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 $S$ which satisfies certain order-theoretic conditions, then there exists a complete order embedding of $S$ into $B(H)$ mapping $p$ to a projection operator. Moreover, every abstract projection in an operator system $S$ is an honest projection in the C*-envelope of $S$. Using this abstract 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.

This is joint work with R. Araiza at Purdue University.