Journal of Operator Theory

Volume 62, Issue 1, Summer 2009 pp. 33-64.

Induced ideals and purely infinite simple Toeplitz algebras

Authors: Qingxiang Xu
Author institution: Department of Mathematics, Shanghai Normal University, Shanghai 200234, P.R. China

Summary: Let

$(G,G_+)$ be a quasi-lattice ordered group,

$\mathbf{\Omega}$ be the collection of hereditary and directed subsets of

$G_+$ , and

$\mathbf{\Omega}_\infty$ be the collection of the maximal elements of

$\mathbf{\Omega}$ . For any

$H\in\mathbf{\Omega}$ , let

$S(H)$ be the closed

$\theta$ -invariant subset of

$\mathbf{\Omega}$ generated by

$H$ , and denote by

${\mathcal T}^{G_H}$ the associated Toeplitz algebra, where

$G_H=G_+\cdot H^{-1}$ . In this paper, the concrete structure of

$S(H)$ is clarified. As a result, it is proved that the induced ideals of the Toeplitz algebra

${\mathcal T}^{G_+}$ studied by Laca, Nica et al.\ can be expressed as the intersections of such kernels as Ker

$\gamma^{G_H,G_+}$ for some

$H\in\mathbf{\Omega}$ , where

$\gamma^{G_H,G_+}$ is the natural morphism from the Toeplitz algebra

${\mathcal T}^{G_+}$ onto

${\mathcal T}^{G_H}$ . A condition is given under which the Toeplitz algebras

${\mathcal T}^{G_H}$ (

$H\in\mathbf{\Omega}_\infty)$ become purely infinite simple. When applied to the free groups with finite or countably infinite generators, this gives a unified proof that the simplicity of the Cuntz algebras

${\mathcal O}_n \, (n\geqslant 2)$ ,

${\mathcal O}_\infty$ implies the purely infinite simplicity of their tensor products.

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