For any finite-dimensional representation V of a compact quantum group acting freely on a unital C*-algebra A, we can form an associated finitely-generated projective module A_V over the fixed-point subalgebra B of this action. The module A_V is the section module of the associated noncommutative vector bundle. Given an equvariant C*-homomorphism f from A to A', we get an induced K-theory map f* from K_0(B) to K_0(B'), where B' is the fixed-point subalgebra of A'. Using Chern-Galois theory or applying coflatness and faithful-flatness arguments, we show that f*([A_V])=[A'_V]. This result allows us to move index pairing calculations from K_0(B) to K_0(B'). As an application, we show that any finite equivariant join SUq(2)*...*SUq(2) is not trivializable as an SUq(2)compact quantum principal bundle. (Based on joint work with Tomasz Maszczyk.)
THE EQUIVARIANT-PULLBACK INVARIANCE OF ASSOCIATING NONCOMMUTATIVE VECTOR BUNDLES Sponsored by the Meyer Fund