Quantum symmetry groups of C*-algebras equipped with orthogonal filtrations

Abstract

Motivated by the work of Goswami on quantum isometry groups of noncommutative manifolds we define the quantum symmetry group of a unital C*-algebra A equipped with an orthogonal filtration as the universal object in the category of compact quantum groups acting on A in a filtration preserving fashion. The existence of such a universal object is proved and several examples discussed. In particular we study the universal quantum group acting on the dual of the free group and preserving both the word length and the block length.

31 pages, v2 corrects a few minor points and updates the list of references. The final version of the paper will appear in the Proceedings of the London Mathematical Society

References in corpus (3)

• Spectral triples for AF C*-algebras and metrics on the Cantor set
• The Property of Rapid Decay for Discrete Quantum Groups
• Isometric coactions of compact quantum groups on compact quantum metric spaces

Cited by in corpus (14)

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• A Connection between Easy Quantum Groups, Varieties of Groups and Reflection Groups
• Quantum automorphism groups of finite quantum groups are classical
• Existence and examples of quantum isometry group for a class of compact metric spaces
• Braided quantum symmetries of graph ${\mathrm{C}}^{\ast }$-algebras
• Quantum symmetries of the twisted tensor products of C*-algebras
• Rigidity on Quantum Symmetry for a Certain Class of Graph C*-algebras
• Existence and rigidity of quantum isometry groups for compact metric spaces
• Quantum Isometry group of dual of finitely generated discrete groups and quantum groups
• Projective limits of quantum symmetry groups and the doubling construction for Hopf algebras
• An example of explicit dependence of quantum symmetry on KMS states
• Equivariant ${\mathrm{C}}^{\ast }$-correspondences and compact quantum group actions on Pimsner algebras
• Quantum isomorphism of $2$-graphs
• Quantum symmetry groups of noncommutative tori