  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 S_qU(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 S_qU(2) is
discussed.
