A question on $C^*$ algebra

Question

Let $0\leq x\leq a$ for $x,a\in B(H)$ for some complex Hilbert space $H.$ Then we know that there exists a contraction $u$ such that $x^{1/2}=ua^{1/2}.$ Now if we know that $x,a\in A$ where $A\subset B(H)$ for some $C^*$ algebra $A$ can we also say that $u$ can be chosen in $A$?

Answer 1

No, not always. For example, let $A=C([-1,1])$, $x(t)=t_+^2$ and $a(t)=t^2$. If there were $u\in C([0,1])$ such that $x^{1/2}=ua^{1/2}$, we would have $u(t)=0$ for $t<0$ and $u(t)=1$ for $t>0$. Clearly there is no continuous function with these properties.