The natural morphisms between Toeplitz algebras on discrete groups

摘要

Let G be a discrete group and (G,G(+)) be a quasi-ordered group. Set G(+)(0) = G(+) boolean AND (G(+))(-1) and G(1) = (G(+)G(+)(0)) boolean OR {e} Let T-G1(G) and TG+(G) be the corresponding Toeplitz algebras; In the paper, a necessary and sufficient condition for a representation of TG+(G) to be faithful is given. It is proved that when G is abelian, there exists a natural C*-algebra morphism from T-G1(G) to TG+(G). As an application, it is shown that when G = Z(2) and G(+) = Z(+) x Z, the K-groups K-0(T-G1(G)) congruent to Z(2), K-1(T-G1(G)) congruent to Z and all Fredholm operators in T-G1(G) are of index zero. [References: 14]