Positive Elements in C Star-Algebras

Positive Element

$a\in \mathcal{A}$ is positive if $a=b^{\ast }b$ for some $b\in \mathcal{A}$.

Proposition

If $\mathcal{A}$ is a C*-algebra, then $a\in \mathcal{A}$ is positive if and only if $a$ is s.a. and $\sigma (a)\subseteq {\mathbb{R}}_{+}=[0,\infty )$.

e.g. Let $X$ be a locally compact Hausdorff space. Then $f\in C_0(X)$ is positive if and only if either $f(x)\ge 0$ for all $x\in X$ (i.e. $\sigma (f)\subset {\mathbb{R}}_{+}$) or $f=\overline{g}g$ for some $g\in C_0(X)$.