Bass--Serre trees of amalgamated free product C*-algebras

Abstract

For any reduced amalgamated free product $\mathrm{C}^*$-algebra $(A,E)=(A_1, E_1) \ast_D (A_2,E_2)$, we introduce and study a canonical ambient $\mathrm{C}^*$-algebra $\Delta\mathbf{T}(A,E)$ of $A$ which generalizes the crossed product arising from the canonical action of an amalgamated free product group on the compactification of the associated Bass--Serre tree. Using an explicit identification of $\Delta \mathbf{T}(A,E)$ with a Cuntz--Pimsner algebra we prove two kinds of "amenability" results for $\Delta \mathbf{T}(A,E)$; nuclearity and universality. As applications of our framework, we provide new conceptual, and simpler proofs of several known theorems on approximation properties, embeddability, and $KK$-theory for reduced amalgamated free product $\mathrm{C}^*$-algebras.