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C*-endomorphisms arising from superselection structures with non-trivialcentre define a 'rank' and a 'first Chern class'. Crossed products by suchendomorphisms involve the Cuntz-Pimsner algebra of a vector bundle having theabove-mentioned rank and first Chern class, and can be used to construct aduality for abstract (nonsymmetric) tensor categories vs. group bundles actingon (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object(i.e., the 'gauge' group bundle) are not ensured: we give a description of thisphenomenon in terms of a certain moduli space associated with the givenendomorphism. The above-mentioned Hilbert bimodules are noncommutativeanalogues of gauge-equivariant vector bundles in the sense of Nistor-Troitsky.

GAUGE-EQUIVARIANT HILBERT BIMODULES AND CROSSED PRODUCTS BY ENDOMORPHISMS