Since convex sets are important in the study of ordered or normed vector spaces, it is only natural to ask for a non-commutative version of convexity that is more suited for vector spaces of operators. Thus matrix convex sets were introduced, which play the same role in operator space theory as ordinary convex sets in classical functional analysis.
Some time earlier 
-convex sets were defined for -algebras. Both notions are different but similar, so a section about -convex sets is also included in this survey.
Some publications about non-commutative convexity are [EW97b], [WW99], [FZ98], [Mor94], [Fuj94].