Maps between positive cones of operator algebras preserving a measure of the difference between arithmetic and geometric means

Abstract

On the set of positive invertible elements in a finite von Neumann algebra carrying a faithful normalized trace $\tau$ the numerical quantity

$$\begin{aligned} d_{\tau }(A,B)=\tau (A + B)/2 - \tau \left( A^{1/2}\left( A^{-1/2}BA^{-1/2}\right) ^{1/2}A^{1/2}\right) \end{aligned}$$

can be viewed as a measure of the difference of the arithmetic and the geometric mean. In this paper, we study maps between the positive definite cones of operator algebras which respect the above distance measure. We obtain the interesting fact that any such map originates from a trace-preserving Jordan ${}^*$-isomorphisms (either algebra ${}^*$-isomorphism or algebra ${}^*$-antiisomorphism in the more restrictive case of factors) between the underlying von Neumann algebras.