The notion of noncommutative set is used for the intuition as the noncommutative analog of a set, as the von Neumann algebras or the ${\rm C}^*$-algebras are for the measurable or topological spaces. But unlike these notions of noncommutative topological or measurable space which are well-defined in the operator algebras framework, the notion of noncommutative set is not (yet) (well-)defined. See the post: What's a noncommutative set?
Sebastien Palcoux
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