Let G be a classical compact Lie group and G_\mu the associated compact
matrix quantum group deformed by a positive parameter \mu (or a nonzero and
real \mu in the type A case). It is well known that the category Rep(G_\mu) of
unitary f.d. representations of G_\mu is a braided tensor C*-category. We show
that any braided tensor *-functor from Rep(G_\mu) to another braided tensor
C*-category with irreducible tensor unit is full if |\mu|\neq 1. In particular,
the functor of restriction to the representation category of a proper compact
quantum subgroup, cannot be made into a braided functor. Our result also shows
that the Temperley--Lieb category generated by an object of dimension >2 can
not be embedded properly into a larger category with the same objects as a
braided tensor C*-subcategory.Comment: 19 pages; published version, to appear in CMP; for a more detailed
exposition see v
