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

$E$ .

Contents Full-Text PDF