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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Xu Qingxiang and Chen Xiaoman
Journal: Proc. Amer. Math. Soc. 127 (1999), 553-561
MSC (1991): Primary 47B35
MathSciNet review: 1487347
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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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Additional Information

Xu Qingxiang
Affiliation: Department of Mathematics, Shanghai Normal University, Shanghai, 200234, People’s Republic of China
Email: mathsci@dns.shtu.edu.cn

Chen Xiaoman
Affiliation: Laboratory of Mathematics for Non-linear Sciences and Institute of Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China
Email: xchen@fudan.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04774-7
PII: S 0002-9939(99)04774-7
Keywords: Toeplitz operator, discrete abelian quasily ordered group, $K$-group
Received by editor(s): March 19, 1997
Received by editor(s) in revised form: June 3, 1997
Additional Notes: The first author was supported by the Science and Technology Foundation of Shanghai Higher Education
The second author was supported by the National Science Foundation of China and Doctoral Program Foundation of Institute of Higher Education.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society