Line bundles and the Thom construction in noncommutative geometry

Abstract

The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we can construct Z- and Ngraded algebras, the Z-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 N-graded algebra. In this case, we can 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 & Nomizu.