The Natural Morphisms between Toeplitz Algebras on Discrete Groups

Abstract

Let G be a discrete group and (G, G+) be a quasi-ordered group. Set G + 0 = G + ∩ ( G + ) − 1 and G 1 = ( G + ∖ G + 0 ) ∪ { e } . Let 𝒯 G 1 ( G ) and 𝒯 G + ( G ) be the corresponding Toeplitz algebras. In the paper, a necessary and sufficient condition for a representation of 𝒯 G + ( G ) to be faithful is given. It is proved that when G is abelian, there exists a natural C*-algebra morphism from 𝒯 G 1 ( G ) to 𝒯 G + ( G ). As an application, it is shown that when G = ℤ2 and G+ = ℤ+ × ℤ, the K-groups K 0 ( 𝒯 G 1 ( G ) ) ≅ ℤ 2 , K 1 ( 𝒯 G 1 ( G ) ) ≅ ℤ and all Fredholm operators in 𝒯 G 1 ( G ) are of index zero.