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

TL;DR

This paper introduces the concept of quantum symmetry groups for unital C*-algebras with orthogonal filtrations, establishing their existence and exploring specific examples like free group duals.

Contribution

It defines and proves the existence of quantum symmetry groups for C*-algebras with orthogonal filtrations, extending Goswami's work on noncommutative manifolds.

Findings

Existence of universal quantum symmetry groups for filtered C*-algebras.

Construction of quantum groups acting on duals of free groups.

Examples demonstrating preservation of word and block lengths.

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.