The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we construct - and -graded algebras, the -graded algebra being a Hopf–Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the -graded algebra. In this case, we carry out the associated circle bundle and Thom constructions. Starting with a C*-algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C*-algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi and Nomizu.
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Edwin Beggs, Tomasz Brzeziński, Line bundles and the Thom construction in noncommutative geometry. J. Noncommut. Geom. 8 (2014), no. 1, pp. 61–105DOI 10.4171/JNCG/149