2006/07/07 by Claudia Pinzari, John E. Roberts, Pinzari, Claudia +1 · 1 citation
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #math.OA #math.QA
paper · pdf · doi:10.48550/arxiv.math/0607188
55 pages, written in Latex, section 5 and Theorem 11.4 improved, subsection 2.5 and section 6 added, references added, typos correced
arxiv created 2006/12/12 · arxiv updated 2009/12/01
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. If the inclusion arises from a quantum subgroup of G, the associated G-system is just the quantum quotient space. If G is a group and the category has permutation symmetry, the associated system is commutative, and therefore isomorphic to the classical quotient space by a closed subgroup of G. If a tensor C*-category has a Hecke symmetry making an object of dimension d and q-quantum determinant one then there is an ergodic action of SqU(d) on a unital C*-algebra, having the spaces of intertwiners from the tensor unit to powers of the object as its spectral subspaces. The special case od SqU(2) is discussed.