KK-equivalence for amalgamated free product C*-algebras

TL;DR

This paper demonstrates that reduced and full amalgamated free product C*-algebras are KK-equivalent, utilizing geometric Fredholm modules and Connes's representation theory insights.

Contribution

It establishes KK-equivalence between reduced and full amalgamated free product C*-algebras, introducing a novel proof based on geometric and representation-theoretic methods.

Findings

Reduced and full amalgamated free product C*-algebras are KK-equivalent.

The proof employs Julg--Valette's geometric Fredholm modules.

Connes's perspective on operator algebra representations is integral to the argument.

Abstract

We prove that any reduced amalgamated free product C*-algebra is KK-equivalent to the corresponding full amalgamated free product C*-algebra. The main ingredient of its proof is Julg--Valette's geometric construction of Fredholm modules with Connes's view for representation theory of operator algebras.