简介:Weintroducethenotionofproperty(RD)foralocallycompact,Hausdorffandr-discretegroupoidG,andshowthatthesetS2^l(G)ofrapidlydecreasingfunctionsonGwithrespecttoacontinuouslengthfunctionIisadensespectralinvariantandFreehet*-subalgebraofthereducedgroupoidC^*-algebraCr^*.(G)ofGwhenGhasproperty(RD)withrespecttol,sotheK-theoriesofbothalgebrasareisomorphicunderinclusion.EachnormalizedcoeycleconG,togetherwithaninvariantprobabilitymeasureontheunitspaceGoofG,givesrisetoacanonicalmapTconthealgebraCo(G)ofcomplexcontinuousfunctionswithcompactsupportonG.WeshowthatthemapTccanbeextendedcontinuouslytoS2^l(G)andplaysthesameroleasann-traceonCr^*(G)whenGhasproperty(RD)andcisofpolynomialgrowthwithrespecttol,sotheConnes'fundamentparingbetweentheK-theoryandthecycliccohomologygivesustheK-theoryinvariantsonCr^*(G).