简介：LetGbeasecondcountablelocallycompactgroupoidwithHaarsystem{λuH},HbeaclosedsubgroupoidcontainingG^0withHaarsystem{λuH}.T.MasudaintroducedtheC^*-Groupoiddynamicsystem(G,A,α)anditsreducedcrossedproductCrea(G,A)in[2]：whereAisC^*-algebraαisthekomomorphismG→Aut(A).InthispapertheauthorintroducestheC^*-groupoiddynamicsystems(H,A,α/H)and(H\G2,A,^-α)inducedby(G,A,α)andprovesthatC*red(H,A)andC^*red(H\G^2,A)arestronglyMoritaequivalentMoreovertheinducedrepresentationfromC^*red(H,A)toC^*red(G,A)isgot.

简介：LetAbeaunitalC*-algebra,n∈N∪{∞}.Itisprovedthattheisomorphism△n:Un0(A)/DUn0(A)→AffT(A)/△n0(π1(Un0(A)))isisometricforsomesuitabledistances.Asanapplication,theauthorhasthesplitexactsequence0→AffT(A)/△n0(π1(Un0(A)))iA→Un(A)/DUn(A)πA→Un(A)/Un0(A)→0withiAcontractive(andisometricifn=∞)undercertainconditionofA.