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

Abstract

For any reduced amalgamated free product ${\mathrm{C}}^{\ast }$-algebra $(A,E)=(A_1,E_1){\ast }_D(A_2,E_2)$, we introduce and study a canonical ambient ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-algebras.

16 pages

References in corpus (2)

• K-amenability for amalgamated free products of amenable discrete quantum groups
• The KK-Theory of Fundamental C*-Algebras