A remark about a theorem of Skandalis

TL;DR

This paper analyzes Skandalis's theorem on $K$-nuclearity, revealing that the failure in his examples is due to the injectivity of the canonical map, even rationally.

Contribution

It clarifies the nature of the failure in Skandalis's examples, showing that injectivity, not surjectivity, is the issue in $K$-nuclearity.

Findings

The canonical map $K_*(A\otimes_{max}A) \to K_*(A\otimes_{min}A)$ fails injectivity in Skandalis's examples.

The failure occurs even at the rational level.

The paper provides insight into the structure of $C^*$-algebras related to $K$-nuclearity.

Abstract

Georges Skandalis exhibited in his work on $K$-nuclearity the first class of $C^*$-algebras $A$ for which the canonical map $K_*(A\otimes_{max}A)\to K_*(A\otimes_{min}A)$ is not an isomorphism. We show that it is the injectivity that fails (even rationally) in his examples.