Rank metric isometries and determinant-preserving mappings on II${}_1$-factors

We fully describe the general form of a linear (or conjugate-linear) rank metric isometry on the Murray–von Neumann algebra associated with a II${}_1$-factor. As an application, we establish Frobenius’ theorem in the setting of II${}_1$-factors, by showing that every determinant-preserving linear bijection between two II${}_1$-factors is necessarily an isomorphism or an anti-isomorphism. This confirms the Harris–Kadison conjecture (1996).