C*-ALGEBRAS GENERATED BY A PROJECTION AND THE REDUCED GROUP C*-ALGEBRAS ${C^*_r\Bbb Z*\Bbb Z_n}$ AND ${C^*_r\Bbb Z_m*\Bbb Z_n}$

Abstract

Let Γ=ℤm * ℤn or ℤ * ℤn, and let Γ(h) be the subtree consisting of all reduced words starting with any reduced word h ∈ Γ\{e}. We prove that the C*-algebra [Formula: see text] generated by [Formula: see text] and the projection Ph onto the subspace ℓ2(Γ(h)) has a unique nontrivial closed ideal ℐ, ℐ is *-isomorphic to [Formula: see text], and the quotient algebra [Formula: see text] is *-isomorphic to either [Formula: see text] or [Formula: see text] depending on the last letter of h. We also prove that [Formula: see text] is a purely infinite, simple C*-algebra if the last letter of h is a generator of ℤ, and that [Formula: see text] has a unique nontrivial closed ideal [Formula: see text] if the last letter of h is a generator of ℤn; furthermore, [Formula: see text] is *-isomorphic to [Formula: see text] and [Formula: see text] is again a purely infinite, simple C*-algebra. As consequences, all the C*-algebras above have real rank zero, and [Formula: see text] is nuclear for any h ≠ e.