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Quantum families of maps and quantum semigroups on finite quantum spaces. (English) Zbl 1160.58007

Summary: Quantum families of maps between quantum spaces are defined and studied. We prove that quantum semigroup (and sometimes quantum group) structures arise naturally on such objects out of more fundamental properties. As particular cases, we study quantum semigroups of maps preserving a fixed state and quantum commutants of given quantum families of maps.

MSC:

58B32 Geometry of quantum groups
46L85 Noncommutative topology
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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