Spectral Invariant Subalgebras of Reduced Groupoid C^＊-algebras  

作　　者：Cheng Jun HOU 

机构地区：[1]School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2017年第4期526-544,共19页数学学报（英文版）

基　　金：Supported by the NNSF of China(Grant Nos.11271224 and 11371290)

摘　　要：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).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 Fréchet *-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 τ_c on the algebra C_c(G) of complex continuous functions with compact support on G. We show that the map τ_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).

关 键 词：Groupoid C*-algebra property(RD) spectral invariance 

分 类 号：O177.5[理学—数学]