# 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.

**Keywords:**Quasi-lattice ordered group, hereditary and directed set, $\theta$-invariant set, $\alpha$-invariant ideal, induced ideal, Toeplitz algebra, Cuntz algebra, purely infinite simplicity.

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