A minimal completion theorem and almost everywhere equivalence for Completely Positive maps

TL;DR

This paper proves a minimal completion theorem for linear maps on C*-algebras and demonstrates that under broad conditions, maps almost everywhere equivalent to quasi-pure maps are actually equal.

Contribution

It introduces a unique minimal completion theorem for completely positive maps and establishes conditions under which almost everywhere equivalence implies equality.

Findings

Existence of a unique minimal completion when feasible

Almost everywhere equivalence to a quasi-pure map implies equality under certain conditions

Provides a theoretical foundation for map completion and equivalence in C*-algebras

Abstract

A problem of completing a linear map on C*-algebras to a completely positive map is analyzed. It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some very general conditions a completely positive map almost everywhere equivalent to a quasi-pure map is actually equal to that map.