On a class of determinant preserving maps for finite von Neumann algebras

TL;DR

This paper characterizes determinant-preserving maps on invertible positive elements in finite von Neumann algebras, showing they derive from trace-preserving Jordan automorphisms, and provides new insights into the Fuglede-Kadison determinant's additive properties.

Contribution

It identifies all maps preserving the Fuglede-Kadison determinant under addition, linking them to trace-preserving Jordan automorphisms, and offers a novel proof of a key determinant inequality.

Findings

Maps are from trace-preserving Jordan automorphisms.

Established a new proof of the determinant inequality.

Characterized when equality holds in the determinant sum inequality.

Abstract

Let $\mathscr{R}$ be a finite von Neumann algebra with a faithful tracial state $\tau $ and let $\Delta$ denote the associated Fuglede-Kadison determinant. In this paper, we characterize all unital bijective maps $\phi$ on the set of invertible positive elements in $\mathscr{R}$ which satisfy $$\Delta(\phi(A)+\phi(B)) = \Delta(A+B).$$ We show that any such map originates from a $\tau$-preserving Jordan $*$-automorphism of $\mathscr{R}$ (either $*$-automorphism or $*$-anti-automorphism in the more restrictive case of finite factors). In establishing the aforementioned result, we make crucial use of the solutions to the equation $\Delta(A + B) = \Delta(A) + \Delta(B)$ in the set of invertible positive operators in $\mathscr{R}$. To this end, we give a new proof of the inequality $$\Delta(A+B) \ge \Delta(A) + \Delta(B),$$ using a generalized version of the Hadamard determinant inequality and conclude that equality holds for invertible $B$ if and only if $A$ is a nonnegative scalar multiple of $B$.