= Solution
The <Commutative Gelfand--Naimark theorem> says that every commutative unital <C-star algebra> $A$ is isometrically star-isomorphic to $C(\Delta(A))$, where $\Delta(A)$ is its compact <character space> and the map is the <Gelfand transform>
$$
\Gamma(a)(\chi)=\chi(a).
$$
Indeed, maximal ideals give enough characters to identify $\sigma(a)$ with the range of $\widehat a$. The C-star identity and the <spectral radius formula> give
$$
\lVert\widehat a\rVert_\infty^2
=r(a^*a)=\lVert a^*a\rVert=\lVert a\rVert^2,
$$
so $\Gamma$ is isometric and has closed range. Characters send $a^*$ to $\overline{\chi(a)}$, so the range is self-conjugate; it contains constants and separates distinct characters. The complex <Stone-Weierstrass theorem> makes the range dense in $C(\Delta(A))$, and closedness makes it all of that algebra. This proves the theorem.
An element $x$ of a C-star algebra is positive when it is self-adjoint and $\sigma(x)\subseteq[0,\infty)$. Consider the commutative C-star subalgebra $C^*(1,x)$. Under its Gelfand–Naimark isomorphism, $x$ becomes a nonnegative continuous function $\widehat x$. The function $\sqrt{\widehat x}$ is continuous and nonnegative, so its inverse image $y$ is positive and satisfies
$$
\boxed{y^2=x.}
$$
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