Let $A$ be a $C^\*$ algebra. Let $a\in A$ be such that $a^\*a-aa^\*\geq 0$. Doe this imply that the spectral radius of $a$ is equal to $\|a\|$?



Regarding spectral radius
Yes, see T. Ando, On hyponormal operators, *Proc. Amer. Math. Soc.* **14** (1963), 290-291. His main result states that $\|T^n\| = \|T\|^n$ for any hyponormal operator, which implies the conclusion by Gelfand's spectral radius formula.


