Quantum families of maps and quantum semigroups on finite quantum spaces

Abstract

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.

update: last section generalized to non-tracial states

Cited by in corpus (35)

• A compositional approach to quantum functions
• Notes on "quantum gravity" and non-commutative geometry
• Noncommutative homogeneous spaces: the matrix case
• Moduli spaces of noncommutative instantons: gauging away noncommutative parameters
• Quantum sets
• Quantum isometry groups of symmetric groups
• On quantum maps into quantum semigroups
• Quantum Isometry Group for Spectral Triples with Real Structure
• Quantum Group of Orientation preserving Riemannian Isometries
• Quantum symmetries and quantum isometries of compact metric spaces
• Families of Monads and Instantons from a Noncommutative ADHM Construction
• Partial actions of C*-quantum groups I: Restriction and Globalization
• Quantum Isometry Groups: Examples and Computations
• Quantum Functor $Mor$
• Existence and examples of quantum isometry group for a class of compact metric spaces
• Quantum Semigroups from Synchronous Games
• Extended de Finetti theorems for boolean independence and monotone independence
• Quantum $\mathrm{SO}(3)$ groups and quantum group actions on $M_2$
• De Finetti Theorems for the Unitary Dual Group
• A state-space approach to quantum permutations
• A Kind of Compact Quantum Semigroups
• A category of quantum posets
• General de Finetti type theorems in noncommutative probability
• Quantum Isometry Group of the n tori
• Noncommutative families of instantons
• On the structure of noncommutative mapping schemes
• Categorical aspects of compact quantum groups
• Non existence of group structure on some quantum spaces
• Quantum semigroups: A survey
• Quantum semigroups generated by locally compact semigroups
• Quantum semigroup structures on quantum families of maps
• Quantum families of maps
• On the problem of classifying simple compact quantum groups
• Transitive Nonlocal Games
• Duals of finite quantum permutation groups