  Following a result of Hatori, Miura and Tagaki ([4]) we give here a spectral
characterization of an isomorphism from a $C^\star$-algebra onto a Banach
algebra. We then use this result to show that a $C^\star$-algebra $A$ is
isomorphic to a Banach algebra $B$ if and only if there exists a surjective
function $\phi:A\rightarrow B$ satisfying (i)
$\sigma\left(\phi(x)\phi(y)\phi(z)\right)=\sigma\left(xyz\right)$ for all
$x,y,z\in A$ (where $\sigma$ denotes the spectrum), and (ii) $\phi$ is
continuous at $\mathbf 1$. A simple example shows that (i) cannot be relaxed to
products of two elements, as is the case with commutative Banach algebras. Our
results also elaborate on a paper ([3]) of Bre\v{s}ar and \v{S}penko.
