Spectral isometries of unital $C^*$-algebras

AI involvement AI-led

Colbrook and Stepaniants explicitly state that GPT-6 Astra found the proof after a sequence of simple prompts. They then discussed it with ChatGPT to understand the mechanism and wrote the exposition themselves. Separate model agents developed and checked the Lean formalization under human direction. This supports AI-led work; the prompting details do not establish task-only autonomy.

• Colbrook and Stepaniants: Spectral isometries of unital $C^*$-algebras — Theorem 1.1, proof mechanism and AI disclosure

Importance Notable

Resolves the stated spectral-isometry conjecture for arbitrary complex unital $C^*$-algebras, extending earlier commutative, properly infinite and real-rank-zero cases. This is a clear advance in operator algebra; a broader field-wide consequence is not established in the reviewed sources.

Technical verification Formal (Lean)

The pinned artifact reports a fresh complete build, 507 consolidated axiom reports, and both Comparator configurations replayed with all six bundled kernels on 28 October. Its main theorem has the same unital, surjective, complex-linear and spectral-radius hypotheses, with no boundedness assumption. Only propext, Classical.choice and Quot.sound are used. This is source-reported formal checking; the separate AI reviews do not establish independent human proof review.

• Exact spectral-isometry statements, provenance and completed checks — Theorem 1.1, proof mechanism and AI disclosure
• Independent statement of the spectral-isometry theorem

Human understanding Explained

Colbrook and Stepaniants explain the proof through entire one-parameter groups of spectral isometries. They split the bidual into finite and properly infinite parts, use trace identities and square-zero preservation, and control invariance with holomorphic dependence and a resolvent bound. Conjugating inner automorphisms reduces the result to the commutative case.