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

TL;DR

This paper introduces a canonical $ ext{C}^*$-algebra associated with amalgamated free product $ ext{C}^*$-algebras, generalizing group actions on Bass--Serre trees, and proves key properties like nuclearity and universality.

Contribution

It constructs and studies a new ambient $ ext{C}^*$-algebra for amalgamated free products, providing simpler proofs of known approximation and embeddability results.

Findings

Proves nuclearity and universality of the constructed algebra.

Provides new proofs for approximation properties of amalgamated free products.

Establishes a connection with Cuntz--Pimsner 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.