Theorem 2

Let A be a stellar algebra. An element $x$ of A is positive if and only if there exists $y\in A$ such that $x=y^*y$.

Suppose that $x$ is positive. Let $y=x^{1/2}$; it is a hermitian element of A and one has $y^*y=y^2=x$.

Conversely, let $y$ be an element of A and put $x=y^*y$. It is a hermitian element of A. Let us show that $x$ is positive. For this, let us write $z=x^-$ and put $w=yz$. One then has

Since $z\geqslant 0$, we have $z^3\geqslant 0$, and it follows that Sp$'_A(w^*w)\subset \mathbf{R}_-$. Let us write $w=w_1+iw_2$ where $w_1$ and $w_2$ are hermitian (Lemma 2 of I, p. 96). The elements $w^2_1$ and $w_2^2$ are positive. We have $ww^*+w^*w=$ $2w^2_1+ 2w_2^2$, and Prop. 14 therefore shows that $ww^*= 2w_1^2+ 2w^2_2+ (-w^*w)$ is positive. Since Sp$'_A(ww^*) =$ Sp$'_A(w^*w)$ (I, p. 5, Prop. 1), which is contained in $\mathbf{R}_-$, we conclude that Sp$'_A(w^*w) =\{0\}$. Since $w^*w$ is hermitian, this implies (Cor. 1 of I, p. 108) that $\|w^*w\|=\varrho (w^*w) = 0$, hence that $z^3= 0$. Since $z$ is hermitian, we have $z= 0$. Thus, $x=x^+$ is positive.