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?**



Can C*-algebras be characterized among Banach *-algebras by the spectral radius?
[Updated to include Nik Weaver's correction / improvement from the comments.]


I think this follows from the spectral radius formula:


$$\left\| a \right\|^2 = \rho(a^\*a) = \lim\_{k \to \infty} \left\| (a^\*a)^k \right\|^{1/k} \leq \left\| a^\*a \right\| \leq \left\| a^\* \right\| \left\| a \right\|$$


This gives $\left\| a \right\|^2 \leq \left\| a^\* \right\| \left\| a \right\|$, implying $\left\| a \right\| \leq \left\| a^\* \right\|$. Applying the same argument to $a^\*$ we get $\left\| a \right\| = \left\| a^\* \right\|$, so the inequality above becomes:


$$\left\| a \right\|^2 \leq \left\| a^\* a \right\| \leq \left\| a \right\|^2$$


which yields the C\* identity, as desired.


(The inequalities above follow from the Banach algebra axiom $\left\| ab \right\| \leq \left\| a \right\| \left\| b \right\|$.)


