Bulletin of the
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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.

Wiener-Hopf $C^*$-algebras of strongly perforated semigroups

Sun Young Jang

University of Ulsan

Abstract

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

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