# Journal of Operator Theory

Volume 50, Issue 2, Fall 2003 pp. 221-247.

Quasi-lattice ordered groups and Toeplitz algebras**Authors**: John Lorch (1) and Qingxiang Xu (2)

**Author institution:**(1) Department of Mathematical Sciences, Ball State University, Muncie, IN 47306-0490, USA

(2) Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R. China

**Summary:**Let $(G,G_+)$ be a quasi-lattice ordered group and ${\cal T}^{G_+}$ the corresponding Toeplitz algebra. First, we show that for $G_+\subseteq E\subseteq G$, the natural $C^*$-morphism $\gamma^{E,G_+}$ from ${\cal T}^{G_+}$ to ${\cal T}^E$ exists if and only if $E=G_+\cdot H^{-1}$, where $H$ is a hereditary and directed subset of $G_+$. Next, if $E$ is a semigroup, then necessary and sufficient conditions for a representation of ${\cal T}^{E}$ to be faithful are obtained. By applying these results, diagonal invariant ideals of ${\cal T}^{G_+}$ are characterized, conditions under which ${\cal T}^{G_+}$ contains a minimal ideal are established, and finally, in the case when $E$ is a semigroup and $G$ is amenable, it is shown that ${\cal T}^{E}$ has the universal property for covariant isometric representations of $E$.

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