A spectral characterization of isomorphisms on C^⋆ -algebras

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^⋆-algebra onto a Banach algebra. We then use this result to show that a C^⋆-algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function ϕ: A→ B satisfying (i) σ(ϕ(x) ϕ(y) ϕ(z)) = σ(xyz) for all x, y, z∈ A (where σ denotes the spectrum), and (ii) ϕ is continuous at 1. In particular, if (in addition to (i) and (ii)) ϕ(1) = 1, then ϕ 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).