A Spectral Characterization of Isomorphisms on $C^{\star }$-Algebras

Abstract

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 $\varphi :A\to B$ satisfying (i) $\sigma \left(\varphi (x)\varphi (y)\varphi (z)\right)=\sigma \left(xyz\right)$ for all $x,y,z\in A$ (where $\sigma$ denotes the spectrum), and (ii) $\varphi$ is continuous at $1$. 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 Brešar and Špenko.