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How transversally elliptic pseudo-differential operator naturally induces a K-homology class in KK(A, C), where the algebra A is the crossed product algebra C(M) ⋊ G, where M is compact manifold and G is compact Lie group. Do you have any reference paper about this work? Thanks.

And what if M is no longer compact?

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up vote 4 down vote accepted

Since $G$ is compact, averaging over $G$ you may assume that your operator is $G$-invariant. If you assume that $G$ acts freely on your manifold, then $C_0(M)\rtimes G$ is Morita equivalent to $C_0(M/G)$, and what you want boils down to the standard fact that an elliptic pseudo-differential operator defines a $K$-homology class on a compact manifold. You also see that it does not work any longer when $M$ is not compact. For the general case, have you checked in Atiyah's Springer Lecture Notes 401, "Elliptic operators and compact groups"?

Note: your title mentions the fundamental class, but your question doesn't.

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