Toeplitz 𝐶*-algebras on ordered groups and their ideals of finite elements

Qingxiang, Xu; Xiaoman, Chen

doi:10.1090/S0002-9939-99-04774-7

Toeplitz $C^*$-algebras on ordered groups and their ideals of finite elements
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by Xu Qingxiang and Chen Xiaoman PDF

Proc. Amer. Math. Soc. 127 (1999), 553-561 Request permission

Abstract:

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 $K_{0}(\mathcal {T}^{G_{+}})\cong \mathbb {Z}$.

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