Group Extensions and the Primitive Ideal Spaces of Toeplitz Algebras

Let Γ be a totally ordered abelian group and I an order ideal in Γ. We prove a theorem which relates the structure of the Toeplitz algebra T (Γ) to the structure of the Toeplitz algebras T ( I ) and T (Γ/ I ). We then describe the primitive ideal space of the Toeplitz algebra T (Γ) when the set Σ(Γ) of order ideals in Γ is well-ordered, and use this together with our structure theorem to deduce information about the ideal structure of T (Γ) when 0→ I → Γ→ Γ/ I → 0 is a non-trivial group extension.