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.

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