In three previous Chaps. 13–15, we have developed the theory of G-equivariant vector bundles over a compact space X for a compact group G. In Chap. 3, vector bundles over a compact space X were related to finitely generated projective modules over the algebra C(X). The K-theory of X was described in terms of projections in the C*-algebra C(X) tensored with the compact operators in 4(5.14), that is, K0(X) is naturally isomorphic to C*-algebra K-theory K(C(X)).
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Husemöller, D., Joachim, M., Jurčo, B., Schottenloher, M. (2008). Topological Algebras: G-Equivariance and KK-Theory. In: Basic Bundle Theory and K-Cohomology Invariants. Lecture Notes in Physics, vol 726. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74956-1_18
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