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