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Kasparov (1988) defined a distinguished K -homology fundamental class, so called the Dirac element. We prove a localization formula for the Dirac element in K -homology of crossed product of C ∗ -algebras. Then we define the quantization of Hamiltonian G -spaces as a push-forward of the Dirac element. With this, we develop a K -homological approach to the quantization commutes with reduction...
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