= Solution
The <continuous functional calculus> for a <Hermitian element of a C-star algebra> identifies its generated unital <C-star algebra> with <continuous functions> on its real spectrum, preserves the <involution> and <norm>, and sends functions to the corresponding spectral values by spectral mapping. Consequently an element is a square of a <Hermitian element of a C-star algebra> exactly when it is a <Hermitian element of a C-star algebra> with nonnegative spectrum: the reverse direction uses the continuous square root on that spectrum.
Nonnegative scalar multiples preserve this property directly. For positive $a,b$ put $C=\|a\|+\|b\|$. Functional calculus gives $\|\|a\|1-a\|\leq\|a\|$ and similarly for $b$, so $\|C1-(a+b)\|\leq C$. The sum is a <Hermitian element of a C-star algebra>, and spectral mapping now places its spectrum in $[0,2C]$. It therefore has a <positive square root in a C-star algebra>. If $C=0$ both elements are zero. Thus \b[$A_+$ is a cone]. It is pointed because an element positive with its negative has spectrum contained in $\{0\}$ and hence is zero by the self-adjoint <norm> formula. It is closed as well: for a converging sequence of <Positive elements of a C-star algebra> use a common <norm> bound $C$ and pass to the limit in $\|C1-a_n\|\leq C$.
For the two-by-two <matrix> the exact criterion is
$$
\boxed{a_{11},a_{22}\in[0,\infty),\qquad a_{21}=\overline{a_{12}},\qquad a_{11}a_{22}\geq|a_{12}|^2.}
$$
A positive <matrix> is Hermitian with nonnegative <eigenvalues>; evaluating its quadratic form at the coordinate vectors gives the diagonal conditions, and its <determinant>, the <eigenvalue> product, gives the final inequality. Conversely if $a_{11}>0$, completing the square writes the quadratic form as $a_{11}|z_1+(a_{12}/a_{11})z_2|^2+(a_{22}-|a_{12}|^2/a_{11})|z_2|^2$, which is nonnegative. If $a_{11}=0$, the <determinant> inequality forces $a_{12}=0$ and the conclusion is immediate. The <spectral theorem> then gives nonnegative <eigenvalues> and their Hermitian square root, proving positivity in the question's sense.
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