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

We introduce the fundamental group $\mathring{F}(\stackrel{¸}{A})$ of a unital $C^{\ast }$-algebra $\stackrel{¸}{A}$ 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 $C^{\ast }$-algebras such that $\mathring{F}(\stackrel{¸}{A})=\{I_n\}$. Our study is due to the results of fundamental group by Nawata and Watatani.