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

Abstract

Let $\mathcal{R}$ 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 $\varphi$ on the set of invertible positive elements in $\mathcal{R}$ which satisfy $$\Delta (\varphi (A)+\varphi (B))=\Delta (A+B).$$ We show that any such map originates from a $\tau$-preserving Jordan $\ast$-automorphism of $\mathcal{R}$ (either $\ast$-automorphism or $\ast$-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 $\mathcal{R}$. 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 $B$.

11 pages