Prove that every element $a$ of a C*-algebra $A$ is a finite linear combination of unitary elements of $A$

Question

Prove that every element $a$ of a C*-algebra $A$ is a finite linear combination of unitary elements of $A$.

I have no idea to figure it out, any help would be appreciated.

Answer 1

Hint: If $a\in A$ is self-adjoint and $\|a\|\leq1$, then $0\leq1-a^2$ and $a+i\sqrt{1-a^2}$ is a unitary such that \begin{align} a&=\frac{1}{2}\left(a+i\sqrt{1-a^2}\right)+\frac{1}{2}\left(a-i\sqrt{1-a^2}\right) \\ &=\frac{1}{2}\left(a+i\sqrt{1-a^2}\right)+\frac{1}{2}\left(a+i\sqrt{1-a^2}\right)^*. \end{align} Thus by rescaling, any self-adjoint in $A$ can be written as a linear combination of two unitaries. Since any element of $A$ is a linear combination of two self-adjoints, we know...