An abstract characterization for projections in operator systems

TL;DR

This paper provides an abstract characterization of projections within operator systems, enabling their detection solely through the system's intrinsic data, with applications in quantum information theory.

Contribution

It introduces a method to identify projections in operator systems using order-theoretic conditions and relates them to embeddings into B(H), advancing the understanding of operator system structure.

Findings

Projections can be characterized abstractly via order-theoretic conditions.

Every abstract projection corresponds to an actual projection in the C*-envelope.

Application to operator systems generated by commuting projection-valued measures.

Abstract

We show that the set of projections in an operator system can be detected using only the abstract data of the operator system. Specifically, we show that if $p$ is a positive contraction in an operator system $V$ which satisfies certain order-theoretic conditions, then there exists a complete order embedding of $V$ into $B(H)$ mapping $p$ to a projection operator. Moreover, every abstract projection in an operator system $V$ is an honest projection in the C*-envelope of $V$. Using this characterization, we provide an abstract characterization for operator systems spanned by two commuting families of projection-valued measures and discuss applications in quantum information theory.