Spectrum-preserving linear maps on semisimple Banach algebras

Determine whether every bijective unital linear map between semisimple Banach algebras that preserves the spectrum of every element is necessarily a Jordan isomorphism.

Background

Within the linear preserver problems in functional analysis, Kaplansky raised questions about when invertibility-preserving maps on Banach algebras are Jordan homomorphisms. Subsequent work led to a sharper unresolved question concerning spectrum preservation. The paper highlights this problem as a central open direction connecting linear preserver theory and Jordan structure in Banach algebras.

Resolving this question would unify several partial results implying Jordan structure under stronger hypotheses (e.g., von Neumann setting, isometries) and clarify the exact boundary between spectral conditions and Jordan isomorphisms in the semisimple Banach algebra framework.