Inductive Limits of Toeplitz Algebras

Abstract:

      Let $(G_1, E_1)$, $(G_2, E_2)$ be two quasi-lattice ordered groups, and ${\cal T}^{E_1}$, ${\cal T}^{E_2}$ be the associated Toeplitz algebras. Let $\phi: G_1\to G_2$ be a unital group homomorphism such that $\phi(E_1)\subseteq E_2$. This paper gives a necessary and sufficient condition under which the natural morphism from ${\cal T}^{E_1}$ to ${\cal T}^{E_2}$ becomes an injective $C^*$-algebra morphism. As an application, inductive limits of Toeplitz algebras are clarified. In particular, we show that Toeplitz algebras over free groups are always amenable.