# Journal of Operator Theory

Volume 41, Issue 2, Spring 1999 pp. 421-435.

Purely infinite simple Toeplitz algebras**Authors**: Marcelo Laca

**Author institution:**Department of Mathematics, University of Newcastle, NSW 2308, Australia

**Summary:**The Toeplitz $C^*$-algebras associated to quasi-lattice ordered\break groups $(G,P)$ studied by Nica in \cite{12} were shown by Laca and Raeburn~(\cite{7}) to be crossed products of an abelian $C^*$-algebra $B_P$ by a semigroup of endomorphisms. Here we define a natural boundary for the semigroup $P$ as a subset of the maximal ideal space (or spectrum) of $B_P$ and prove that the Toeplitz $C^*$-algebra associated to $P$ is simple exactly when this boundary is all of the spectrum of ${B_P}$, in which case the Toeplitz $C^*$-algebra is actually purely infinite. We also prove that when the boundary is a proper subset of the spectrum, it induces an ideal of the Toeplitz $C^*$-algebra which is maximal among induced id eals.

**Keywords:**Toeplitz algebras, quasi-lattice order, semigroup of isometries, semigroup crossed product

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