Spectral Isometry

A spectral isometry between unital complex Banach algebras and is a complex-linear map satisfying for all . Here the spectral radius is the largest modulus of a for which is noninvertible. For example, a unital algebra isomorphism preserves invertibility and hence is a spectral isometry. A related term is isometric element of a C^*-algebra, defined by .

Mathieu's spectral isometry conjecture asks whether every unital surjective spectral isometry between unital complex C^*-algebras is a Jordan isomorphism, meaning a complex-linear bijection that preserves squares (Mathieu and Schick 2002). Colbrook and Stepaniants (2026a) report this result for arbitrary unital C^*-algebras. They further report that a surjective spectral isometry without the unitality assumption has the form , where is a central unitary and is a unital Jordan isomorphism. The authors credit GPT-6 Astra with finding the proof. Palomar records machine checks of Lean proofs of the unital theorem and the central-unitary normalization (Colbrook and Stepaniants 2026b).