Positive element

Remarks

The following are equivalent for $a \in A$: $a$ is positive; $a = b^* b$ for some $b \in A$; $a = c^2$ for some self-adjoint $c \in A$. The positive elements form a closed convex cone $A_+$ with $A_+ \cap (-A_+) = \set{0}$. For self-adjoint $a, b$ one writes $a \le b$ when $b - a \ge 0$.

Example: in $C(X)$ with $X$ compact Hausdorff, $\sigma(f) = f(X)$, so $f \ge 0$ exactly when $f$ is real-valued with $f(x) \ge 0$ for all $x$. In $M_n(\C)$ the positive elements are the positive semidefinite matrices.

Non-example: $a = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ in $M_2(\C)$ has $\sigma(a) = \set{0} \subseteq [0, \infty)$, but $a^* \ne a$, so clause 1 fails. Also $\operatorname{diag}(1, -1)$ is self-adjoint but has $-1$ in its spectrum, so clause 2 fails.