INDUCED IDEALS AND PURELY INFINITE SIMPLE TOEPLITZ ALGEBRAS

Let (G, G(+)) be a quasi-lattice ordered group, Omega be the collection of hereditary and directed subsets of G(+), and Omega(infinity) be the collection of the maximal elements of Omega. For any H is an element of Omega, let S(H) be the closed theta-invariant subset of Omega generated by H, and denote by T-GH, the associated Toeplitz algebra, where G(H) = G(+) . 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 TG+ studied by Laca, Nica et at. can be expressed as the intersections of such kernels as Ker gamma(GH,G+) for some H is an element of Omega, where gamma(GH,G+) is the natural morphism from the Toeplitz algebra TG+ onto T-GH. A condition is given under which the Toeplitz algebras T-GH (H is an element of Omega(infinity)) 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 O-n (n >= 2), O-infinity implies the purely infinite simplicity of their tensor products.