Spectral invariant subalgebras of reduced groupoid C*-algebras

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We introduce the notion of property (RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S2l(G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Frechet *-subalgebra of the reduced groupoid C*-algebra Cr*(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 G0 of G, gives rise to a canonical map τc on the algebra Cc(G) of complex continuous functions with compact support on G. We show that the map τc can be extended continuously to S2l(G) and plays the same role as an n-trace on Cr*(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 Cr*(G).