We introduce the notion of property(RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S~l(G) of rapidly decreasing functions on G with respect to a continuous length function l is a...We introduce the notion of property(RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S~l(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~*(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 τon the algebra C(G) of complex continuous functions with compact support on G. We show that the map τcan be extended continuously to S~l(G) and plays the same role as an n-trace on C~*(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~*(G).展开更多
基金Supported by the NNSF of China(Grant Nos.11271224 and 11371290)
文摘We introduce the notion of property(RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S~l(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~*(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 τon the algebra C(G) of complex continuous functions with compact support on G. We show that the map τcan be extended continuously to S~l(G) and plays the same role as an n-trace on C~*(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~*(G).
