  This paper addresses the problem of describing the structure of tensor
C*-categories M with conjugates and irreducible tensor unit. No assumption on
the existence of a braided symmetry or on amenability is made. Our assumptions
are motivated by the remark that these categories often contain non-full tensor
C*-subcategories with conjugates and the same objects admitting an embedding
into the Hilbert spaces. Such an embedding defines a compact quantum group by
Woronowicz duality. An important example is the Temperley--Lieb category
canonically contained in a tensor C*-category generated by a single real or
pseudoreal object of dimension bigger than 2. The associated quantum groups are
the universal orthogonal quantum groups of Wang and Van Daele. Our main result
asserts that there is a full and faithful tensor functor from M to a category
of Hilbert bimodule representations of the compact quantum group. In the
classical case, these bimodule representations reduce to the G-equivariant
Hermitian bundles over compact homogeneous G-spaces, with G a compact group.
Our structural results shed light on the problem of whether there is an
embedding functor of M into the Hilbert spaces. We show that this is related to
the problem of whether a classical compact Lie group can act ergodically on a
non-type I von Neumann algebra. In particular, combining this with a result of
Wassermann shows that an embedding exists if M is generated by a pseudoreal
object of dimension 2.
