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

TL;DR

This paper characterizes rank metric isometries on II$_1$-factors and proves that determinant-preserving bijections are isomorphisms or anti-isomorphisms, confirming a longstanding conjecture in operator algebra.

Contribution

It provides a complete description of rank metric isometries on II$_1$-factors and establishes Frobenius' theorem in this setting, confirming the Harris--Kadison conjecture.

Findings

Rank metric isometries are fully characterized.

Determinant-preserving bijections are isomorphisms or anti-isomorphisms.

Confirmed the Harris--Kadison conjecture (1996).

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 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).