A duality is discussed for Lie group bundles vs. certain tensor categories
with non-simple unit. A classification of such tensor categories is
given in terms of principal bundles, and a cohomological invariant is
assigned, representing the obstruction to perform an embedding into the
category of vector bundles. For each of these categories, we construct a
K-group coinciding with the Nistor-Troitsky gauge-equivariant K-theory
when the above-mentioned obstruction vanishes. Such K-groups can be used
to compute the K-theory of a certain class of C*-algebras.