Spectrum-shrinking maps and nonlinear preservers on matrix domains

Abstract: The celebrated Kaplansky–Aupetit conjecture asks whether every surjective linear map between unital semisimple Banach algebras that shrinks the spectrum must be a Jordan homomorphism. While the conjecture has been resolved in specific settings (most notably for von Neumann algebras by Aupetit and for algebras of bounded linear operators on Banach spaces by Sourour), it remains widely open, even for C*-algebras. In contrast, spectrum-preserving maps are often more accessible, and a natural question is whether results in that setting can be extended to the spectrum-shrinking case. However, existing results indicate that such generalizations are typically far more delicate. Motivated by this, the talk investigates continuous spectrum-shrinking maps from various subsets Xₙ of the complex matrix algebra Mₙ with values in another matrix algebra Mₘ. The classes of domains Xₙ under consideration include structural matrix algebras (i.e. subalgebras of Mₙ containing all diagonal matrices), the sets of normal and singular matrices, and matrix Lie groups such as GL(n), SL(n), and U(n). Our first objective is to determine when such spectrum-shrinking maps automatically preserve the spectrum. Building on this, and Šemrl’s influential nonlinear characterization of Jordan automorphisms of Mₙ (when n ≥ 3) as continuous maps preserving both spectrum and commutativity, our second objective is to establish an analogous nonlinear preserver theorem for maps Xₙ → Mₙ. This is based on joint work with Alexandru Chirvasitu (University at Buffalo) and Mateo Tomašević (University of Zagreb).

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar