 Square-root resolvent integral in a C-star algebra

For a Positive element of a C-star algebra $x$, the displayed improper integral converges in norm. Near zero its integrand has norm at most $t^{-1/2}$; near infinity it has norm at most $\Vert x\Vert t^{-3/2}$. The scalar identity follows by setting $t=s{u}^2$ for $s>0$ and using ${\int }_0^{\infty }2/(1+u^2)du=\pi$; for $s=0$ both sides vanish. The bounds are uniform on the compact spectrum of an element, so continuous functional calculus gives the operator identity.