Toeplitz $C^{*}$-algebras on ordered groups and their ideals of finite elements

Let $G$ be a discrete abelian group and $(G,G_{+})$ an ordered group. Denote by $(G,G_{F})$ the minimal quasily ordered group containing $(G,G_{+})$. In this paper, we show that the ideal of finite elements is exactly the kernel of the natural morphism between these two Toeplitz $C^{*}$-algebras. When $G$ is countable, we show that if the direct sum of $K$-groups $K_{0}(\mathcal{T}^{G_{+}})\oplus K_{1}(\mathcal{T}^{G_{+}})\cong \mathbb{Z}$, then $K_{0}(\mathcal{T}^{G_{+}})\cong \mathbb{Z}$.