A Gromov-Hausdorff distance between von Neumann algebras and an application to free quantum fields

A distance between von Neumann algebras is introduced, depending on a further norm inducing the w*-topology on bounded sets. Such notion is related both with the Gromov–Hausdorff distance for quantum metric spaces of Rieffel and with the Effros–Maréchal topology on the von Neumann algebras acting on a Hilbert space. This construction is tested on the local algebras of free quantum fields endowed with norms related with the Buchholz–Wichmann nuclearity condition, showing the continuity of such algebras w.r.t. the mass parameter.