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

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 $B$.