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 f : A. B satisfying (i) s (f(x) f(y) f(z)) = s (xyz) for all x, y, z. A (where s denotes the spectrum), and (ii) f is continuous at 1. In particular, if (in addition to (i) and (ii)) f(1) = 1, then f 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 Bre. sar and. Spenko (J Math Anal Appl 393: 144-150, 2012), and a paper of Bourhim et al. (Arch Math 107: 609-621, 2016).