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

Abstract

Let $\mathscr{R}$ be a finite von Neumann algebra with a faithful tracial state $τ$ and let $Δ$ denote the associated Fuglede-Kadison determinant. In this paper, we characterize all unital bijective maps $ϕ$ on the set of invertible positive elements in $\mathscr{R}$ which satisfy $$Δ(ϕ(A)+ϕ(B)) = Δ(A+B).$$ We show that any such map originates from a $τ$-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 $Δ(A + B) = Δ(A) + Δ(B)$ in the set of invertible positive operators in $\mathscr{R}$. To this end, we give a new proof of the inequality $$Δ(A+B) \ge Δ(A) + Δ(B),$$ 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 $B$.