On embeddings of full amalgamated free product C*–algebras

Abstract

We examine the question of when the ∗ * –homomorphism λ : A ∗ D B → A ~ ∗ D ~ B ~ \lambda : A*_D B\to \widetilde {A}*_ {\widetilde {D}}\widetilde {B} of full amalgamated free product C ∗ ^* –algebras, arising from compatible inclusions of C ∗ ^* –algebras A ⊆ A ~ A\subseteq \widetilde {A} , B ⊆ B ~ B\subseteq \widetilde {B} and D ⊆ D ~ D\subseteq \widetilde {D} , is an embedding. Results giving sufficient conditions for λ \lambda to be injective, as well as classes of examples where λ \lambda fails to be injective, are obtained. As an application, we give necessary and sufficient conditions for the full amalgamated free product of finite-dimensional C ∗ ^* –algebras to be residually finite dimensional.