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

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