A Duality Theorem for Ergodic Actions of Compact Quantum Groups on C *-Algebras

Abstract

The spectral functor of an ergodic action of a compact quantum group G on a unital C *-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor *-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C *-algebra.As an application, we associate an ergodic G-action on a unital C *-algebra to an inclusion of Rep(G) into an abstract tensor C *-category \({\mathcal{T}}\) .If the inclusion arises from a quantum subgroup K of G, the associated G-system is just the quotient space K\G. If G is a group and \({\mathcal{T}}\) has permutation symmetry, the associated G-system is commutative, and therefore isomorphic to the classical quotient space by a subgroup of G.If a tensor C *-category has a Hecke symmetry making an object ρ of dimension d and μ-determinant 1, then there is an ergodic action of S μ U(d) on a unital C *-algebra having the \((\iota,\rho^r)\) as its spectral subspaces. The special case of \(S_\mu U(2)\) is discussed.