Spectral isometries of unital $C^*$-algebras

Abstract

This paper provides a human digestion of a proof of the spectral isometry conjecture for $C^*$-algebras found by GPT-6 Astra. We prove that every unital surjective complex-linear spectral isometry between unital $C^*$-algebras is a Jordan isomorphism. Without the assumption that the map preserves the identity, it is a central unitary multiple of a Jordan isomorphism. The main step is to show that an entire one-parameter group of unital spectral isometries consists of algebra automorphisms. To prove this, we extend the group to the bidual, which splits into finite and properly infinite summands. On the finite part, we use quadratic and cubic trace identities and on the properly infinite part, we apply a structure theorem for maps that preserve square-zero elements. To show that both summands are invariant, we use holomorphic dependence on the parameter together with a uniform resolvent bound, which ensures that self-adjoint elements have images with real spectrum. We then apply the group theorem to inner automorphisms conjugated by the original spectral isometry. This reduces the problem to the commutative case, where Nagasawa's theorem completes the proof.