While proving this in class, Our Prof. did it into two cases: One case was if Banach *-algebra has the unit $u$. then he used the following result in his proof:
"if $A$ is Banach *-algebra with unit $u$, then for every $x$ in $A$ with norm of $x$ less than equals to 1, there is a $y$ in $A$ such that $y^2= u-x$ moreover if $x=x^*$ then $y=y^*$"
He skipped the proof of this result and i want to prove this result but i don't have any idea, how to start. Please help. Sorry for bad english!
Hint: define $y=u+\frac {(1/2)} {1} (-x)+\frac {(1/2)(1/2-1)} {(1)(2)} (-x)^{2}+\frac {(1/2)(1/2-1)(1/2-2)} {(1)(2)(3)} (-x)^{3}+\cdots$.