Fundamental group of simple $C^*$-algebras with unique trace

TL;DR

This paper introduces and computes the fundamental group of unital simple $C^*$-algebras with a unique trace, using K-theory and Picard group techniques, providing new insights into their structure.

Contribution

It defines the fundamental group for such algebras and demonstrates how to compute it using K-theoretical obstructions and Picard groups, including for nuclear and non-nuclear cases.

Findings

Computed fundamental groups for several $C^*$-algebras

Established K-theoretical obstructions for fundamental group calculation

Linked Picard groups to the structure of the fundamental group

Abstract

We introduce the fundamental group ${\mathcal F}(A)$ of a unital simple $C^*$-algebra $A$ with a unique normalized trace. We compute fundamental groups ${\mathcal F}(A)$ of several nuclear or non-nuclear $C^*$-algebras $A$. K-theoretical obstruction enables us to compute the fundamental group easily. Our study is essentially based on the computation of Picard groups by Kodaka.