Proposition 8

Let A be a stellar algebra and $a$ a nonzero element of A. There exists a positive linear form $\lambda \in A'_+$ such that $\lambda (a^*a)>0$.

Consider the real Banach space $A_h$ of hermitian elements of A. The set $A_+$ is a salient pointed closed convex cone in $A_h$ (Prop. 14 of I, p. 116). The element $a^*a$ is positive (Theorem 2 of I, p. 118) and nonzero; hence the hermitian element $-a^*a$ is therefore not positive. By EVT, II, p. 42, Cor. 5, there exists a continuous real linear form $\lambda \in A'_h$ such that $\lambda (-a^*a)<0$ and $\lambda (A_+)\subset \mathbf{R}_+$. The linear form $\lambda$ extends to a hermitian $\mathbf{C}$-linear form on A (cf. I, p. 98), which is positive and has the required property.