Spectral isometries of -algebras
Does preserving spectral radii force a linear map between -algebras to preserve their algebraic structure? A -algebra is a normed algebra with an adjoint operation, satisfying the -norm identity. The spectral radius is the largest modulus in an element’s spectrum. The question asks whether a surjective complex-linear map that preserves the identity and all spectral radii must be a Jordan isomorphism: a bijection that preserves squares.
Why it matters and what counts as progress
Why it matters: The question tests how much algebraic structure is encoded by spectral data. It extends the comparison between geometric isometries, which preserve norms, and maps that preserve only spectral radii.
What counts as progress: Prove preservation of squares under the stated hypotheses, construct a counterexample, or establish a precise restricted case. Keep identity preservation, surjectivity, scalar field and algebra class explicit. Distinguish a Jordan isomorphism from a map that also preserves adjoints.
How the problem developed
- 1951 · Norm-isometry theorem [1]
R. V. Kadison Characterized unital surjective complex-linear norm isometries of -algebras as Jordan *-isomorphisms. Spectral-radius preservation asks a different, weaker hypothesis.
M. Mathieu and G. J. Schick Recorded the spectral-isometry conjecture in First results on spectrally bounded operators. Later papers established several special algebra classes.
GPT-6 Astra with Matthew J. Colbrook and George Stepaniants The authors credit Astra with finding the general proof and give a human explanation. The supporting Lean project records a complete check of the exact unital theorem, with hypotheses and proof-route differences stated.
Selected milestones. The paper credits Nagasawa, Aupetit, Mathieu, Schick and other prior structural results; this timeline is not a full history of spectral preservers.
AI contribution
Spectral isometries of unital -algebras
Every unital surjective complex-linear map T between arbitrary complex unital -algebras that preserves the spectral radius of every element is a Jordan isomorphism: T is bijective and . No boundedness or adjoint-preservation hypothesis is imposed.
Sources
Publication history
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