| In the past twenty years, Toeplitz operators denned on discrete groups have been investigated by many mathematicians such as A.Nica[7] M.Laca[4][16]', I.Raeburn[5] etc. The main object of this paper is, based on clarifying the induced ideals, to show that some Toeplitz algebras are purely infinite provided that a certain condition is satisfied.This paper is organized in the following way. In Sections 1 and 2 we will give some basic definitions such as quasi-lattice ordered groups, directed and hereditary sets and so on. The corresponding propositions will at the same time be given respectively. In Section 3, the main task is to clarify the natural morphisms between Toeplitz algebras. For any discrete group G. and any subset E of G, we can define a Toeplitz algebra TE. Given two subsets E\ and E2 with E1 E2, there is a natural morphism rE2,E1 from TE1 to TE2. In some cases, this morphism fails to be a C*-morphism. In this section, we will give some necessary and sufficient conditions, under which the natural morphism between Toeplitz algebras exists as a C*-morphism. First, let (G.G+) be a quasi-lattice ordered group, and suppose E is a subset of G such that G+ E, we will show that the natural C*-morphism rE,G+from TG+ to TE exists if and only if E = G+ . H-1, where H is a hereditary and directed subset of G+ (see Theorem 3.2.1). Next, in Section 3.3, we will also establish another necessary and sufficient conditions from a different side. Let Gbea discrete group, E1 and E2. be two subsets with E1 E2 and e E2 Then the natural morphism rE2,E1 : TE1 → TE2 satisfying rE2,E1(TE1g) = TE2g for any g ∈ G, exists as a C*-morphism if and only if E2 is finitely covariant-lifted by E\ (see Theorem 3.3.1).The Sections 4 and 5 contain the main results of this paper. The main aim of Section 4 is to prove that the intersections of the kernels of the C*-morphisms studied in Section 3 equal to all induced ideals of TG+, and as an application, we prove that TG+ is simple if and only if rGH,G+ is isomorphic for any H ∈Ω . Let (G,G+) be a quasi-lattice ordered group, the collection of all nonempty hereditary and directed subsets of G+ will be denoted by Ω. Let H ∈ Ω and denote by Ω.h the closed θ-invariant subset of Ω generated by H. is a commutative C*-subalgebra of TG+. In this sectionwe will show that and thus (Theorem 4.2.1). The main aim of Section 5 is to study the purely infinite property of the associated Toeplitz algebras. The motivation comes from the fact that the Cuntz algebra On is exactly purely infinite ([3]. theorem V.4.6, pl47). We will denote by Ω# the set of maximal elements of Ω ( i.e. the collection of all maximal hereditary and directed subsets of G+ ). Let G# be the closure of Ω∞ in Ω. If Ω∞ ≠Ω and for any x,y∈G+ with x∈ y, there exists g ∈G+ such thatthen we will show that the Toeplitz algebra TGH is purely infinite (Theorem 5.1). The revise of this property is also studied at the end of this section.
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