Let $a\in A^+$ be a positive element in a $C^*$-algebra $A$. When does there exist $b\in A$ such that $b\neq \sqrt{a}$, $b^*$ and $b$ are $\mathbb{C}$-linear-independent, and $a=b^* b$?
It seems, using $b=\alpha + i\beta$, where $\alpha, \beta$ are self-adjoints, we have $a=(\alpha -i\beta)(\alpha+i\beta)=\alpha^2 +\beta^2 +i\alpha \beta -i \beta\alpha$, and so a pair of $\alpha,\beta$ that commute would suffice. But under what conditions can such a pair exist?
I know a pair of self-adjoints commute iff their product is also a self-adjoint. Thanks in advance.
Unless $A$ is trivial, there are lots and lots of such $b$. For starters you can take $b=ua^{1/2}$ for any unitary $u$ in the unitization of $A$.