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

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.