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Journal of Operator Theory

Volume 90, Issue 1,  Summer 2023  pp. 41-72.

An abstract characterization for projections in operator systems

Authors:  Roy Araiza (1) and Travis B. Russell (2)
Author institution: (1) Department of Mathematics, University of Illinois Urbana-Champaign, Urbana Illinois, 61801, U.S.A.
(2) Army Cyber Institute, United States Military Academy, West Point New York, 10996, U.S.A.

Summary:  We show that the set of projections in an operator system can be detected using only abstract data. Specifically, we show that if $p$ is a positive contraction in an operator system $\mathcal V$ which satisfies certain order-theoretic conditions, then there exists a complete order embedding of $\mathcal V$ into $B(H)$ mapping $p$ to a projection operator. We provide an abstract characterization for operator systems spanned by two commuting families of projection-valued measures. This yields a new characterization for quantum commuting correlations in terms of abstract operator systems.

Keywords:  Operator systems, projections, quantum correlations.

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