BURES DISTANCE FOR COMPLETELY POSITIVE MAPS

Abstract

Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between C*-algebras by Kretschmann, Schlingemann and Werner. We present a Hilbert C*-module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples. We also prove a rigidity theorem, showing that representation modules of completely positive maps which are close to the identity map contain a copy of the original algebra.