an involutive Banach algebra. The set $A'_+$ is a pointed convex cone in the real vector space of $\mathbf{C}$-linear forms on A.