A spectral characterization of isomorphisms on $C^\star$-algebras

Abstract

Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), 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 $\mathbf 1$. In particular, if (in addition to (i) and (ii)) $\phi (\mathbf 1)=\mathbf 1$, then $\phi$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Brešar and Špenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).