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

Abstract

Let 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 R which satisfyΔ(ϕ(A)+ϕ(B))=Δ(A+B). We show that any such map originates from a τ-preserving Jordan ⁎-automorphism of 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 R. To this end, we give a new proof of the inequalityΔ(A+B)≥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.