Free Fisher information for non-tracial states

Dimitri Shlyakhtenko

Vol. 211 (2003), No. 2, 375–390

DOI: 10.2140/pjm.2003.211.375

Abstract

We extend Voiculescu’s microstates-free definitions of free Fisher information and free entropy to the non-tracial framework. We explain the connection between these quantities and free entropy with respect to certain completely positive maps acting on the core of the non-tracial non-commutative probability space. We give a condition on free Fisher information of an infinite family of variables, which guarantees factoriality of the von Neumann algebra they generate.

Milestones

Received: 5 March 2002

Revised: 22 July 2002

Published: 1 October 2003