Every 2-positive order zero map between C*-algebras is completely positive, and for unital separable C*-algebras finite decomposition rank can be characterized using 2-positive maps in place of completely positive ones.
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2-positive almost order zero maps and decomposition rank
Every 2-positive order zero map between C*-algebras is completely positive, and for unital separable C*-algebras finite decomposition rank can be characterized using 2-positive maps in place of completely positive ones.