Spectral Invariant Subalgebras of Reduced Groupoid C*-algebras

We introduce the notion of property (RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S-2(l) (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 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 tau(c) on the algebra C-c(G) of complex continuous functions with compact support on G. We show that the map tau(c) can be extended continuously to S-2(l) (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).