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 [24] and with the Effros-Maréchal topology [10,19] 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 [3], showing the continuity of such algebras w.r.t. the mass parameter.