Bull. Korean Math. Soc. 2010; 47(6): 1275-1283
Printed November 1, 2010
https://doi.org/10.4134/BKMS.2010.47.6.1275
Copyright © The Korean Mathematical Society.
Sun Young Jang
University of Ulsan
If the Wiener-Hopf $C^*$-algebra ${\mathcal W}(G,M)$ for a discrete group $G$ with a semigroup $M$ has the uniqueness property, then the structure of it is to some extent independent of the choice of isometries on a Hilbert space. In this paper we show that if the Wiener-Hopf $C^*$-algebra ${\mathcal W}(G, M)$ of a partially ordered group $G$ with the positive cone $M$ has the uniqueness property, then $(G, M)$ is weakly unperforated. We also prove that the Wiener-Hopf $C^*$-algebra $ {\mathcal W}(\Bbb Z, M)$ of subsemigroup $M$ generating the integer group $\Bbb Z$ is isomorphic to the Toeplitz algebra, but $ {\mathcal W}(\Bbb Z, M)$ does not have the uniqueness property except the case $M = \Bbb N$.
Keywords: left regular isometric representation, Wiener-Hopf $C^*$-algebra, unperforated semigroup, Toeplitz algebra
MSC numbers: 46L05, 47C15
2006; 43(2): 333-341
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd