Let $(A,\,^\ast,\lVert\cdot\rVert)$ be a Banach $\ast$-algebra with the property that $\lVert a \rVert^2=\rho(a^\ast a)$ holds for all $a\in A$, 
where $\rho(x)$ denotes the spectral radius of an element $x\in A$.

**Is $(A,\,^\ast,\lVert\cdot\rVert)$ then a $C^\ast$-algebra?**