Rank metric isometries and determinant-preserving mappings on II₁-factors

2025/06/13 by Jinghao Huang, Huang, Jinghao, Karimbergen Kudaybergenov +3
Mathematics · Decision Sciences · #Fixed Point Theorems Analysis #Advanced Topics in Algebra #Fuzzy and Soft Set Theory

paper · pdf · doi:10.48550/arxiv.2506.11428

Abstract

We fully describe the general form of a linear (or conjugate-linear) rank metric isometry on the Murray--von Neumann algebra associated with a II₁-factor. As an application, we establish Frobenius' theorem in the setting of II₁-factors, by showing that every determinant-preserving linear bijection between two II₁-factors is necessarily an isomorphism or an anti-isomorphism. This confirms the Harris--Kadison conjecture (1996).

Citations

• Invertibility preserving mappings onto finite C*-algebras
• Hermitian operators and isometries on symmetric operator spaces
• Simultaneous upper triangular forms for commuting operators in a finite\n von Neumann algebra
• Determinants associated to traces on operator bimodules
• Algebras of log-integrable functions and operators
• A decomposition theorem in II₁-factors
• Upper triangular Toeplitz matrices and real parts of quasinilpotent operators
• Entropy for expansive algebraic actions of residually finite groups
• Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents
• Invariant Subspaces for Operators in a General II₁-factor
• Brown Measures of Unbounded Operators Affiliated with a Finite von Neumann Algebra
• L²-cohomology for von Neumann algebras
• Brown measures of sets of commuting operators in a type II₁ factor

Related

• Rank one perturbations of 2-isometries
• The classification problem for von Neumann factors
• A Folner Invariant for Type II₁ Factors
• Ring isomorphisms of ∗-subalgebras of Murray-von Neumann factors
• On the extension of isometries between the unit spheres of a JBW^*-triple and a Banach space