If a $C^*$ algebra have a faithful tracial state ,we can construct a representation $(\pi_{\tau},H_{\tau})$ of $A$.$H_{\tau}$ can be obtained as following:
Let $N=\{a\in A,\tau(a^*a)=0\}$,then $H_{\tau}$ is the completion of $A/N$.
My question:if $\tau$ is a faithful tracial state,then $N=0$,$A/N=A$.Since $A$ is complete,the completion of $A/N$ is $A$.That is to say,the GNS space is $A$.Is my understanding true?
If $A$ is infinite-dimensional, it will usually not be complete in the $\|\cdot\|_2$-norm. For a trivial example, take $A=L^\infty[0,1]$ and $\tau$ integration against Lebesgue measure. Then $\tau$ is a faithful trace, and $H_\tau=L^2[0,1]$.