Fundamental group of $C^{*}$-algebras with finite dimensional trace space

TL;DR

This paper introduces the fundamental group of unital $C^*$-algebras with finite dimensional trace space, exploring its restrictions and constructing examples with specific fundamental groups, extending previous work by Nawata and Watatani.

Contribution

It defines the fundamental group for such $C^*$-algebras, investigates its properties under K-theoretical and positivity constraints, and constructs uncountably many nonisomorphic simple algebras with a given fundamental group.

Findings

Fundamental group elements are constrained by K-theory and positivity.

Existence of uncountably many simple $C^*$-algebras with trivial fundamental group.

Extension of Nawata and Watatani's results on fundamental groups.

Abstract

We introduce the fundamental group $\r{F}(\c{A})$ of a unital $C^*$-algebra $\c{A}$ with finite dimensional trace space. The elements of fundamental group are restricted by K-theoretical obstruction and positivity. Moreover we show there are uncountably many mutually nonisomorphic simple $C^{*}$-algebras such that $\r{F}(\c{A})=\set{I_{n}}$. Our study is due to the results of fundamental group by Nawata and Watatani.