Proposition 14

Let A be an involutive algebra. The set $A_+$ is a salient closed pointed convex cone in the real Banach space $A_h$ (EVT, II, p. 11).

Let $\widetilde{A}$ be the star-algebra deduced from A by adjunction of a unit element. Since $A_+= A\cap \widetilde{A}_+$, it is enough to prove the proposition for $\widetilde{A}$. We may therefore suppose that A has a unit element.

We have $0\in A_+$. For every $\lambda \in \mathbf{R}_+^*$ and every $x\in A$, we have Sp$'_A(\lambda x) =$ $\lambda$ Sp$'_A(x)$, which implies that $A_+$ is a cone in the real Banach space $A_h$.

To show that $A_+$ is convex, it is enough to show that if $x$ and $y$ are positive, then $x+y\geqslant 0$ (EVT, II, p. 11, prop. 10). By homothety, it is enough to prove that if $x\geqslant 0$ and $y\geqslant 0$ satisfy moreover $\|x\|\leqslant 1$, $\|y\|\leqslant 1$, then the element $\frac{1}{2}(x+y)$ is positive. Now we have

by assertion b) of lemma 12, and that same assertion then shows that $\frac{1}{2}(x+y)$ is positive.

Finally, assertion a) of lemma 12 also implies that $A_+$ is closed.

Since $A_+$ is a pointed cone in $A_h$, it is salient if and only if $A_+\cap (-A_+)$ is reduced to 0. But if $x\in A_+\cap (-A_+)$, we have Sp$'_A(x) =$ $\{0\}$, hence $\varrho (x) = 0$, and $\|x\|= 0$ since $x$ is hermitian (lemma 7, (2) of I, p. 104), whence $x= 0$.

Proposition 14 means that the relation “ $x\geqslant y$ ” is an order relation on $A_h$ (EVT, II, p. 13, prop. 13).