Abstract
We introduce the notion of property (RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S l2 (G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Fréchet *-subalgebra of the reduced groupoid C*-algebra C * r (G) of G when G has property (RD) with respect to l, so the K-theories of both algebras are isomorphic under inclusion. Each normalized cocycle c on G, together with an invariant probability measure on the unit space G 0 of G, gives rise to a canonical map τ c on the algebra C c (G) of complex continuous functions with compact support on G. We show that the map τ c can be extended continuously to S l2 (G) and plays the same role as an n-trace on C * r (G) when G has property (RD) and c is of polynomial growth with respect to l, so the Connes’ fundament paring between the K-theory and the cyclic cohomology gives us the K-theory invariants on C * r (G).
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Acknowledgements
The paper comes partly from the author’s doctoral thesis in Fudan University. He would like to thank his advisor Professor Xiaoman Chen for his fruitful discussions and suggestions. He also thanks the editors/referees for their helpful comments.
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Supported by the NNSF of China (Grant Nos. 11271224 and 11371290)
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Hou, C.J. Spectral invariant subalgebras of reduced groupoid C*-algebras. Acta. Math. Sin.-English Ser. 33, 526–544 (2017). https://doi.org/10.1007/s10114-016-6264-y
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DOI: https://doi.org/10.1007/s10114-016-6264-y