A note on trace scaling actions and fundamental groups of $C^*$-algebras

TL;DR

This paper explores the limitations of trace scaling actions on tensor products of certain AF algebras, revealing that some simple AF algebras with unique trace do not admit such actions.

Contribution

It demonstrates, using classification theorems, that there exist unital simple AF algebras with unique trace whose tensor products lack trace scaling actions of their fundamental groups.

Findings

Existence of a unital simple AF algebra with unique trace

Tensor product with compact operators admits no trace scaling action

Utilizes Effros-Handelman-Shen and Elliott's classification theorems

Abstract

Using Effros-Handelman-Shen theorem and Elliott's classification theorem of AF algebras, we show that there exists a unital simple AF algebra A with unique trace such that $A\otimes \mathbb{K}$ admits no trace scaling action of the fundamental group of A.