Decomposable and atomic projection maps

TL;DR

This paper characterizes when trace invariant projection maps on finite-dimensional C*-algebras are non-decomposable, showing they are atomic if and only if they cannot be expressed as sums of 2-positive and 2-copositive maps, with implications for projections onto high-dimensional spin factors.

Contribution

It provides a complete characterization of non-decomposable trace invariant projection maps as atomic, linking their structure to positivity properties and spin factor projections.

Findings

Non-decomposable projection maps are atomic.

Projections onto spin factors of dimension > 6 are atomic.

Atomicity is equivalent to non-decomposability for these maps.

Abstract

It is shown that a trace invariant projection map, i.e. a positive unital idempotent map, of a finite dimensional C*-algebra into itself is non-decomposable if and only if it is atomic, or equivalently not the sum of a 2-positive and a 2-copositive map. In particular projections onto spin factors of dimension greater than 6 are atomic.